Brand-New-Goldbach, proof of it and all its extensions

[Archimedes Plutonium]

Alright, so, let us go back in time, a time in which no Goldbach was around to make a conjecture. A time in which only dinosaurs were around, no math professors to fill a mind with blither blather nonsense.

So here we are in fresh new math, and we have Arithmetic and we have Unique Prime Factorization.

So we ask the simple question of from 0 to 100 how many even numbers are there and there are 50, but we pare away the 2, 4, because we want only oven numbers that are the sum of two primes, so that leaves us with 48 even numbers from 6 to 100.

Now we start the program and we abbreviate unique prime factorization as UPFAT.

Now, we include a column of Unique Prime Addition, where we borrow one of the multiplication primes and craft a number equal to the even number, call it UPADD. Now if there is no prime in UPFAT to craft UPADD we take the next higher prime and use it. Now if the primes of UPFAT do not work we bump up one of the primes to the next higher until something does work.

Even UPFAT UPADD
6 2*3 3+3

8 2*2*2 3+5

10 2*5 5*5

12 2*2*3 5+7

14 2*7 7+7

16 2*2*2*2 3+13

18 2*3*3 5+13

20 2*2*5 7+13

22 2*11 11+11

24 2*2*2*3 5+19


Now, we know the odd primes from 3 to 100 is this list of 24 primes 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97.

So there are 24 primes to work with to build all the even numbers from 6 to 100.

Now we ask, what are all the Two Prime Composites from 9 to 100, deleting all 2s

3*3 = 9 3+3= 6
3 * 5  = 15 3+5 = 8
3 * 7  = 21 3+7 = 10
5 * 5  = 25 5+5 =10
3 * 11 = 33 3+11 = 14
5 * 7  = 35 5+ 7 = 12
3 * 13 = 39 3+13 = 16
7 *  7  = 49 7+7 = 14
3 * 17 = 51 3+17 = 20
5 * 11 = 55 5+11 = 16
3 * 19 = 57   3+19 = 22
5 * 13  = 65 5+13 = 18
3 * 23  = 69   3+23 = 26
7 * 11  = 77 7+11 = 28
5 *17  = 85 5+17 = 22
3 * 29  = 87 3+29 = 32
7 * 13  =  91 7+13 = 20
3 * 31  =  93 3+ 31 = 34
5 * 19  =  95 5+ 19 = 24

There are 19 such Two Prime Composites and all the evens from 6 to 34 are there except missing 30. The 30 will be picked up by 7*23 or by 11*19.

Alright we are ready for a preliminary Theorem proof of Goldbach

Preliminary Theorem Statement:: given any even number from 6 onwards, it has to have at least two primes that sum to that even number

Preliminary Proof Statement:: Every even number is dissolvable into unique prime factors. Every two odd numbers when added produce an even number. Every even number N, starting with 6, has a prime number P smaller than itself, thus, we find a second prime number Q to add on, giving P+ Q = N, and if not, then the number P*Q is also, nonexistent, violating the Fundamental Theorem of Arithmetic.

This is the Goldbach conjecture and proof, where we find just one solution set for each even number.

Now, to Prove the Goldbach extensions and Legendre and Legendre extensions we need the Columns of Even Numbers and the Bertrand Postulate, between N and 2N exists a prime.

The Legendre Extension Conjecture that between all Perfect Squares Interval, rests a Two Prime Composite, starting with 4 to 9, a Two Prime Composite all of which have form P*Q and where the P is always 3.

What we do that is new, is define Monotone Increase as per a sequence of counts where there is no zero count in an interval such as this shown below:


9
Counter = 1
16
Counter = 1
25
Counter = 1
36
Counter = 1
49
Counter = 2
64
Counter = 1
81
Counter = 2
100
Counter = 1
121
Counter = 3
144
Counter = 1
169
Counter = 2
196
Counter = 3
225
Counter = 2
256
Counter = 1
289
Counter = 4
324
Counter = 2
361
Counter = 2
400
Counter = 2
441
Counter = 3
484
Counter = 3
529
Counter = 3
576
Counter = 3
625
Counter = 2
676
Counter = 5
729
Counter = 2
784
Counter = 4
841
Counter = 3
900
Counter = 4
961
Counter = 2
1024
Counter = 4
1089
Counter = 4
1156
Counter = 3
1225
Counter = 4
1296
Counter = 4
1369
Counter = 5
1444
Counter = 4
1521
Counter = 3
1600
Counter = 3
1681
Counter = 5
1764
Counter = 5
1849
Counter = 5
1936
Counter = 5
2025
Counter = 4
2116
Counter = 4
2209
Counter = 5
2304
Counter = 4
2401
Counter = 6
2500
Counter = 5
2601
Counter = 4
2704
Counter = 4
2809
Counter = 6
2916
Counter = 4
3025
Counter = 7
3136
Counter = 5
3249
Counter = 7
3364
Counter = 4
3481
Counter = 5
3600
Counter = 7
3721
Counter = 4
3844
Counter = 9
3969
Counter = 2
4096
Counter = 4
4225
Counter = 8
4356
Counter = 8
4489
Counter = 4
4624
Counter = 8
4761
Counter = 8
4900
Counter = 5
5041
Counter = 6
5184
Counter = 5
5329
Counter = 7
5476
Counter = 6
5625
Counter = 6
5776
Counter = 5
5929
Counter = 9
6084
Counter = 5
6241
Counter = 9
6400
Counter = 6
6561
Counter = 6
6724
Counter = 8
6889
Counter = 8
7056
Counter = 8
7225
Counter = 7
7396
Counter = 5
7569
Counter = 7
7744
Counter = 6
7921
Counter = 11
8100
Counter = 9
8281
Counter = 8
8464
Counter = 7
8649
Counter = 7
8836
Counter = 7
9025
Counter = 8
9216
Counter = 6
9409
Counter = 7
9604
Counter = 9
9801
Counter = 9
10000
Counter = 8
10201
Counter = 8
10404
Counter = 8
10609
Counter = 9
10816
Counter = 10
11025
Counter = 9
11236
Counter = 7
11449

...

4277814025
Counter = 2064
4277944836
Counter = 2077
4278075649
Counter = 2108
4278206464
Counter = 2129
4278337281
Counter = 2039
4278468100
Counter = 2025
4278598921
Counter = 2098
4278729744
Counter = 2014
4278860569
Counter = 2065
4278991396
Counter = 2056
4279122225
Counter = 2061
4279253056
Counter = 2012
4279383889
Counter = 2055
4279514724
Counter = 2112
4279645561
Counter = 2066
4279776400
Counter = 2050
4279907241
Counter = 2063
4280038084
Counter = 2014
4280168929
Counter = 2089
4280299776
Counter = 2036
4280430625
Counter = 2050
4280561476
Counter = 2000
4280692329
Counter = 2099
4280823184
Counter = 2093
4280954041
Counter = 2062
4281084900
Counter = 2084
4281215761
Counter = 2052
4281346624
Counter = 2106
4281477489
Counter = 2095
4281608356
Counter = 2073
4281739225
Counter = 2052
4281870096
Counter = 2064
4282000969
Counter = 2026
4282131844
Counter = 2080
4282262721
Counter = 2055
4282393600
Counter = 2071
4282524481
Counter = 2058
4282655364
Counter = 2043
4282786249
Counter = 2088
4282917136
Counter = 2071
4283048025
Counter = 2073
4283178916
Counter = 2069
4283309809
Counter = 2056
4283440704
Counter = 2097
4283571601
Counter = 2061
4283702500
Counter = 2120
4283833401
Counter = 2091
4283964304
Counter = 2091
4284095209
Counter = 2098
4284226116
Counter = 2023
4284357025
Counter = 2052
4284487936
Counter = 2093
4284618849
Counter = 2042
4284749764
Counter = 2074
4284880681
Counter = 2086
4285011600
Counter = 2041
4285142521
Counter = 2057
4285273444
Counter = 2088
4285404369
Counter = 2049
4285535296
Counter = 2104
4285666225
Counter = 2067
4285797156
Counter = 2061
4285928089
Counter = 2068
4286059024
Counter = 2073
4286189961
Counter = 2036
4286320900
Counter = 2077
4286451841
Counter = 2044
4286582784
Counter = 2086
4286713729
Counter = 2099
4286844676
Counter = 2108
4286975625
Counter = 2083
4287106576
Counter = 2051
4287237529
Counter = 2083
4287368484
Counter = 2060
4287499441
Counter = 2068
4287630400
Counter = 2086
4287761361
Counter = 2053
4287892324
Counter = 2105
4288023289
Counter = 2087
4288154256
Counter = 2065
4288285225
Counter = 2074
4288416196
Counter = 2020
4288547169
Counter = 2126
4288678144
Counter = 2090
4288809121
Counter = 2115
4288940100
Counter = 2090
4289071081
Counter = 2088
4289202064
Counter = 2077
4289333049
Counter = 2125
4289464036
Counter = 2008
4289595025
Counter = 2080
4289726016
Counter = 2042
4289857009
Counter = 2086
4289988004
Counter = 2088
4290119001
Counter = 2058
4290250000
Counter = 2036
4290381001
Counter = 2149
4290512004
Counter = 2067
4290643009
Counter = 2078
4290774016
Counter = 2060
4290905025
Counter = 2089
4291036036
Counter = 2058
4291167049
Counter = 2118
4291298064
Counter = 1992
4291429081
Counter = 2031
4291560100
Counter = 2078
4291691121
Counter = 2090
4291822144
Counter = 2046
4291953169
Counter = 2117
4292084196
Counter = 2091
4292215225
Counter = 2098
4292346256
Counter = 2074
4292477289
Counter = 2047
4292608324
Counter = 2087
4292739361
Counter = 2090
4292870400
Counter = 2065
4293001441
Counter = 2058
4293132484
Counter = 2086
4293263529
Counter = 2059
4293394576
Counter = 2045
4293525625
Counter = 2064
4293656676
Counter = 2058
4293787729
Counter = 2049
4293918784
Counter = 2013
4294049841
Counter = 2126
4294180900
Counter = 2125
4294311961
Counter = 2038
4294443024
Counter = 2093
4294574089
Counter = 2095
4294705156
Counter = 2089
4294836225
(source Vinicius)

AP

[Archimedes Plutonium]

Alright, about this juncture point we need to review the Fundamental Theorem of Arithmetic-- unique prime factorization theorem. Important theorem for used daily, but its proof is one of the ugliest proofs of elementary math. FTArith is about multiplication of primes, while Goldbach is about addition of primes.

I am setting up Goldbach as a FTArith only Add instead of multiply.

The Goldbach is true because it is linked to FTArith Multiply

Every prime that exists (omit 2) when taken in pairs and multiplied produces a Two Prime Composite

Every two primes when added yields an even number. What Goldbach pondered is whether all evens 6 and beyond are the sum of two primes.

Something in math says it is unacceptable to be even beyond 6 and not be the sum of two primes.

AP

[Archimedes Plutonium]

Post by Archimedes Plutonium
Alright, about this juncture point we need to review the Fundamental Theorem of Arithmetic-- unique prime factorization theorem. Important theorem for used daily, but its proof is one of the ugliest proofs of elementary math. FTArith is about multiplication of primes, while Goldbach is about addition of primes.
I am setting up Goldbach as a FTArith only Add instead of multiply.
The Goldbach is true because it is linked to FTArith Multiply
Every prime that exists (omit 2) when taken in pairs and multiplied produces a Two Prime Composite
Every two primes when added yields an even number. What Goldbach pondered is whether all evens 6 and beyond are the sum of two primes.
Something in math says it is unacceptable to be even beyond 6 and not be the sum of two primes.

Alright, good great progress. And when this is over with, I am speculating that I will have a better proof of the Fundamental Theorem of Arithmetic, unique prime factorization. Old Math's is far too much of a tourist guide yappity yap about factorization. A true proof would be a mere two or three sentences, not some long winded yarn. Why is the right triangle area 1/2 base times height, because it is 1/2 a rectangle, QED. And FTArith should be like that, one two sentences, finished.

Now, I have the answer to the question of the above post. Why is it unacceptable for an even number to not be the sum of two primes? Why.

The answer is the question, why is it unacceptable for a number to have a composite number as its factor and which that composite number cannot be reduced to primes. Sounds alien. That you have a number N to factor into primes, and yet you come to a composite inside N and you cannot factor that composite into primes.

That is the same question as why cannot a even number come along in which it is unable to have two primes add up to the number, only a prime and a composite or unit can add up to the number.

So we know from Fundamental Theorem of Arithmetic-Multiply, that every number N is dissolved down into being a list of primes. And actually, that this the brunt, the mainstay of FTArithMult. Its analog of Fundamental Theorem of Arithmetic-Addition (yes this is new to math) is that every Even Number must be able to be reduced to the addition of 2 primes.

Fundamental Theorem of Arithmetic-Multiply:: this is our commonly known Unique Prime Factorization theorem which says that given any number N, it is reduced to a multiplication of a set of primes and that set is unique.

Fundamental Theorem of Arithmetic-Addition;; this is only just discovered today, and says that given any Even Number N, 4 and beyond is the sum of at least one pair of primes. The larger the number N is, the more prime pairs exist. The prime pairs are unique in multiplication but not unique in addition.

In other words, what Goldbach conjecture was, was a missing of a Fundamental theorem of Arithmetic-Add.

The proof of FTA-multiply is similar to the proof of FTA-Add and involves Algebra of Group theory in Columns.

0 2
1 1


0 3
1 2

0 4
1 3
2 2


0 5
1 4
2 3


0 6
1 5
2 4
3 3

0 7
1 6
2 5
3 4


0 8
1 7
2 6
3 5
4 4

0 9
1 8
2 7
3 6
4 5


0 10
1 9
2 8
3 7
4 6
5 5

So, modern math, New Math is going to have a proof of FTA-multiply along with FTA-add all from the uniqueness of these Group Algebra columns. Multiplication and addition are unique to the Columns. By unique, I mean 2*8 can only be found in 10 Column for it is also 2+8. And 3*6 can only be found in 9 Column, for it is also 3+6.

What the Columns of Group theory Algebra does, is lump addition with multiplication uniquely. Uniqueness is ingrained in the Columns. Every number exists in the columns somewhere, because every number is going to be a 1*N. So that 55 will appear as 1*55 in the 56 Column and then 55 will appear in the 16 Column as 5*11 which is also 5+11.

You see, these Columns are all about Uniqueness. They are the Group Theory of Arithmetic, that is how important these columns are. And Group theory should start out with these columns.

So, now, we know the Old Math's messy proof of Unique Prime Factorization. long and messy, and rather crumby in my opinion, for it lacks the force of proof, the force of convincing. It is more like a museum tour, look at the p1, p2, p3, look at the q1, q2, q3. No force of logic is forthcoming, just look look look.

More latter,,,,,

AP

[Archimedes Plutonium]

Alright, let me get a start on New Maths proof of Fundamental Theorem of Arithmetic.

Proof:: given any number N, we divide it by all the numbers smaller than N, and keep only those that divide N evenly, discarding all others. We always omit 1. For example if we had 12 then divide it by 11,10,9,8,7,6,5,4,3,2. Keeping only the whole number results -- 6, 4, 3, 2. Now in turn we factor thosenumbers leaving 3, 2. Now we clean up by multiplying our list of all possible factors and see that 12 = 2*2*3. Finally, uniqueness is assured because any other set of primes would not add to 2+2+3 = 7. Uniqueness comes from switching back and forth between add or multiply. QED

AP

[b***@gmail.com]

AP brain farto, famous for posting nonsense, already
for 30 years on sci.math. Not a single line of math.

Post by Archimedes Plutonium
Alright, let me get a start on New Maths proof of Fundamental Theorem of Arithmetic.
Proof:: given any number N, we divide it by all the numbers smaller than N, and keep only those that divide N evenly, discarding all others. We always omit 1. For example if we had 12 then divide it by 11,10,9,8,7,6,5,4,3,2. Keeping only the whole number results -- 6, 4, 3, 2. Now in turn we factor thosenumbers leaving 3, 2. Now we clean up by multiplying our list of all possible factors and see that 12 = 2*2*3. Finally, uniqueness is assured because any other set of primes would not add to 2+2+3 = 7. Uniqueness comes from switching back and forth between add or multiply. QED
AP

[b***@gmail.com]

Some coffee and a cake is enough
to see that addition doesnt have
unique prime number summands:

24 = 11 + 13 = 17 + 7

[Archimedes Plutonium]

Post by Archimedes Plutonium
Alright, let me get a start on New Maths proof of Fundamental Theorem of Arithmetic.
Proof:: given any number N, we divide it by all the numbers smaller than N, and keep only those that divide N evenly, discarding all others. We always omit 1. For example if we had 12 then divide it by 11,10,9,8,7,6,5,4,3,2. Keeping only the whole number results -- 6, 4, 3, 2. Now in turn we factor thosenumbers leaving 3, 2. Now we clean up by multiplying our list of all possible factors and see that 12 = 2*2*3. Finally, uniqueness is assured because any other set of primes would not add to 2+2+3 = 7. Uniqueness comes from switching back and forth between add or multiply. QED

Of course, that can be pared down to a proof of the Fundamental Theorem of Arithmetic-Multiply::

Proof:: Given any number N, we divide by all N less than N until no more even divisions (factors) occur. For example 12 = 2*6 = 2*2*3. So now we have a product of primes, and the only question is Uniqueness. We take the product of primes and add them 2+2+3 = 7. Any other multiplication violates addition uniqueness, hence multiplication is unique.

Fundamental Theorem of Arithmetic- Addition

Proof:: Given any number N, we add together all N less than N until we have a Algebra Group Column of N. For example N= 8

0 8
1 7
2 6
3 5
4 4

The Algebra Group Column is all possible additions of N. Algebra theory always places Addition with Multiplication, for one is the symmetrical equivalent of the other. For 2*4 is the same as 4+4.

Now, in the proof of FTArith-Multiply, we reduced N to primes. In FTArith-Addition, since Multiply and Addition are Symmetrical Equivalents we reduce N in addition to its Prime Addition N = P + Q, where P and Q are primes. Justification:: Algebra of multiply and add are symmetrical equivalents, hence N = P+Q exists. So, for 8, knowing two odd add to a even, we start with the smallest odd prime 3 and keep adding all the primes smaller than N until we reach a P+Q=N. QED

So, here in math history we have a full proof of Goldbach, Now the question is, does mathematics need a Axiom that given any number N, 4 or larger, that N has at least one P+Q, or, does the proof of FTArith-Multiply justify N = P+Q.

So, most math professors cannot nor will not understand the question posed.

The problem in Old Math, is that they could prove FTArith-Multiply, but, sloppily, they could not prove its uniqueness. They gave a Induction argument using Euclid if P divides N, then P divides its factors. That was horribly sloppy for it does not prove uniqueness. So Old Math never really had a proof of Fundamental Theorem Multiply, there's was a discussion of uniqueness but not a proof of uniqueness.

The only means of proving Uniqueness in FTArith-Multiply is that N = p1*q1*p2*q2*...pm*qm is for A= p1+q1+p2+q2+...+pm +qm is unique, confering uniqueness upon N.

You see, a huge gap and hole in logic in Old Math, that uniqueness is only conferred upon multiplication because the addition is unique.

Now in the proof of FTArith-Add, the symmetry plays full circle, in that the uniqueness of multiplication is used to prove that every number N is the sum of two primes.

Now, does Math need an axiom to state that, or, does the proof of FTArith-Multiply allow for the statement, that N is pared down to a sum of two primes.

I think the axiom is required, for it was assumed that addition must go along with multiplication.

When we say 12 = 3*4 which is 4+4+4, we recognize that they are one and the same, but, we fail to recognize they are different. That add is not multiply, but that two of them are different, otherwise we could get rid of multiply and have an algebra completely based on add alone. We see that Algebra cannot be based upon add alone and that we require this multiply to always be present with add. Present not because they are one and the same, but because they are very much different from one another.

So, when Algebra of Group theory came to be, it was noticed that always a Group theory required both-- add with multiply. No-one in Old Math could say, "why does add always have to have multiply present"And the answer is that the two are different.

So, a proper proof of Fundamental Theorem of Arithmetic-Multiply, can exist only when we prove uniqueness by addition that 12 = 2*2*3 because 2+2+3 = 7. The uniqueness of multiply is guaranteed because of the uniqueness of addition of those factors. Not by some tourist walk around that you fully divided all terms.

And then, Old Math missed the theorem that if Multiply has a Fundamental Theorem, thus, Addition must have also, its Fundamental Theorem of Arithmetic Add, that given N as 4 or larger, N = prime P + prime Q.

AP

[b***@gmail.com]

Any proof of the fundamental theorem of algebra, that
looks at all possible divisors of a number, is already
the wrong approach.

You don't need to add the other nonsense of additive
uniqueness, you already revealed yourself a complete
idiot in the intro.

What does it buy you looking at 6, 4, 3, 2 in factoring
12? Nothing. You dont factor like this. Either you
pick the pair 6*2=12 or the pair 4*3=12,

and from this you can workout a proof.

Post by Archimedes Plutonium

Post by Archimedes Plutonium
Alright, let me get a start on New Maths proof of Fundamental Theorem of Arithmetic.
Proof:: given any number N, we divide it by all the numbers smaller than N, and keep only those that divide N evenly, discarding all others. We always omit 1. For example if we had 12 then divide it by 11,10,9,8,7,6,5,4,3,2. Keeping only the whole number results -- 6, 4, 3, 2. Now in turn we factor thosenumbers leaving 3, 2. Now we clean up by multiplying our list of all possible factors and see that 12 = 2*2*3. Finally, uniqueness is assured because any other set of primes would not add to 2+2+3 = 7. Uniqueness comes from switching back and forth between add or multiply. QED

Proof:: Given any number N, we divide by all N less than N until no more even divisions (factors) occur. For example 12 = 2*6 = 2*2*3. So now we have a product of primes, and the only question is Uniqueness. We take the product of primes and add them 2+2+3 = 7. Any other multiplication violates addition uniqueness, hence multiplication is unique.
Fundamental Theorem of Arithmetic- Addition
Proof:: Given any number N, we add together all N less than N until we have a Algebra Group Column of N. For example N= 8
0 8
1 7
2 6
3 5
4 4
The Algebra Group Column is all possible additions of N. Algebra theory always places Addition with Multiplication, for one is the symmetrical equivalent of the other. For 2*4 is the same as 4+4.
Now, in the proof of FTArith-Multiply, we reduced N to primes. In FTArith-Addition, since Multiply and Addition are Symmetrical Equivalents we reduce N in addition to its Prime Addition N = P + Q, where P and Q are primes. Justification:: Algebra of multiply and add are symmetrical equivalents, hence N = P+Q exists. So, for 8, knowing two odd add to a even, we start with the smallest odd prime 3 and keep adding all the primes smaller than N until we reach a P+Q=N. QED
So, here in math history we have a full proof of Goldbach, Now the question is, does mathematics need a Axiom that given any number N, 4 or larger, that N has at least one P+Q, or, does the proof of FTArith-Multiply justify N = P+Q.
So, most math professors cannot nor will not understand the question posed.
The problem in Old Math, is that they could prove FTArith-Multiply, but, sloppily, they could not prove its uniqueness. They gave a Induction argument using Euclid if P divides N, then P divides its factors. That was horribly sloppy for it does not prove uniqueness. So Old Math never really had a proof of Fundamental Theorem Multiply, there's was a discussion of uniqueness but not a proof of uniqueness.
The only means of proving Uniqueness in FTArith-Multiply is that N = p1*q1*p2*q2*...pm*qm is for A= p1+q1+p2+q2+...+pm +qm is unique, confering uniqueness upon N.
You see, a huge gap and hole in logic in Old Math, that uniqueness is only conferred upon multiplication because the addition is unique.
Now in the proof of FTArith-Add, the symmetry plays full circle, in that the uniqueness of multiplication is used to prove that every number N is the sum of two primes.
Now, does Math need an axiom to state that, or, does the proof of FTArith-Multiply allow for the statement, that N is pared down to a sum of two primes.
I think the axiom is required, for it was assumed that addition must go along with multiplication.
When we say 12 = 3*4 which is 4+4+4, we recognize that they are one and the same, but, we fail to recognize they are different. That add is not multiply, but that two of them are different, otherwise we could get rid of multiply and have an algebra completely based on add alone. We see that Algebra cannot be based upon add alone and that we require this multiply to always be present with add. Present not because they are one and the same, but because they are very much different from one another.
So, when Algebra of Group theory came to be, it was noticed that always a Group theory required both-- add with multiply. No-one in Old Math could say, "why does add always have to have multiply present"And the answer is that the two are different.
So, a proper proof of Fundamental Theorem of Arithmetic-Multiply, can exist only when we prove uniqueness by addition that 12 = 2*2*3 because 2+2+3 = 7. The uniqueness of multiply is guaranteed because of the uniqueness of addition of those factors. Not by some tourist walk around that you fully divided all terms.
And then, Old Math missed the theorem that if Multiply has a Fundamental Theorem, thus, Addition must have also, its Fundamental Theorem of Arithmetic Add, that given N as 4 or larger, N = prime P + prime Q.
AP

[Archimedes Plutonium]

Post by Archimedes Plutonium

Post by Archimedes Plutonium
Alright, let me get a start on New Maths proof of Fundamental Theorem of Arithmetic.
Proof:: given any number N, we divide it by all the numbers smaller than N, and keep only those that divide N evenly, discarding all others. We always omit 1. For example if we had 12 then divide it by 11,10,9,8,7,6,5,4,3,2. Keeping only the whole number results -- 6, 4, 3, 2. Now in turn we factor thosenumbers leaving 3, 2. Now we clean up by multiplying our list of all possible factors and see that 12 = 2*2*3. Finally, uniqueness is assured because any other set of primes would not add to 2+2+3 = 7. Uniqueness comes from switching back and forth between add or multiply. QED

Proof:: Given any number N, we divide by all N less than N until no more even divisions (factors) occur. For example 12 = 2*6 = 2*2*3. So now we have a product of primes, and the only question is Uniqueness. We take the product of primes and add them 2+2+3 = 7. Any other multiplication violates addition uniqueness, hence multiplication is unique.

Sad, sad sad, that Wikipedia under its Fundamental Theorem of Arithmetic opens up to Gauss's Fundamental Theorem of Algebra cover text. I suppose the author of that entry just plain wants to confuse everyone. Wants to advertise Gauss, when Gauss should be over in the Fundamental Theorem of Algebra, not arithmetic. But this is typical example of a mind in mathematics these days-- attention only to minutia detail, not much attention to "did I get the overall big picture all correct and straight and clear"

The lack of logic in Old Math is a horrendous lack.

Post by Archimedes Plutonium
Fundamental Theorem of Arithmetic- Addition
Proof:: Given any number N, we add together all N less than N until we have a Algebra Group Column of N. For example N= 8
0 8
1 7
2 6
3 5
4 4
The Algebra Group Column is all possible additions of N. Algebra theory always places Addition with Multiplication, for one is the symmetrical equivalent of the other. For 2*4 is the same as 4+4.
Now, in the proof of FTArith-Multiply, we reduced N to primes. In FTArith-Addition, since Multiply and Addition are Symmetrical Equivalents we reduce N in addition to its Prime Addition N = P + Q, where P and Q are primes. Justification:: Algebra of multiply and add are symmetrical equivalents, hence N = P+Q exists. So, for 8, knowing two odd add to a even, we start with the smallest odd prime 3 and keep adding all the primes smaller than N until we reach a P+Q=N. QED

Alright, what I am focused on here, is the concept that in mathematics, addition goes always with multiplication being present and vice versa. These two are COMPLIMENTS, not equals. And that you cannot have a math that has just only one without the other.

So, that if math has a Fundamental Theorem of Arithmetic that is Multiplication, then, math must require another Fundamental Theorem of Arithmetic that is Addition. The two are vastly different but required and essential.

Post by Archimedes Plutonium
So, here in math history we have a full proof of Goldbach, Now the question is, does mathematics need a Axiom that given any number N, 4 or larger, that N has at least one P+Q, or, does the proof of FTArith-Multiply justify N = P+Q.
So, most math professors cannot nor will not understand the question posed.
The problem in Old Math, is that they could prove FTArith-Multiply, but, sloppily, they could not prove its uniqueness. They gave a Induction argument using Euclid if P divides N, then P divides its factors. That was horribly sloppy for it does not prove uniqueness. So Old Math never really had a proof of Fundamental Theorem Multiply, there's was a discussion of uniqueness but not a proof of uniqueness.

Alright, let me use Wikipedia's page on Fund Theor Arithm Multiply to expose how Old Math's proof is not a proof at all, but a discussion at best, they fail in proving uniqueness.

Now I cited Euclid above, and my citation is vague.

Wikipedia calls it the Euclid Lemma, and is proposition 30:

If two numbers by multiplying one another make some number, and any prime number measure the product, it will also measure one of the original numbers. Euclid Elements

So, what Euclid said was that if A*B = N and that prime P divides N, then P will divide either A or B or both.

And what I am going to argue, is that, such a lemma is not going to prove uniqueness. The use of such a lemma is just a distraction-walk about- discussion, not a proof that A*B is unique.

What proves A*B is unique, is that A+B is unique, thus, a transfer of uniqueness is conferred upon A*B.

So what I argue is that to prove uniqueness of multiplication, you need to insert addition. Vice versa, to prove uniqueness of addition, you must enter with the uniqueness of multiplication.

So, Goldbach conjecture was never proven, because, well, it cannot be proven without recognizing a whole chapter of mathematics was missing-- a fundamental theorem of arithmetic addition.

Post by Archimedes Plutonium
The only means of proving Uniqueness in FTArith-Multiply is that N = p1*q1*p2*q2*...pm*qm is for A= p1+q1+p2+q2+...+pm +qm is unique, confering uniqueness upon N.
You see, a huge gap and hole in logic in Old Math, that uniqueness is only conferred upon multiplication because the addition is unique.
Now in the proof of FTArith-Add, the symmetry plays full circle, in that the uniqueness of multiplication is used to prove that every number N is the sum of two primes.
Now, does Math need an axiom to state that, or, does the proof of FTArith-Multiply allow for the statement, that N is pared down to a sum of two primes.
I think the axiom is required, for it was assumed that addition must go along with multiplication.

I remember to this very day, in math class, forgotten what class, where the teacher remarks "we have group theory, and given elementary algebra, addition, multiplication.." and a student interrupts by asking why always the two operators together? And the professor really not being able to justify why always add with multiply exists together.

Well, here we know why, because add and multiply are not equals, but are Compliments for they are vastly different from one another, but necessary in being together, for compliment means different yet together.

Post by Archimedes Plutonium
When we say 12 = 3*4 which is 4+4+4, we recognize that they are one and the same, but, we fail to recognize they are different. That add is not multiply, but that two of them are different, otherwise we could get rid of multiply and have an algebra completely based on add alone. We see that Algebra cannot be based upon add alone and that we require this multiply to always be present with add. Present not because they are one and the same, but because they are very much different from one another.
So, when Algebra of Group theory came to be, it was noticed that always a Group theory required both-- add with multiply. No-one in Old Math could say, "why does add always have to have multiply present"And the answer is that the two are different.
So, a proper proof of Fundamental Theorem of Arithmetic-Multiply, can exist only when we prove uniqueness by addition that 12 = 2*2*3 because 2+2+3 = 7. The uniqueness of multiply is guaranteed because of the uniqueness of addition of those factors. Not by some tourist walk around that you fully divided all terms.
And then, Old Math missed the theorem that if Multiply has a Fundamental Theorem, thus, Addition must have also, its Fundamental Theorem of Arithmetic Add, that given N as 4 or larger, N = prime P + prime Q.

So, Old Math's proof of Fundamental Theorem of Arithmetic is a fake proof, a shame, because it does not prove uniqueness. The Euclid Lemma, can only discuss on whether a factor is a prime factor or whether a composite factor. That is all the Euclid Lemma does. And the Old Math FTA alleged proof fails because it fails to prove uniqueness.

Here is where complimentarity of addition with multiplication is vital. The addition of a number reduced to prime factors when added is a unique addition number, that is what confers uniqueness to multiplication of primes.

AP

[Archimedes Plutonium]

Alright, so, in New Math, the proof of Goldbach is the proof of Fundamental Theorem of Arithmetic-Add, where every Even Number has at least one decomposition into two prime numbers. Where 2 = 1+1, 4= 2+2, 8= 3+5. Does it need a axiom saying all numbers, whether even or odd, have a addition decomposition into primes, of just two primes, except 1, unless we take 0 as being prime, then 1 = 1+0.

This makes total sense, in that Fundamental Theorem of Arithmetic-Multiply is decomposition of every number M into prime factors.

So, does Goldbach need a axiom, that proves== all Numbers N have at least one pair of primes that sum to the number, or can math prove it without such an axiom. Here I believe math needs an axiom. One that does not state the Goldbach directly but is more general and easily proves the Goldbach. An axiom like this, for it is what was constantly holding me up in prior proof attempts of Goldbach, where I constantly had to say-- if 3+5 does not exist in 8 Matrix, then 3*5 does not exist in the Matrix Columns. That point needed an axiom, otherwise, I was grappling at straws.

So, if mathematics has an axiom in Arithmetic, that the Matrix Columns of Numbers are Uniqueness property of addition and multiplication. Simply stated as such. Then I can prove Goldbach in a snap. Because when I reach the point in the proof argument that the Bertrand primes exist on rightside and leftside and wondering if they line up. So, if they do not line up, then I apply the AXIOM of Uniqueness telling me the Bertrand Primes P and Q, cause the number P*Q to not exist, in violation of that axiom.

So the axiom is more general than just proving Goldbach, the Axiom is a distillation of the fact that the Matrix Columns force the arithmetic of add and multiply to be Unique results.

So that way back when in very early 1970s, sitting in Math Class at college, and the professor saying that the Matrix has both add and multiply, and another student questioning why both, why not algebra be just add. Well, if Math had this Axiom, then there would not be that "classroom hand waving"

If there had been this axiom, then I no longer need to say in a Goldbach proof, if 3 and 5 did not line up in 8 Column, means 3*5 =15 does not exist, for all I needed is to point to this axiom.

Here that AXIOM should say this-- the sums of two numbers A+B is unique as well as the multiplication of those two numbers A*B is unique and they are related.

So, given the Matrix Columns and the axiom of uniqueness, would easily prove Goldbach.

By the way, a valid proof of Fundamental Theorem of Arithmetic-Multiplication needs to use that same Axiom of Uniqueness.

So, we have Goldbach out of the way, and Goldbach, remember says a even number N has at least one pair of primes, but we can see, that most even numbers have two or more pair of primes, such as 10 is 3+7 and also 5+5. As the number N gets large, the number of Goldbach solutions gets large. So, if my proof of Goldbach is true and valid, it should also lead to easy proofs of extensions to Goldbach such as after 14, Goldbach has two solutions, after 70, Goldbach has 3 solutions, after 130, Goldbach has 4 solutions.

So when a person of Logic sees that data, and here chasing after just 1 solution, makes the person of logic realize, a Major Overhaul is needed, so that the Goldbach of 1 solution and the Goldbach of myriad solutions all are tackled and conquered in one fell swoop.

AP

[b***@gmail.com]

Brain farto, even not able to understand what a prime
number is. Prime numbers are defined based on multi-
plication, and so is unique factoring.

There is no such thing as an addition prime number,
and there is also no unique summands. So your
"Fundamental Theorem of Arithmetic-Multiply"

is nonsense like your oval conic sections. Somebody
needs to really beat the shit out of your ass some
time, since you are producing so much nonsense.

30 years, only spamming, not a single line of math.

Post by Archimedes Plutonium
Alright, so, in New Math, the proof of Goldbach is the proof of Fundamental Theorem of Arithmetic-Add, where every Even Number has at least one decomposition into two prime numbers. Where 2 = 1+1, 4= 2+2, 8= 3+5. Does it need a axiom saying all numbers, whether even or odd, have a addition decomposition into primes, of just two primes, except 1, unless we take 0 as being prime, then 1 = 1+0.
This makes total sense, in that Fundamental Theorem of Arithmetic-Multiply is decomposition of every number M into prime factors.
So, does Goldbach need a axiom, that proves== all Numbers N have at least one pair of primes that sum to the number, or can math prove it without such an axiom. Here I believe math needs an axiom. One that does not state the Goldbach directly but is more general and easily proves the Goldbach. An axiom like this, for it is what was constantly holding me up in prior proof attempts of Goldbach, where I constantly had to say-- if 3+5 does not exist in 8 Matrix, then 3*5 does not exist in the Matrix Columns. That point needed an axiom, otherwise, I was grappling at straws.
So, if mathematics has an axiom in Arithmetic, that the Matrix Columns of Numbers are Uniqueness property of addition and multiplication. Simply stated as such. Then I can prove Goldbach in a snap. Because when I reach the point in the proof argument that the Bertrand primes exist on rightside and leftside and wondering if they line up. So, if they do not line up, then I apply the AXIOM of Uniqueness telling me the Bertrand Primes P and Q, cause the number P*Q to not exist, in violation of that axiom.
So the axiom is more general than just proving Goldbach, the Axiom is a distillation of the fact that the Matrix Columns force the arithmetic of add and multiply to be Unique results.
So that way back when in very early 1970s, sitting in Math Class at college, and the professor saying that the Matrix has both add and multiply, and another student questioning why both, why not algebra be just add. Well, if Math had this Axiom, then there would not be that "classroom hand waving"
If there had been this axiom, then I no longer need to say in a Goldbach proof, if 3 and 5 did not line up in 8 Column, means 3*5 =15 does not exist, for all I needed is to point to this axiom.
Here that AXIOM should say this-- the sums of two numbers A+B is unique as well as the multiplication of those two numbers A*B is unique and they are related.
So, given the Matrix Columns and the axiom of uniqueness, would easily prove Goldbach.
By the way, a valid proof of Fundamental Theorem of Arithmetic-Multiplication needs to use that same Axiom of Uniqueness.
So, we have Goldbach out of the way, and Goldbach, remember says a even number N has at least one pair of primes, but we can see, that most even numbers have two or more pair of primes, such as 10 is 3+7 and also 5+5. As the number N gets large, the number of Goldbach solutions gets large. So, if my proof of Goldbach is true and valid, it should also lead to easy proofs of extensions to Goldbach such as after 14, Goldbach has two solutions, after 70, Goldbach has 3 solutions, after 130, Goldbach has 4 solutions.
So when a person of Logic sees that data, and here chasing after just 1 solution, makes the person of logic realize, a Major Overhaul is needed, so that the Goldbach of 1 solution and the Goldbach of myriad solutions all are tackled and conquered in one fell swoop.
AP

[Archimedes Plutonium]

Alright, the proof of Goldbach is the proof of the Fundamental Theorem of Arithmetic-Addition. We we prove all the many extensions of Goldbach, and we actively use the Axiom of Complimentarity of Arithmetic-- that add is connected to multiply in a symmetry of uniqueness as displayed by this Matrix Columns of numbers

0  1

0  2
1  1


0  3
1  2

0  4
1  3
2  2


0  5
1  4
2  3


0  6
1  5
2  4
3  3

0  7
1  6
2  5
3  4


0  8
1  7
2  6
3  5
4  4

0  9
1  8
2  7
3  6
4  5


0  10
1   9
2   8
3   7
4   6
5   5


0   11
1   10
2    9
3    8
4    7
5    6


0    12
1    11
2    10
3      9
4      8
5      7
6      6

So if 3+5 is not a Goldbach solution to 8, then the multiplication 3*5 = 15 does not exist in violation of the axiom.

So, several days ago
Vinicius Claudino Ferraz        
1:01 PM (9 hours ago) wrote:

Extended Gold_Bach

Conjectures:

  4) Every even number n >= 4 is a sum of two primes.
  6) Every even number n >= 6 is a sum of two odd primes.
 14) Every even number n >= 14 is a sum of two odd primes twice.
     12 = 5 + 7
 70) Every even number n >= 70 is a sum of two odd primes three times.
     68 = 7 + 61 = 31 + 37
130) Every even number n >= 130 is a sum of two odd primes four times.
     128 = 19 + 109
     128 = 31 + 97
     128 = 61 + 67
154) Every even number n >= 154 is a sum of two odd primes five times.
     152 = 3 + 149
     152 = 13 + 139
     152 = 43 + 109
     152 = 73 + 79
190) Every even number n >= 190 is a sum of two odd primes six times.
     188 = 7 + 181
     188 = 31 + 157
     188 = 37 + 151
     188 = 61 + 127
     188 = 79 + 109
334) Every even number n >= 334 is a sum of two odd primes seven times.
     332 = 19 + 313
     332 = 61 + 271
     332 = 103 + 229
     332 = 109 + 223
     332 = 139 + 193
     332 = 151 + 181
400) Every even number n >= 400 is a sum of two odd primes nine times.
     398 = 19 + 379
     398 = 31 + 367
     398 = 61 + 337
     398 = 67 + 331
     398 = 127 + 271
     398 = 157 + 241
     398 = 199 + 199
     398 is the last time for seven times.
   > 368 is the last time for eight times. this is weird.
     488 is the last time for nine times.
     632 is the last time for ten times.

490) Every even number n >= 490 is a sum of two odd primes ten times.
634) Every even number n >= 634 is a sum of two odd primes eleven times.

--- end of Vinicius Claudino Ferraz wrote ---

So the Fundamental Theorem of Arithmetic-Addition proves Goldbach, now what proves all those Goldbach Extensions? Well, quite easily the axiom proves all those extensions by the sheer weight of the correspondence with Two Prime Composites:


 Now we ask, what are all the Two Prime Composites from 9 to 100, deleting all 2s (and 1 as unit prime)

3*3 = 9                 3+3= 6
3 * 5  = 15           3+5 = 8
3 * 7  = 21           3+7 = 10
5 * 5  = 25           5+5 =10
3 * 11 = 33          3+11 = 14
5 * 7  = 35           5+ 7 = 12
3 * 13 = 39          3+13 = 16
7 *  7  = 49          7+7 = 14
3 * 17 = 51          3+17 = 20
5 * 11 = 55          5+11 = 16
3 * 19 = 57          3+19 = 22
5 * 13  = 65         5+13 = 18
3 * 23  = 69         3+23 = 26
7 * 11  = 77         7+11 = 18
5 *17  = 85          5+17 = 22
3 * 29  = 87        3+29 = 32
7 * 13  =  91       7+13 = 20
3 * 31  =  93        3+ 31 = 34
5 * 19  =  95        5+ 19 = 24

So, if those primes progression did not exist, it violates the axiom.

Will simply note that the Two Prime Composites follows the pattern of the Goldbach Extensions. We have to include in the proof that the Two Prime Composites is a Monotone Increase progression, so that at some point in the progression we cease to see 1 counter, then, cease to see 2 counters, then cease to see 3 counters, duplicating the Extension Conjectures.

P.S. now I need to cover the mistake in Old Math's Fundamental Theorem of Arithmetic-Multiply, where they used Euclid Lemma to try to prove uniqueness, but that was a error of logic, for the lemma cannot prove uniqueness. That is why a Axiom was needed that proves uniqueness via a comparison of sum with product. So that 12 = 2*2*3 is unique set of primes because 2+2+3 = 7, that no other set of three primes can add to 7. So I need to re-work the Fundamental Theorem of Arithmetic-- Unique Prime Factorization and fix it so it is valid.

AP

[Don Redmond]

Post by Archimedes Plutonium
And what I am going to argue, is that, such a lemma is not going to prove uniqueness. The use of such a lemma is just a distraction-walk about- discussion, not a proof that A*B is unique.
What proves A*B is unique, is that A+B is unique, thus, a transfer of uniqueness is conferred upon A*B.
So what I argue is that to prove uniqueness of multiplication, you need to insert addition. Vice versa, to prove uniqueness of addition, you must enter with the uniqueness of multiplication.
AP

You can prove uniqueness of multiplication by induction. See Landau, Foundations of Analysis.

You can prove FT Arithmetic by induction (actually Well-Ordering, but the two are equivalent). See Hardy and Wright, Intro to Number Theory.

What could be more solid than an induction proof? Didn't claim this once a while back?

The main objection to your proof of Goldbach is that you give no procedure just lots of examples. We need a process that takes 2n and finds primes p and q such that p + q = 2n and a proof that that it always works. So if I take 2n = 10^1000, your general procedure should be guaranteed to work and not just because it worked on all the evens less than 10^100, say.

Don

[Archimedes Plutonium]

Post by Don Redmond

Post by Archimedes Plutonium
And what I am going to argue, is that, such a lemma is not going to prove uniqueness. The use of such a lemma is just a distraction-walk about- discussion, not a proof that A*B is unique.
What proves A*B is unique, is that A+B is unique, thus, a transfer of uniqueness is conferred upon A*B.
So what I argue is that to prove uniqueness of multiplication, you need to insert addition. Vice versa, to prove uniqueness of addition, you must enter with the uniqueness of multiplication.
AP

You can prove uniqueness of multiplication by induction. See Landau, Foundations of Analysis.

Not really for the Lemma to prove FTA.
Wikipedia calls it the Euclid Lemma, and is proposition 30:

If two numbers by multiplying one another make some number, and any prime number measure the product, it will also measure one of the original numbers. Euclid Elements

So, what Euclid said was that if A*B = N and that prime P divides N, then P will divide either A or B or both.

Let us say we had the numbers 2*5 =10, and 10*10 = 100. Now a prime P=5 measures 10 and measures 5. And P =5 measures 100. Hence by that tool 2*5 = 2*5*2*5

Where is the uniqueness in that.

So, when we install the AXIOM of uniqueness into Mathematics itself, that 2*5 compliments 2+5 =7 whereas 2+5+2+5= 14 can we ever prove Uniqueness in mathematics

Post by Don Redmond
You can prove FT Arithmetic by induction (actually Well-Ordering, but the two are equivalent). See Hardy and Wright, Intro to Number Theory.
What could be more solid than an induction proof? Didn't claim this once a while back?
The main objection to your proof of Goldbach is that you give no procedure just lots of examples. We need a process that takes 2n and finds primes p and q such that p + q = 2n and a proof that that it always works. So if I take 2n = 10^1000, your general procedure should be guaranteed to work and not just because it worked on all the evens less than 10^100, say.
Don

No, you are wrong, the trouble with Old Math FTA, is the mathematician never realized that to prove Uniqueness in mathematics, requires a axiom that embodies uniqueness. If you have no axiom that expresses uniqueness, in mathematics, you cannot expect all the other axioms-- lacking in the concept of uniqueness, to, all of a sudden dredge up uniqueness with some silly lemma.

The only reason Arithmetic can count, is that it has a counting axiom-- successor function. Without that axiom of counting, you cannot expect the rest of math to be able to muster counting.

If Geometry had no axiom of what it means to be parallel, you cannot expect the rest of the axioms of geometry to conjure up the meaning of parallel.

This is why Goldbach is unprovable, because the Fundamental Theorem of Arithmetic, based upon a Lemma for uniqueness, is not only incapable of proving FTA, but is incapable of ever proving Goldbach, for the only proof of Goldbach rests on math having a concept of UNIQUENESS. When we reach the point in the proof of Goldbach, that the Bertrand primes line up. Old Math cannot make the next jump-- that p and q line up, because Old Math has no axiom that expresses uniqueness. The Bertrand primes line up because of an axiom-- if p*q exists, then p+q exists.

So Don, why did you not see, in all your years in math, did not see that your Fundamental Theorem of Arithmetic falls inadequately short of a proof of uniqueness. The lemma proves that 10 is the same as 100.

If you had a Axiom of Uniqueness, then FTA is simple to prove. Any number N = p*q*r*s*t is unique because, well the axiom has p+q+r+s+t is a unique add. Why did not you catch that Don, instead of continuing to preach a gossipal of fakery.

Then, in class Don, whenever students ask, why is Algebra always have operators add multiply, why not just add alone or multiply alone? What sort of fakery answer did you give them Don? Go see so and so, or read so and so.

The reason that Algebra theory, group, field, Galois etcetera, the reason that they always have add and multiply together, is because of the missing axiom that math has missed since Ancient Greek times. The missing axiom of uniqueness-- in math, the only means of measuring uniqueness-- is by the compliments of add with multiply. 2*5 is unique, because 2+5 forces 2*5 to be unique, and vice versa.

So, Don, are you going to continue to preach your shoddy math fakery, or, time to change for the better.

AP

[b***@gmail.com]

The missing axiom of mongo math. Brain farto is even
too stupid to understand what a unique factorization
means.

For the integers multiplication is a ring, and not
a group. This means there is no inverse x^(-1) in
the integers.

On the other hand for the integers addition this
is a group, and not only a ring. We have here an
inverse, namely (-x).

So a number has always dozen of additive splits
into summands. When you have a number n, and
another number p, with p<n, you can automatically
split it into:

n = p + (n-p)

Integer multiplication doesn't have this. A number
has only a few factors. But the integers must be
positive. This uniquesness doesn't

hold anymore if we allow negative integers, because
we then have for example:

6 = 2 * 3 = -2 * -3

Not unique anymore.

Post by Archimedes Plutonium

Post by Don Redmond

Post by Archimedes Plutonium
And what I am going to argue, is that, such a lemma is not going to prove uniqueness. The use of such a lemma is just a distraction-walk about- discussion, not a proof that A*B is unique.
What proves A*B is unique, is that A+B is unique, thus, a transfer of uniqueness is conferred upon A*B.
So what I argue is that to prove uniqueness of multiplication, you need to insert addition. Vice versa, to prove uniqueness of addition, you must enter with the uniqueness of multiplication.
AP

You can prove uniqueness of multiplication by induction. See Landau, Foundations of Analysis.

Not really for the Lemma to prove FTA.
If two numbers by multiplying one another make some number, and any prime number measure the product, it will also measure one of the original numbers. Euclid Elements
So, what Euclid said was that if A*B = N and that prime P divides N, then P will divide either A or B or both.
Let us say we had the numbers 2*5 =10, and 10*10 = 100. Now a prime P=5 measures 10 and measures 5. And P =5 measures 100. Hence by that tool 2*5 = 2*5*2*5
Where is the uniqueness in that.
So, when we install the AXIOM of uniqueness into Mathematics itself, that 2*5 compliments 2+5 =7 whereas 2+5+2+5= 14 can we ever prove Uniqueness in mathematics

Post by Don Redmond
You can prove FT Arithmetic by induction (actually Well-Ordering, but the two are equivalent). See Hardy and Wright, Intro to Number Theory.
What could be more solid than an induction proof? Didn't claim this once a while back?
The main objection to your proof of Goldbach is that you give no procedure just lots of examples. We need a process that takes 2n and finds primes p and q such that p + q = 2n and a proof that that it always works. So if I take 2n = 10^1000, your general procedure should be guaranteed to work and not just because it worked on all the evens less than 10^100, say.
Don

No, you are wrong, the trouble with Old Math FTA, is the mathematician never realized that to prove Uniqueness in mathematics, requires a axiom that embodies uniqueness. If you have no axiom that expresses uniqueness, in mathematics, you cannot expect all the other axioms-- lacking in the concept of uniqueness, to, all of a sudden dredge up uniqueness with some silly lemma.
The only reason Arithmetic can count, is that it has a counting axiom-- successor function. Without that axiom of counting, you cannot expect the rest of math to be able to muster counting.
If Geometry had no axiom of what it means to be parallel, you cannot expect the rest of the axioms of geometry to conjure up the meaning of parallel.
This is why Goldbach is unprovable, because the Fundamental Theorem of Arithmetic, based upon a Lemma for uniqueness, is not only incapable of proving FTA, but is incapable of ever proving Goldbach, for the only proof of Goldbach rests on math having a concept of UNIQUENESS. When we reach the point in the proof of Goldbach, that the Bertrand primes line up. Old Math cannot make the next jump-- that p and q line up, because Old Math has no axiom that expresses uniqueness. The Bertrand primes line up because of an axiom-- if p*q exists, then p+q exists.
So Don, why did you not see, in all your years in math, did not see that your Fundamental Theorem of Arithmetic falls inadequately short of a proof of uniqueness. The lemma proves that 10 is the same as 100.
If you had a Axiom of Uniqueness, then FTA is simple to prove. Any number N = p*q*r*s*t is unique because, well the axiom has p+q+r+s+t is a unique add. Why did not you catch that Don, instead of continuing to preach a gossipal of fakery.
Then, in class Don, whenever students ask, why is Algebra always have operators add multiply, why not just add alone or multiply alone? What sort of fakery answer did you give them Don? Go see so and so, or read so and so.
The reason that Algebra theory, group, field, Galois etcetera, the reason that they always have add and multiply together, is because of the missing axiom that math has missed since Ancient Greek times. The missing axiom of uniqueness-- in math, the only means of measuring uniqueness-- is by the compliments of add with multiply. 2*5 is unique, because 2+5 forces 2*5 to be unique, and vice versa.
So, Don, are you going to continue to preach your shoddy math fakery, or, time to change for the better.
AP

[Don Redmond]

Post by Archimedes Plutonium

Post by Don Redmond

Post by Archimedes Plutonium
And what I am going to argue, is that, such a lemma is not going to prove uniqueness. The use of such a lemma is just a distraction-walk about- discussion, not a proof that A*B is unique.
What proves A*B is unique, is that A+B is unique, thus, a transfer of uniqueness is conferred upon A*B.
So what I argue is that to prove uniqueness of multiplication, you need to insert addition. Vice versa, to prove uniqueness of addition, you must enter with the uniqueness of multiplication.
AP

You can prove uniqueness of multiplication by induction. See Landau, Foundations of Analysis.

Not really for the Lemma to prove FTA.
If two numbers by multiplying one another make some number, and any prime number measure the product, it will also measure one of the original numbers. Euclid Elements
So, what Euclid said was that if A*B = N and that prime P divides N, then P will divide either A or B or both.
Let us say we had the numbers 2*5 =10, and 10*10 = 100. Now a prime P=5 measures 10 and measures 5. And P =5 measures 100. Hence by that tool 2*5 = 2*5*2*5
Where is the uniqueness in that.

I understand the problem here. When you said multiplication was unique I took you to mean that you you get a unique result. 2*3 = 6 and not 8 or 10 or something else.

If you mean unique factorization, then just jump to the proof in Hardy and Wright.

Post by Archimedes Plutonium
AP

Don

[Don Redmond]

Post by Archimedes Plutonium
The reason that Algebra theory, group, field, Galois etcetera, the reason that they always have add and multiply together, is because of the missing axiom that math has missed since Ancient Greek times. The missing axiom of uniqueness-- in math, the only means of measuring uniqueness-- is by the compliments of add with multiply. 2*5 is unique, because 2+5 forces 2*5 to be unique, and vice versa.
AP

Just so you know groups are only concerned with one operation, not two. Does this mean groups are fake mathematics too?

Don

[Vinicius Claudino Ferraz]

Don Dan, ding dong

it's high time for AP to ask me some basic program to write,
and to keep running all night long.

I HAVE ALL PRIME NUMBERS <= 5,118,742,463
MUAHAHAHAHA

Em quinta-feira, 10 de agosto de 2017 14:33:26 UTC-3, Don Redmond escreveu:

[Archimedes Plutonium]

Post by Don Redmond

Post by Archimedes Plutonium
The reason that Algebra theory, group, field, Galois etcetera, the reason that they always have add and multiply together, is because of the missing axiom that math has missed since Ancient Greek times. The missing axiom of uniqueness-- in math, the only means of measuring uniqueness-- is by the compliments of add with multiply. 2*5 is unique, because 2+5 forces 2*5 to be unique, and vice versa.
AP

Just so you know groups are only concerned with one operation, not two. Does this mean groups are fake mathematics too?
Don

When I say Group theory, I actually am sloppy, for I really mean Field theory of Algebra. Probably Ring theory is the same

Show me field theory that in the end is one operator

[Archimedes Plutonium]

Post by Archimedes Plutonium

Post by Don Redmond

Post by Archimedes Plutonium
The reason that Algebra theory, group, field, Galois etcetera, the reason that they always have add and multiply together, is because of the missing axiom that math has missed since Ancient Greek times. The missing axiom of uniqueness-- in math, the only means of measuring uniqueness-- is by the compliments of add with multiply. 2*5 is unique, because 2+5 forces 2*5 to be unique, and vice versa.
AP

Just so you know groups are only concerned with one operation, not two. Does this mean groups are fake mathematics too?
Don

When I say Group theory, I actually am sloppy, for I really mean Field theory of Algebra. Probably Ring theory is the same
Show me field theory that in the end is one operator

Quoting from Wikipedia on Ring;;

"In mathematics, a ring is one of the fundamental algebraic structures used in abstract algebra. It consists of a set equipped with two binary operations that generalize the arithmetic operations of addition and multiplication."

Now, what "abstract algebra" means is not that it is loose and airy or misty and unintelligible, or difficult and strange. What "abstract" means is that it is fundamental and what fundamental means is that it is nearby to the axioms of mathematics, close to home of what the axioms of that subject of math say and do.

So in the proof of Goldbach we come to a Matrix Column in specific of 8

0 8
1 7
2 6
3 5
4 4

and that column is all the possible ways of getting 8= a +b in Counting Numbers. All the possible ways.

So in a Goldbach proof, what happens is that the left column has to have a Bertrand Postulate prime, call it P, and the rightside column has to have another Bertrand prime call it Q, in our case it is 3 and 5. In our case, they are lined up. But, in general are the Bertrand primes lined up?

So we write the Column in general, or, as the author above would say "abstract".

0 N
1 a
2 b
3 c
4 d
.
.
.
.5N .5N

Now, our Bertrand prime on leftside is still P and on rightside is still Q

And we come to that juncture in the proof. Now we need to say that the Bertrand prime on left lines up with the one on right. Is there anything in mathematics that guarantees they line up?

Yes, there is the guarantee from Arithmetic, that if P lines up with Q as that of P+Q = N, then the number P*Q exists. But that is a axiom that is missing in Old Math. A missing axiom that says all numbers in the Matrix Columns are unique additions and unique multiplications, because addition is the compliment of multiplication. If A*B is unique, then so is A+B is unique.

Uniqueness of product or uniqueness of sum, does not occur by some silly Lemma, such as Euclid's lemma for Fundamental Theorem of Arithmetic. Uniqueness occurs because math has a missing Axiom that says sums and products are both unique, because of this axiom.

So, in the proof of Goldbach, really a simpleton proof, is that you form Bertrand primes, then you apply axiom of uniqueness which lines up the primes. For if 3+5 did not exist, then 3*5 is nonexistent.

So, what Old Math missed was not only this Uniqueness Axiom, but missed the fact that all of Algebra starts with the display of Matrix Columns.

To start Algebra, on day one, what we start with is a display of this::

0  1

0  2
1  1


0  3
1  2

0  4
1  3
2  2


0  5
1  4
2  3


0  6
1  5
2  4
3  3

0  7
1  6
2  5
3  4


0  8
1  7
2  6
3  5
4  4

0  9
1  8
2  7
3  6
4  5


0  10
1   9
2   8
3   7
4   6
5   5


0   11
1   10
2    9
3    8
4    7
5    6


0    12
1    11
2    10
3      9
4      8
5      7
6      6

Now, those above Matrix Columns are Algebra and they show uniqueness of every addition and every multiplication, so that if a Bertrand primes P, Q exist in a Column, then P*Q exists and P+Q exists in that column, hence Goldbach proven.

So, we can say Axiom of Uniqueness, or we can say -true because of Algebra Matrix Columns.

In my proofs of Goldbach in early 1990s, I just kept saying if 3+5 does not exist, then 3*5 does not exist, violation of arithmetic, hence Goldbach.

So, what axiom was missing?

Was it just Uniqueness or was it Complimentarity, or Symmetry. Did we miss one axiom, or three?

AP

[q***@gmail.com]

Post by Archimedes Plutonium

Post by Archimedes Plutonium

Post by Don Redmond

Post by Archimedes Plutonium
The reason that Algebra theory, group, field, Galois etcetera, the reason that they always have add and multiply together, is because of the missing axiom that math has missed since Ancient Greek times. The missing axiom of uniqueness-- in math, the only means of measuring uniqueness-- is by the compliments of add with multiply. 2*5 is unique, because 2+5 forces 2*5 to be unique, and vice versa.
AP

Just so you know groups are only concerned with one operation, not two. Does this mean groups are fake mathematics too?
Don

When I say Group theory, I actually am sloppy, for I really mean Field theory of Algebra. Probably Ring theory is the same
Show me field theory that in the end is one operator

Quoting from Wikipedia on Ring;;
"In mathematics, a ring is one of the fundamental algebraic structures used in abstract algebra. It consists of a set equipped with two binary operations that generalize the arithmetic operations of addition and multiplication."
Now, what "abstract algebra" means is not that it is loose and airy or misty and unintelligible, or difficult and strange. What "abstract" means is that it is fundamental and what fundamental means is that it is nearby to the axioms of mathematics, close to home of what the axioms of that subject of math say and do.
So in the proof of Goldbach we come to a Matrix Column in specific of 8
0 8
1 7
2 6
3 5
4 4
and that column is all the possible ways of getting 8= a +b in Counting Numbers. All the possible ways.
So in a Goldbach proof, what happens is that the left column has to have a Bertrand Postulate prime, call it P, and the rightside column has to have another Bertrand prime call it Q, in our case it is 3 and 5. In our case, they are lined up. But, in general are the Bertrand primes lined up?
So we write the Column in general, or, as the author above would say "abstract".
0 N
1 a
2 b
3 c
4 d
.
.
.
.5N .5N
Now, our Bertrand prime on leftside is still P and on rightside is still Q
And we come to that juncture in the proof. Now we need to say that the Bertrand prime on left lines up with the one on right. Is there anything in mathematics that guarantees they line up?
Yes, there is the guarantee from Arithmetic, that if P lines up with Q as that of P+Q = N, then the number P*Q exists. But that is a axiom that is missing in Old Math. A missing axiom that says all numbers in the Matrix Columns are unique additions and unique multiplications, because addition is the compliment of multiplication. If A*B is unique, then so is A+B is unique.

There is no "missing axiom."
Peano axioms are the standard for addition and multiplication.

https://en.wikipedia.org/wiki/Peano_axioms#Arithmetic

All your sums "exist", whether Goldbach is true or false.
But your sums are not unique, all your sums are a+(N-a)=N for 0<=a<=N/2.
This is trivial.

All your products "exist", whether Goldbach is true or false.
All your products are unique, each is a*(N-a) for 0<=a<=N/2.
This is trivial.

What is not trivial and what you have never been able to prove is
whether there must exist for every even N out to infinity and beyond
at least one 'a' and 'N-a' such that both 'a' and 'N-a' are prime
(or that they "line up", to use your term for this).
(but declaring "it is like a*(N-a) doesn't exist" is completely meaningless.

Post by Archimedes Plutonium
Uniqueness of product or uniqueness of sum, does not occur by some silly Lemma, such as Euclid's lemma for Fundamental Theorem of Arithmetic. Uniqueness occurs because math has a missing Axiom that says sums and products are both unique, because of this axiom.
So, in the proof of Goldbach, really a simpleton proof, is that you form Bertrand primes, then you apply axiom of uniqueness which lines up the primes. For if 3+5 did not exist, then 3*5 is nonexistent.

a+(N-a) exists and a*(N-a) exist whether or not 'a' and 'N-a'
are prime or not, whether Goldbach is true or it is not.

If you are fumbling around trying to find words to somehow claim some
contradiction then you are going to have to do a LOT better than this.

People might try to help you write the needed sentence, but like you have
said to several others recently, if you are incapable of doing the simplest
things correctly, why should they spend time trying to do your work for you
when we all know this is a completely hopeless crusade.

Post by Archimedes Plutonium
So, what Old Math missed was not only this Uniqueness Axiom, but missed the fact that all of Algebra starts with the display of Matrix Columns.

Your "columns" or "arrays" or "matricies", or whatever other made up
name you want to create in your honor, are trivial. You have demonstrated
nothing (yet) that this idea does anything to demonstrate a proof
or even to advance the field slightly in the direction towards a proof.

Definition: Proof, an argument which convinces others.

https://en.wikipedia.org/wiki/Proof_(truth)

To prove Goldbach you must construct some argument that holds for
every even N out to infinity and beyond and that is correct and
clear and simple and convincing to others. You don't count for many reasons.

Looking at three examples, thinking you see a pattern and screeching
PROOF!!!! will never convince anyone else in the world except you.
That is your classic "AP Induction" nonsense that you use again and again.

Getting someone else to calculate 20 examples or 20 million examples
will never convince anyone else in the world except you.

What will convince others, at least if you don't make any mistakes
and you produce a clear simple detailed correct proof is:

Goldbach is true for n=4. (the easy part)

then IF Goldbach is true for even N then it is true for N+2. (the hard part)

That will do it. And you don't have a chance in the universe of proving that.

But your "monotone" ramblings lead nowhere and convince no one.
This is for two reasons.

The first and more trivial is that if you find some number of
prime pairs for some particular N then it is usually easy to
find some slightly larger N that has FEWER prime pairs.
That violates monotone increasing by DECREASING and if it does
that even once then your monotone increasing argument is demolished.

The second and far more important is that thus far you have not
been able to make a single convincing argument about the number
of such prime pairs for every even N out to infinity and beyond.
This has nothing to do with 20 or 20 million examples, this has
to do with every even N out to infinity and beyond.

Never ever bring a crank to a brain fight

Post by Archimedes Plutonium
0  1
0  2
1  1
0  3
1  2
0  4
1  3
2  2
0  5
1  4
2  3
0  6
1  5
2  4
3  3
0  7
1  6
2  5
3  4
0  8
1  7
2  6
3  5
4  4
0  9
1  8
2  7
3  6
4  5
0  10
1   9
2   8
3   7
4   6
5   5
0   11
1   10
2    9
3    8
4    7
5    6
0    12
1    11
2    10
3      9
4      8
5      7
6      6
Now, those above Matrix Columns are Algebra and they show uniqueness of every addition and every multiplication, so that if a Bertrand primes P, Q exist in a Column, then P*Q exists and P+Q exists in that column, hence Goldbach proven.
So, we can say Axiom of Uniqueness, or we can say -true because of Algebra Matrix Columns.
In my proofs of Goldbach in early 1990s, I just kept saying if 3+5 does not exist, then 3*5 does not exist, violation of arithmetic, hence Goldbach.
So, what axiom was missing?
Was it just Uniqueness or was it Complimentarity, or Symmetry. Did we miss one axiom, or three?
AP

[Vinicius Claudino Ferraz]

A number N = 2k exists.
A prime P < N exists.
What does guarantee N - P = Q is prime?

Is there any algorithm like Fund Theo of Arithmetics?
Divide k by 2, 3, 5, ... until >= square root of k.
Finally divide by itself, because k may be prime.

I know.
Verify if Q = N - 3 is prime.
Verify if Q = N - 5 is prime.
Verify if Q = N - 7 is prime.
Until Q >= N/2

What does guarantee that "no Q's are prime"?

You claim that forall n >= 6, there is a prime P <= N/2
and there is a prime Q >= N/2
such that P + Q = N

I want a construction. :-)

Try for N = 1,000,000 . Who are (P, Q)? Formula?

Post by Archimedes Plutonium
Preliminary Theorem Statement:: given any even number from 6 onwards, it has to have at least two primes that sum to that even number
Preliminary Proof Statement:: Every even number is dissolvable into unique prime factors. Every two odd numbers when added produce an even number. Every even number N, starting with 6, has a prime number P smaller than itself, thus, we find a second prime number Q to add on, giving P+ Q = N, and if not, then the number P*Q is also, nonexistent, violating the Fundamental Theorem of Arithmetic.

[b***@gmail.com]

999983 + 17 = 1000000

Post by Vinicius Claudino Ferraz
A number N = 2k exists.
A prime P < N exists.
What does guarantee N - P = Q is prime?
Is there any algorithm like Fund Theo of Arithmetics?
Divide k by 2, 3, 5, ... until >= square root of k.
Finally divide by itself, because k may be prime.
I know.
Verify if Q = N - 3 is prime.
Verify if Q = N - 5 is prime.
Verify if Q = N - 7 is prime.
Until Q >= N/2
What does guarantee that "no Q's are prime"?
You claim that forall n >= 6, there is a prime P <= N/2
and there is a prime Q >= N/2
such that P + Q = N
I want a construction. :-)
Try for N = 1,000,000 . Who are (P, Q)? Formula?

Post by Archimedes Plutonium
Preliminary Theorem Statement:: given any even number from 6 onwards, it has to have at least two primes that sum to that even number
Preliminary Proof Statement:: Every even number is dissolvable into unique prime factors. Every two odd numbers when added produce an even number. Every even number N, starting with 6, has a prime number P smaller than itself, thus, we find a second prime number Q to add on, giving P+ Q = N, and if not, then the number P*Q is also, nonexistent, violating the Fundamental Theorem of Arithmetic.

[Vinicius Claudino Ferraz]

OK. A bird in hands is better than 5,402 = f(1,000,000) flying.

1000000 = 100019 + 899981
1000000 = 200009 + 799991
1000000 = 300137 + 699863
1000000 = 400067 + 599933
1000000 = 499943 + 500057

So are there

Post by b***@gmail.com
999983 + 17 = 1000000

Post by Vinicius Claudino Ferraz
A number N = 2k exists.
A prime P < N exists.
What does guarantee N - P = Q is prime?
Is there any algorithm like Fund Theo of Arithmetics?
Divide k by 2, 3, 5, ... until >= square root of k.
Finally divide by itself, because k may be prime.
I know.
Verify if Q = N - 3 is prime.
Verify if Q = N - 5 is prime.
Verify if Q = N - 7 is prime.
Until Q >= N/2
What does guarantee that "no Q's are prime"?
You claim that forall n >= 6, there is a prime P <= N/2
and there is a prime Q >= N/2
such that P + Q = N
I want a construction. :-)
Try for N = 1,000,000 . Who are (P, Q)? Formula?

Post by Archimedes Plutonium
Preliminary Theorem Statement:: given any even number from 6 onwards, it has to have at least two primes that sum to that even number
Preliminary Proof Statement:: Every even number is dissolvable into unique prime factors. Every two odd numbers when added produce an even number. Every even number N, starting with 6, has a prime number P smaller than itself, thus, we find a second prime number Q to add on, giving P+ Q = N, and if not, then the number P*Q is also, nonexistent, violating the Fundamental Theorem of Arithmetic.

[Vinicius Claudino Ferraz]

Sorry: "mouse keys".

How many odd primes did we use? 5402 + 5402 = 10804

How many odd primes are there < 1000000? 78497

How many odd primes did we not use? 78497 - 10804 = 67693

Most primes do not work.

Post by Vinicius Claudino Ferraz
OK. A bird in hands is better than 5,402 = f(1,000,000) flying.
1000000 = 100019 + 899981
1000000 = 200009 + 799991
1000000 = 300137 + 699863
1000000 = 400067 + 599933
1000000 = 499943 + 500057

Post by b***@gmail.com
999983 + 17 = 1000000

[b***@gmail.com]

Is your f(n) the same as this R(n) or r(n) here?

Goldbach conjecture verification - Tomás Oliveira e Silva
http://sweet.ua.pt/tos/goldbach.html

Post by Vinicius Claudino Ferraz
Sorry: "mouse keys".
How many odd primes did we use? 5402 + 5402 = 10804
How many odd primes are there < 1000000? 78497
How many odd primes did we not use? 78497 - 10804 = 67693
Most primes do not work.

Post by Vinicius Claudino Ferraz
OK. A bird in hands is better than 5,402 = f(1,000,000) flying.
1000000 = 100019 + 899981
1000000 = 200009 + 799991
1000000 = 300137 + 699863
1000000 = 400067 + 599933
1000000 = 499943 + 500057

Post by b***@gmail.com
999983 + 17 = 1000000

[b***@gmail.com]

TOS used peta flops. What is next?

A Path to Capable Exascale Computing
Paul Messina, Argonne National Laboratory

Post by b***@gmail.com
Is your f(n) the same as this R(n) or r(n) here?
Goldbach conjecture verification - Tomás Oliveira e Silva
http://sweet.ua.pt/tos/goldbach.html

Post by Vinicius Claudino Ferraz
Sorry: "mouse keys".
How many odd primes did we use? 5402 + 5402 = 10804
How many odd primes are there < 1000000? 78497
How many odd primes did we not use? 78497 - 10804 = 67693
Most primes do not work.

Post by Vinicius Claudino Ferraz
OK. A bird in hands is better than 5,402 = f(1,000,000) flying.
1000000 = 100019 + 899981
1000000 = 200009 + 799991
1000000 = 300137 + 699863
1000000 = 400067 + 599933
1000000 = 499943 + 500057

Post by b***@gmail.com
999983 + 17 = 1000000

[Vinicius Claudino Ferraz]

I'm afraid of that study.

f(n) = r(n) = R(n) /2 xor [R(n) + 1] /2

Is this link workin? https://drive.google.com/open?id=0B2oJGg3QLrWDNE9tOUNCcVdwZjA

Post by b***@gmail.com
Is your f(n) the same as this R(n) or r(n) here?
Goldbach conjecture verification - Tomás Oliveira e Silva
http://sweet.ua.pt/tos/goldbach.html

Post by Vinicius Claudino Ferraz
Sorry: "mouse keys".
How many odd primes did we use? 5402 + 5402 = 10804
How many odd primes are there < 1000000? 78497
How many odd primes did we not use? 78497 - 10804 = 67693
Most primes do not work.

Post by Vinicius Claudino Ferraz
OK. A bird in hands is better than 5,402 = f(1,000,000) flying.
1000000 = 100019 + 899981
1000000 = 200009 + 799991
1000000 = 300137 + 699863
1000000 = 400067 + 599933
1000000 = 499943 + 500057

Post by b***@gmail.com
999983 + 17 = 1000000

[Vinicius Claudino Ferraz]

That sounds funny.
1) fundamental theorem of sum.
2) fundamental theorem of product. OK
3) fundamental theorem of exponentiation.
4) fundamental theorem of tetration.
...
n) fundamental theorem of n-ation.

[Peter Percival]

Post by Vinicius Claudino Ferraz
That sounds funny.
1) fundamental theorem of sum.
2) fundamental theorem of product. OK
3) fundamental theorem of exponentiation.
4) fundamental theorem of tetration.
...
n) fundamental theorem of n-ation.

The fundamental theorem of a nation ought to be its constitution.
Didn't Gödel find a logical flaw in the US constitution?

[Archimedes Plutonium]

Post by Vinicius Claudino Ferraz
That sounds funny.
1) fundamental theorem of sum.
2) fundamental theorem of product. OK
3) fundamental theorem of exponentiation.
4) fundamental theorem of tetration.
...
n) fundamental theorem of n-ation.

If you think about it, the Fund. Theorem of Calculus is the complimentarity of derivative with integral

So the Fundamental Theorem of Arithmetic should be complimentarity of addition versus multiplication.

Now, the only place in mathematics where we need 1 to be not prime is the Old Math's Unique Prime Factorization.

But, what do we say when Old Math has a screwed up proof of Unique Prime Factorization, in that Euclid's Lemma if P divides N, then P divides one of its factors. So, that Old Math screwed up on FTA proof, for that Lemma does not, and can not prove uniqueness. What proves uniqueness in Arithmetic facctorization is addition, its compliment.

So, the number 12 in Old Math is 2*2*3 and it is unique to 12, not because of the silly Euclid Lemma (remember, in Old Math, those cads, could not even see that the conic section is a oval, not an ellipse, so with cads like that on Fundamental Theorem of Arithmetic, it is no wonder, the cads thought Euclid Lemma proves uniqueness). So, in New Math the Unique Prime Factors of 12 are really this 1*2*2*3 = 12 and unique because 1+2+2+3 = 9. If you throw another 1 in there it is 10, so you can throw only one 1 in there.

Now, does that improve Mathematics any if we allow for 1 to be Unit Prime.

Well it improves math in that Goldbach now becomes true for all Even Numbers and no longer have to say starting with 4. Math is always bad, if there are exceptions. Math is best when it is "for all". Just as physics is not a law, if it has exceptions.

So, 2 is the sum of two primes of 1+1.

Now what is the unique prime factorization of 1? It is just 1. In Old Math, you ask them what is the prime factorization of 2, their smallest prime and they would say 2. So if it is okay for 2, then it is okay for 1. In New Math, the unique prime factorization of 2 is 1*2. It is unique because 1+2 is unique.

So now, in New Math, Goldbach primes for 8 are usually 3+5, but we have also 1+7. The Goldbach primes for 2 are 1+1, for 4 are 2+2 and then 3+1 for 6 are 3+3 and also 5+1.

Now, does the fact that 1 is Unit Prime, help the Prime Counting Function x/Ln(x)? Not sure if it does.

But the main point I need to drive home. That Algebra, its Group theory, that Galois started in 1800s, really began in Ancient Greek times with Arithmetic, when the Fundamental Theorem of Arithmetic started as Unique Prime Factorization. Algebra Matrix theory, group theory started then, for the true valid proof of Fundamental Theorem requires addition be the Uniqueness proof.

0 1

0  2
1  1


0  3
1  2

0  4
1  3
2  2


0  5
1  4
2  3


0  6
1  5
2  4
3  3

0  7
1  6
2  5
3  4


0  8
1  7
2  6
3  5
4  4

0  9
1  8
2  7
3  6
4  5


0  10
1   9
2   8
3   7
4   6
5   5


0 11
1 10
2 9
3 8
4 7
5 6


0 12
1 11
2 10
3 9
4 8
5 7
6 6

None of us can expect the Ancient Greeks to have started Arithmetic with the above Algebra Matrix Columns (I used to call them Arrays, but I need the term array for another purpose of math history, so call the above Matrix Columns)

Arithmetic in mathematics should start with Matrix Column, for it drives home the fact that Addition is Complimentary to Multiplication, and uniqueness can only be proven by switching from addition to multiplication and back and forth.

AP

[Vinicius Claudino Ferraz]

Exercise: 1 + 2 + 2 + 3 = ?

Em quarta-feira, 9 de agosto de 2017 02:10:27 UTC-3, Archimedes Plutonium escreveu:
Lemma proves uniqueness). So, in New Math the Unique Prime Factors of 12 are really this 1*2*2*3 = 12 and unique because 1+2+2+3 = 9. If you throw another 1 in there it is 10, so you can throw only one 1 in there.

Post by Archimedes Plutonium
Now, does that improve Mathematics any if we allow for 1 to be Unit Prime.
Well it improves math in that Goldbach now becomes true for all Even Numbers and no longer have to say starting with 4. Math is always bad, if there are exceptions. Math is best when it is "for all". Just as physics is not a law, if it has exceptions.
So, 2 is the sum of two primes of 1+1.
Now what is the unique prime factorization of 1? It is just 1. In Old Math, you ask them what is the prime factorization of 2, their smallest prime and they would say 2. So if it is okay for 2, then it is okay for 1. In New Math, the unique prime factorization of 2 is 1*2. It is unique because 1+2 is unique.
So now, in New Math, Goldbach primes for 8 are usually 3+5, but we have also 1+7. The Goldbach primes for 2 are 1+1, for 4 are 2+2 and then 3+1 for 6 are 3+3 and also 5+1.
Now, does the fact that 1 is Unit Prime, help the Prime Counting Function x/Ln(x)? Not sure if it does.
But the main point I need to drive home. That Algebra, its Group theory, that Galois started in 1800s, really began in Ancient Greek times with Arithmetic, when the Fundamental Theorem of Arithmetic started as Unique Prime Factorization. Algebra Matrix theory, group theory started then, for the true valid proof of Fundamental Theorem requires addition be the Uniqueness proof.
0 1
0  2
1  1
0  3
1  2
0  4
1  3
2  2
0  5
1  4
2  3
0  6
1  5
2  4
3  3
0  7
1  6
2  5
3  4
0  8
1  7
2  6
3  5
4  4
0  9
1  8
2  7
3  6
4  5
0  10
1   9
2   8
3   7
4   6
5   5
0 11
1 10
2 9
3 8
4 7
5 6
0 12
1 11
2 10
3 9
4 8
5 7
6 6
None of us can expect the Ancient Greeks to have started Arithmetic with the above Algebra Matrix Columns (I used to call them Arrays, but I need the term array for another purpose of math history, so call the above Matrix Columns)
Arithmetic in mathematics should start with Matrix Column, for it drives home the fact that Addition is Complimentary to Multiplication, and uniqueness can only be proven by switching from addition to multiplication and back and forth.
AP

[Vinicius Claudino Ferraz]

[x | p => x \in {1, p}] <=> p is prime
[x | q => x \in {1, q}] <=> q is prime

p <> 2, q <> 2
p + q = 2n "covers" {6, 8, 10, 12...}

What have "ax = p, bx = q" to do with sum?

This is as difficult as to factorize n ± 1, n ± 2, ..., n ± j
You disturb n, the prime factors are randomized.

These things well exposed is a math daydream.
Which dog will gnaw that bone?

[Archimedes Plutonium]

Post by Vinicius Claudino Ferraz
Exercise: 1 + 2 + 2 + 3 = ?

Thanks for pointing out that typo error.

I firmly believe that math should count 1 as UNIT PRIME, however, I do not want math to use 1 in unique prime factorization, for it just is too cumbersome. Being unit prime, just use that fact on occasion, such as Goldbach is true for all even numbers since 2 = 1+1.

So math starts with 1, not 0, and the Peano axioms start with 1, not 0, and so the Matrix Columns that starts Algebra and Arithmetic is

0  1

0  2
1  1


0  3
1  2

0  4
1  3
2  2


0  5
1  4
2  3


0  6
1  5
2  4
3  3

0  7
1  6
2  5
3  4


0  8
1  7
2  6
3  5
4  4

0  9
1  8
2  7
3  6
4  5


0  10
1   9
2   8
3   7
4   6
5   5


0   11
1   10
2    9
3    8
4    7
5    6


0    12
1    11
2    10
3      9
4      8
5      7
6      6

AP

[b***@gmail.com]

THAT TYPO. THAT TYPO. THAT TYPO.

30 years not a single line of math. Thats not
a typo. Thats the brain fartism, of AP.

Post by Archimedes Plutonium

Post by Vinicius Claudino Ferraz
Exercise: 1 + 2 + 2 + 3 = ?

Thanks for pointing out that typo error.
I firmly believe that math should count 1 as UNIT PRIME, however, I do not want math to use 1 in unique prime factorization, for it just is too cumbersome. Being unit prime, just use that fact on occasion, such as Goldbach is true for all even numbers since 2 = 1+1.
So math starts with 1, not 0, and the Peano axioms start with 1, not 0, and so the Matrix Columns that starts Algebra and Arithmetic is
0  1
0  2
1  1
0  3
1  2
0  4
1  3
2  2
0  5
1  4
2  3
0  6
1  5
2  4
3  3
0  7
1  6
2  5
3  4
0  8
1  7
2  6
3  5
4  4
0  9
1  8
2  7
3  6
4  5
0  10
1   9
2   8
3   7
4   6
5   5
0   11
1   10
2    9
3    8
4    7
5    6
0    12
1    11
2    10
3      9
4      8
5      7
6      6
AP

[Vinicius Claudino Ferraz]

For n = 3, GoldBach is true
For n = 4, GoldBach is true
...
For n = 1,000,000, GoldBach is true

All we need is an induction
H(n) ==> H(n + 1)
2n = p_0 + q_0 ==> 2n + 2 = p_1(p_0, q_0, n) + q_1(p_0, q_0, n)

_____________________________

Suppose by contradiction that
∃ n > 500,000 such that ∀ prime p <= n, 2n - p is not prime.

That's not possible, by the quantity of primes from 3 to 500,000.
This may be proved by probability.

The probability of non existance 2n = p + q is zero. Why?

[Vinicius Claudino Ferraz]

Every prime p can be represented by an even number 2n minus another prime q.

\forall prime p, \exists n, \exists prime q such that p = 2n - q

no no no this is obvious. let q = p. p = 2n - p => 2p = 2n

ah!

[Archimedes Plutonium]

On Wednesday, August 9, 2017 at 1:36:52 PM UTC-5, Vinicius Claudino Ferraz wrote:
(snip)

Vinicius, you wrote several days ago these Goldbach extensions::

So, several days ago
Vinicius Claudino Ferraz        
wrote:

Extended Gold_Bach

Conjectures:

  4) Every even number n >= 4 is a sum of two primes.
  6) Every even number n >= 6 is a sum of two odd primes.
 14) Every even number n >= 14 is a sum of two odd primes twice.
     12 = 5 + 7
 70) Every even number n >= 70 is a sum of two odd primes three times.
     68 = 7 + 61 = 31 + 37
130) Every even number n >= 130 is a sum of two odd primes four times.
     128 = 19 + 109
     128 = 31 + 97
     128 = 61 + 67
154) Every even number n >= 154 is a sum of two odd primes five times.
     152 = 3 + 149
     152 = 13 + 139
     152 = 43 + 109
     152 = 73 + 79
190) Every even number n >= 190 is a sum of two odd primes six times.
     188 = 7 + 181
     188 = 31 + 157
     188 = 37 + 151
     188 = 61 + 127
     188 = 79 + 109
334) Every even number n >= 334 is a sum of two odd primes seven times.
     332 = 19 + 313
     332 = 61 + 271
     332 = 103 + 229
     332 = 109 + 223
     332 = 139 + 193
     332 = 151 + 181
400) Every even number n >= 400 is a sum of two odd primes nine times.
     398 = 19 + 379
     398 = 31 + 367
     398 = 61 + 337
     398 = 67 + 331
     398 = 127 + 271
     398 = 157 + 241
     398 = 199 + 199
     398 is the last time for seven times.
   > 368 is the last time for eight times. this is weird.
     488 is the last time for nine times.
     632 is the last time for ten times.

490) Every even number n >= 490 is a sum of two odd primes ten times.
634) Every even number n >= 634 is a sum of two odd primes eleven times.

--- end of Vinicius Claudino Ferraz wrote ---

Can your computer go out to N=1000 for this table, to continue this table--


TABLE construction of all the Two Prime Composites from 9 to 1000 (no 2 as prime)

3*3 = 9                 3+3= 6
3 * 5  = 15           3+5 = 8
3 * 7  = 21           3+7 = 10
5 * 5  = 25           5+5 =10
3 * 11 = 33          3+11 = 14
5 * 7  = 35           5+ 7 = 12
3 * 13 = 39          3+13 = 16
7 *  7  = 49          7+7 = 14
3 * 17 = 51          3+17 = 20
5 * 11 = 55          5+11 = 16
3 * 19 = 57          3+19 = 22
5 * 13  = 65         5+13 = 18
3 * 23  = 69         3+23 = 26
7 * 11  = 77         7+11 = 18
5 *17  = 85          5+17 = 22
3 * 29  = 87        3+29 = 32
7 * 13  =  91       7+13 = 20
3 * 31  =  93        3+ 31 = 34
5 * 19  =  95        5+ 19 = 24

[b***@gmail.com]

Obviously AP even doesnt understand the table of
Vinicius Claudino Ferraz. Its just r(n), the number
of Goldbach partitions.

For godssake install Python, and run this code, and you
will get your stupid useless table. Or buy some millimeter
grid paper, and draw an oval conic section.

from sympy import isprime

def A045917(n):
x = 0
for i in range(2, n+1):
if isprime(i) and isprime(2*n-i):
x += 1
return x

http://oeis.org/A045917

Post by Archimedes Plutonium
(snip)
So, several days ago
Vinicius Claudino Ferraz        
Extended Gold_Bach
  4) Every even number n >= 4 is a sum of two primes.
  6) Every even number n >= 6 is a sum of two odd primes.
 14) Every even number n >= 14 is a sum of two odd primes twice.
     12 = 5 + 7
 70) Every even number n >= 70 is a sum of two odd primes three times.
     68 = 7 + 61 = 31 + 37
130) Every even number n >= 130 is a sum of two odd primes four times.
     128 = 19 + 109
     128 = 31 + 97
     128 = 61 + 67
154) Every even number n >= 154 is a sum of two odd primes five times.
     152 = 3 + 149
     152 = 13 + 139
     152 = 43 + 109
     152 = 73 + 79
190) Every even number n >= 190 is a sum of two odd primes six times.
     188 = 7 + 181
     188 = 31 + 157
     188 = 37 + 151
     188 = 61 + 127
     188 = 79 + 109
334) Every even number n >= 334 is a sum of two odd primes seven times.
     332 = 19 + 313
     332 = 61 + 271
     332 = 103 + 229
     332 = 109 + 223
     332 = 139 + 193
     332 = 151 + 181
400) Every even number n >= 400 is a sum of two odd primes nine times.
     398 = 19 + 379
     398 = 31 + 367
     398 = 61 + 337
     398 = 67 + 331
     398 = 127 + 271
     398 = 157 + 241
     398 = 199 + 199
     398 is the last time for seven times.
   > 368 is the last time for eight times. this is weird.
     488 is the last time for nine times.
     632 is the last time for ten times.
490) Every even number n >= 490 is a sum of two odd primes ten times.
634) Every even number n >= 634 is a sum of two odd primes eleven times.
--- end of Vinicius Claudino Ferraz wrote ---
Can your computer go out to N=1000 for this table, to continue this table--
TABLE construction of all the Two Prime Composites from 9 to 1000 (no 2 as prime)
3*3 = 9                 3+3= 6
3 * 5  = 15           3+5 = 8
3 * 7  = 21           3+7 = 10
5 * 5  = 25           5+5 =10
3 * 11 = 33          3+11 = 14
5 * 7  = 35           5+ 7 = 12
3 * 13 = 39          3+13 = 16
7 *  7  = 49          7+7 = 14
3 * 17 = 51          3+17 = 20
5 * 11 = 55          5+11 = 16
3 * 19 = 57          3+19 = 22
5 * 13  = 65         5+13 = 18
3 * 23  = 69         3+23 = 26
7 * 11  = 77         7+11 = 18
5 *17  = 85          5+17 = 22
3 * 29  = 87        3+29 = 32
7 * 13  =  91       7+13 = 20
3 * 31  =  93        3+ 31 = 34
5 * 19  =  95        5+ 19 = 24

[b***@gmail.com]

Vinicius Claudino Ferraz communicated was probably:

Smallest positive even integer that is an
unordered sum of two primes in exactly n ways.
http://oeis.org/A023036

Denoted by a(n).

Post by b***@gmail.com
Obviously AP even doesnt understand the table of
Vinicius Claudino Ferraz. Its just r(n), the number
of Goldbach partitions.
For godssake install Python, and run this code, and you
will get your stupid useless table. Or buy some millimeter
grid paper, and draw an oval conic section.
from sympy import isprime
x = 0
x += 1
return x
http://oeis.org/A045917

Post by Archimedes Plutonium
(snip)
So, several days ago
Vinicius Claudino Ferraz        
Extended Gold_Bach
  4) Every even number n >= 4 is a sum of two primes.
  6) Every even number n >= 6 is a sum of two odd primes.
 14) Every even number n >= 14 is a sum of two odd primes twice.
     12 = 5 + 7
 70) Every even number n >= 70 is a sum of two odd primes three times.
     68 = 7 + 61 = 31 + 37
130) Every even number n >= 130 is a sum of two odd primes four times.
     128 = 19 + 109
     128 = 31 + 97
     128 = 61 + 67
154) Every even number n >= 154 is a sum of two odd primes five times.
     152 = 3 + 149
     152 = 13 + 139
     152 = 43 + 109
     152 = 73 + 79
190) Every even number n >= 190 is a sum of two odd primes six times.
     188 = 7 + 181
     188 = 31 + 157
     188 = 37 + 151
     188 = 61 + 127
     188 = 79 + 109
334) Every even number n >= 334 is a sum of two odd primes seven times.
     332 = 19 + 313
     332 = 61 + 271
     332 = 103 + 229
     332 = 109 + 223
     332 = 139 + 193
     332 = 151 + 181
400) Every even number n >= 400 is a sum of two odd primes nine times.
     398 = 19 + 379
     398 = 31 + 367
     398 = 61 + 337
     398 = 67 + 331
     398 = 127 + 271
     398 = 157 + 241
     398 = 199 + 199
     398 is the last time for seven times.
   > 368 is the last time for eight times. this is weird.
     488 is the last time for nine times.
     632 is the last time for ten times.
490) Every even number n >= 490 is a sum of two odd primes ten times.
634) Every even number n >= 634 is a sum of two odd primes eleven times.
--- end of Vinicius Claudino Ferraz wrote ---
Can your computer go out to N=1000 for this table, to continue this table--
TABLE construction of all the Two Prime Composites from 9 to 1000 (no 2 as prime)
3*3 = 9                 3+3= 6
3 * 5  = 15           3+5 = 8
3 * 7  = 21           3+7 = 10
5 * 5  = 25           5+5 =10
3 * 11 = 33          3+11 = 14
5 * 7  = 35           5+ 7 = 12
3 * 13 = 39          3+13 = 16
7 *  7  = 49          7+7 = 14
3 * 17 = 51          3+17 = 20
5 * 11 = 55          5+11 = 16
3 * 19 = 57          3+19 = 22
5 * 13  = 65         5+13 = 18
3 * 23  = 69         3+23 = 26
7 * 11  = 77         7+11 = 18
5 *17  = 85          5+17 = 22
3 * 29  = 87        3+29 = 32
7 * 13  =  91       7+13 = 20
3 * 31  =  93        3+ 31 = 34
5 * 19  =  95        5+ 19 = 24

[b***@gmail.com]

49500 is the smallest with 1000 Goldbach partitions.

Post by b***@gmail.com
Smallest positive even integer that is an
unordered sum of two primes in exactly n ways.
http://oeis.org/A023036
Denoted by a(n).

Post by b***@gmail.com
Obviously AP even doesnt understand the table of
Vinicius Claudino Ferraz. Its just r(n), the number
of Goldbach partitions.
For godssake install Python, and run this code, and you
will get your stupid useless table. Or buy some millimeter
grid paper, and draw an oval conic section.
from sympy import isprime
x = 0
x += 1
return x
http://oeis.org/A045917

Post by Archimedes Plutonium
(snip)
So, several days ago
Vinicius Claudino Ferraz        
Extended Gold_Bach
  4) Every even number n >= 4 is a sum of two primes.
  6) Every even number n >= 6 is a sum of two odd primes.
 14) Every even number n >= 14 is a sum of two odd primes twice.
     12 = 5 + 7
 70) Every even number n >= 70 is a sum of two odd primes three times.
     68 = 7 + 61 = 31 + 37
130) Every even number n >= 130 is a sum of two odd primes four times.
     128 = 19 + 109
     128 = 31 + 97
     128 = 61 + 67
154) Every even number n >= 154 is a sum of two odd primes five times.
     152 = 3 + 149
     152 = 13 + 139
     152 = 43 + 109
     152 = 73 + 79
190) Every even number n >= 190 is a sum of two odd primes six times.
     188 = 7 + 181
     188 = 31 + 157
     188 = 37 + 151
     188 = 61 + 127
     188 = 79 + 109
334) Every even number n >= 334 is a sum of two odd primes seven times.
     332 = 19 + 313
     332 = 61 + 271
     332 = 103 + 229
     332 = 109 + 223
     332 = 139 + 193
     332 = 151 + 181
400) Every even number n >= 400 is a sum of two odd primes nine times.
     398 = 19 + 379
     398 = 31 + 367
     398 = 61 + 337
     398 = 67 + 331
     398 = 127 + 271
     398 = 157 + 241
     398 = 199 + 199
     398 is the last time for seven times.
   > 368 is the last time for eight times. this is weird.
     488 is the last time for nine times.
     632 is the last time for ten times.
490) Every even number n >= 490 is a sum of two odd primes ten times.
634) Every even number n >= 634 is a sum of two odd primes eleven times.
--- end of Vinicius Claudino Ferraz wrote ---
Can your computer go out to N=1000 for this table, to continue this table--
TABLE construction of all the Two Prime Composites from 9 to 1000 (no 2 as prime)
3*3 = 9                 3+3= 6
3 * 5  = 15           3+5 = 8
3 * 7  = 21           3+7 = 10
5 * 5  = 25           5+5 =10
3 * 11 = 33          3+11 = 14
5 * 7  = 35           5+ 7 = 12
3 * 13 = 39          3+13 = 16
7 *  7  = 49          7+7 = 14
3 * 17 = 51          3+17 = 20
5 * 11 = 55          5+11 = 16
3 * 19 = 57          3+19 = 22
5 * 13  = 65         5+13 = 18
3 * 23  = 69         3+23 = 26
7 * 11  = 77         7+11 = 18
5 *17  = 85          5+17 = 22
3 * 29  = 87        3+29 = 32
7 * 13  =  91       7+13 = 20
3 * 31  =  93        3+ 31 = 34
5 * 19  =  95        5+ 19 = 24

[Vinicius Claudino Ferraz]

In my list1.txt, I saved all goldbach representations from everybody sequentially
from 6,8,10,12,... to 140504

no counter example found.

in my list2.txt, I saved r(n) considering just the count of representations in list1.txt

list1 ~ analytical ~ 1.0 GB
list2 ~ synthetical ~ 1.0 MB

190 MB in google drive .zip

Post by b***@gmail.com
49500 is the smallest with 1000 Goldbach partitions.

[Archimedes Plutonium]

Now one place where 1 is prime, unit prime, is desirable is the staircase conjecture. That conjecture is take the smallest toolbox of primes to append onto existing primes and see how many even numbers can be built. So if our toolbox has just the primes 1,3,5 we are stopped at 30.

What the staircase conjecture does is look for a minimum toolbox and 1 would be important as a prime.

AP

[b***@gmail.com]

Concerning Goldbach Sloan features data for with and
without the assumption 1 is prime. Doesn't change much.

http://oeis.org/wiki/Index_to_OEIS:_Section_Go#Goldbach

Post by Archimedes Plutonium
Now one place where 1 is prime, unit prime, is desirable is the staircase conjecture. That conjecture is take the smallest toolbox of primes to append onto existing primes and see how many even numbers can be built. So if our toolbox has just the primes 1,3,5 we are stopped at 30.
What the staircase conjecture does is look for a minimum toolbox and 1 would be important as a prime.
AP

[Vinicius Claudino Ferraz]

Mail delivered to sender. Too long.

Here's your useless list:

http://claudino.webs.com/temp.txt

Post by Archimedes Plutonium
Can your computer go out to N=1000 for this table, to continue this table--