Fermat’s Last Theorem: A one-operation proof

[Victor Sorokine]

Eight months have passed since the first time when an elementary proof of FLT was found, laid down and proposed to the attention of 70 mathematicians (they have asked for it; most of them are specialized in numbers theory). About 1,000 individuals visited internet sites with the text of the proof. However, no positive nor negative opinions have been expressed so far.

The idea of the elementary proof of the Fermat’s Last Theorem (FLT) proposed to the reader, is extremely simple:
After splitting the numbers a, b, c into pairs of sums, then grouping them in two sums U' and U'' and multiplying the equation a^n + b^n – c^n = 0 by 11^n (i.e. 11 power n, and the numbers a, b, c by 11), the (k+3)-th digit from the right in the number a^n + b^n – c^n (where k is the number of zeroes at the end of the number a + b – c) is not equal to 0 (the numbers U' and U'' are “multiplied” in a different way!). In order to understand the proof, you only need to know the Newton Binomial, the simplest formulation of the Fermat’s Little Theorem (included here), the definition of the prime number, how to add numbers and multiplication of a two-digit number by 11. That’s ALL! The main (and the toughest) task is not to mess up with a dozen of numbers symbolized by letters.
The redaction of the text dates of June 1, 2005 (after a discussion on the Faculty of Mathematics of Moscow University site).

The texts of the proof can be found on following sites:
English version of the demonstration (4kb): Revista Foaie Matematică: www.fmatem.moldnet.md/1_(v_sor_05).htm

Russin version in pdf : http://fox.ivlim.ru/docs/sorokine/vtf.pdf

FORUMS (Russian-language):
http://lib.mexmat.ru/forum/viewforum.php?f=1&sid=3fefd56c6fe2fa0e361464672ea92292 ;
http://forum.dubinushka.ru/index.php?showforum=40 ; http://www.scientific.ru/dforum/altern - page 7.

 

[Hurkyl]

www.fmatem.moldnet.md/1_(v_sor_05).htm

This link does not work for me.

 

[Victor Sorokine]

FLT in PDF

pdf:
http://fox.ivlim.ru/docs/sorokine/flt.pdf

 

[robert Ihnot]

I can not follow the notation. What is the meaning of: Everywhere in the text
a1^10. (Which is written in red.)?

 

[Victor Sorokine]

I can not follow the notation. What is the meaning of: Everywhere in the text
a1^10. (Which is written in red.)?

In the all publications:
"Everywhere in the text a1 ≠ 0."
or: a_1 =/ 0 (the last digit of the number "a" =/ 0).
Thank,
vs

 

[Hurkyl]

I agree, it is difficult to read, and not all statements are justified.

Statement 9, in particular: you've asserted that u''_{k+2} = [v + (a_{k+1} + b_{k+1} - c_{k+1})_2]_1, and the only assumption that's been invoked on a, b, and c is that the last digit of a + b - c is zero, and that the last nonzero digit is 5. You've given no proof of that assertion, and I can provide a counterexample:

In this example, n = 13. All numbers are written in base 13.

a = 400
b = 206
c = 286

so that

u = a + b - c = 350
u' = 0
u'' = 350
v = 4
u''_3 = 3
a_2 + b_2 - c_2 = -8
(a_2 + b_2 - c_2) = 0
v + (a_2 + b_2 - c_2)_2 = 4
[v + (a_2 + b_2 - c_2)_2]_1 = 4
So that u''_3 \neq [v + (a_2 + b_2 - c_2)_2]_1

 

[Artermis]

wow I have no clue what hurkyl just said =(

 

[learningphysics]

What is meant by this in section 0.2:

[Digits in negative numbers are <<negative>>]

 

[Victor Sorokine]

I agree, it is difficult to read, and not all statements are justified.

...you've asserted that ... the only assumption that's been invoked on a, b, and c is that the last digit of a + b - c is zero. ... You've given no proof of that assertion, and I can provide a counterexample:

In this example, n = 13. All numbers are written in base 13.

a = 400
b = 206
c = 286
...
[/itex]

Dear Hurkyl,

1. From a^n + b^n – c^n = 0 and from Fermat’s Little Theorem for (a_1, b_1, c_1) =/ 0 we have: (a_1 + b_1 – c_1)_1 = 0 (= u_1).

2. Your example:
a = 400, b = 206, c = 286 and (abc)_1 = 0
is not Case 1, where (abc)_1 =/ 0.

Your example:
a = 400, b = 206, c = 286, k =2, u = c-b and c-b = 286 – 206 = 80 has 1 zero in the end
is not Case 2, where u = c-b, c-b has (cf. §1) kn-1 = 2x13 – 1 = 25 zeros in the end.

Respectfully yours,
vs.

 

[Victor Sorokine]

What is meant by this in section 0.2:

[Digits in negative numbers are <<negative>>]

Digits in negative numbers are <<negative>> =
= all digits (=/ 0) in negative numbers are negative.
vs

 

[robert Ihnot]

None of this makes any sense to me: Victor Sorokine: From a^n + b^n – c^n = 0 and from Fermat’s Little Theorem for (a_1, b_1, c_1) =/ 0 we have: (a_1 + b_1 – c_1)_1 = 0 (= u_1).

What is (a_1, b_1, c_1) ? or the next statement: (a_1 + b_1 – c_1)_1 = 0 (= u_1)?

 

[learningphysics]

None of this makes any sense to me: Victor Sorokine: From a^n + b^n – c^n = 0 and from Fermat’s Little Theorem for (a_1, b_1, c_1) =/ 0 we have: (a_1 + b_1 – c_1)_1 = 0 (= u_1).

What is (a_1, b_1, c_1) ? or the next statement: (a_1 + b_1 – c_1)_1 = 0 (= u_1)?

robert,
a_1 refers to the last digit of a (in base n)

Fermat's little theorem says that if a number n is prime then: a^n - a is divisible by n (a can be any integer I think). So we have a^n-a divisible by n... b^n-b divisible by n... c^n-c divisible by n... so their sum is divisible by n...

(a^n - a) + (b^n-b) - (c^n-c) is divisible by n so...
(a^n + b^n - c^n) - (a + b - c) is divisible by n...
0 - (a+b-c) is divisible by n...
so
(a + b - c) is divisible by n...
(a + b - c)_1 (last digit of a+b-c) = 0... since the number is in base n and divisible by n the last digit must be zero...

Then it's possible to show that:
a_1 + b_1 + c_1=0

 

[Victor Sorokine]

...Then it's possible to show that:
a_1 + b_1 + c_1=0

Thanks,
vs

 

[robert Ihnot]

Victor Sorokine: Then it's possible to show that:
a_1 + b_1 + c_1=0
WEll, I am not sure if base 11 has something to do with this, and, very possibly, Hurkyl is arguing about: You've given no proof of that assertion, and I can provide a counterexamplle: In this example, n = 13. All numbers are written in base 13.

I am reminded of Able's submission to the French Academy, and after two years Able wondered why he had not heard. http://www.shu.edu/projects/reals/history/abel.html Unfortunately, the Academy picked Legendre and Cauchy as referees to judge it. The former, who was in his seventies, claimed that he could not read the handwriting and left all the work to the latter. The latter, who was much more interested in his own work and possibly just a bit jealous, brought the work home and promptly "misplaced" it.

learning physics: (a + b - c)_1 (last digit of a+b-c) = 0... since the number is in base n and divisible by n the last digit must be zero... Who said anything about writing it in base n, how can that change the fundamental properties? Anyway that does not effect a+b+c, since if a+b-c==a+b+c ==0 Mod n, then for n not 2, c is divisible by n, is that what you are saying? If so no one for centuries thought that obvious, let alone in only a few short steps; if, I have this right, what you mean is to assert that n, the power of the exponent, a prime, must divide one of the terms, particularly c to boot!


You mention 70 mathematicians and 1000 internet hits, possibly if the paper was rewritten and the notation easier to understand, you might get a better response.

 

[Hurkyl]

Your PDF says that case 1 is that (bc)_1 \neq 0.

Anyways, it doesn't matter: how about this one:

a = 507
b = 105
c = 58C (C in base 13 = 12 in base 10)

My calculations again show statement 9 to be in error.

 

[learningphysics]

learning physics: (a + b - c)_1 (last digit of a+b-c) = 0... since the number is in base n and divisible by n the last digit must be zero... Who said anything about writing it in base n

That's part of his paper... writing it in base n... Anyway this is what I meant:

(a+b-c)_1 =0

where (a+b-c)_1 is the last digit of a+b-c... I didn't mean multiplication by the last digit of a+b-c. Sorry about that.

 

[robert Ihnot]

learningphysics: That's part of his paper... writing it in base n... Anyway this is what I meant: (a+b-c)_1 =0

Well if you look at definition 1, 1* let us assume that a_n+b_n-c_n= 0.

From what is above it in the paper, it would mean that we are talking about the nth digit of an expression related by the equation (never even defined as) a^n+b^n=c^n, where n is a prime conventionally expressed as p. (This assumes we have n digits to speak of, or maybe n is just a dummy variable that runs from n=1, n=?)

HOWEVER, he might be talking about modulo n, which has never, as usual, been defined as the meaning. I somehow think it is the second meaning, possibly.

 

[Victor Sorokine]

Victor Sorokine: Then it's possible to show that:
a_1 + b_1 + c_1=0
WEll, I am not sure if base 11 has something to do with this, and, very possibly, Hurkyl is arguing about: You've given no proof of that assertion, and I can provide a counterexamplle: In this example, n = 13. All numbers are written in base 13.

Where is a mstake:
1. (a^(n-1))_1 = 1 (Little FT).
2. [(a^(n-1))_1 x a_1]_1 = a_1 = a^n_1.
3. (a^n_1 + b^n_1 – c^n_1)_1 = (a_1 + b_1 – c-1)_1 = u_1.
vs

 

[Victor Sorokine]

Your PDF says that case 1 is that (bc)_1 \neq 0.

Anyways, it doesn't matter: how about this one:

a = 507
b = 105
c = 58C (C in base 13 = 12 in base 10)

My calculations again show statement 9 to be in error.

Let the digits in base 13: 1, 2, 3, 4, 5, 6, 7, 8, 9, 9+1 = X [=10 in base 10], X+1 = Y [=11], Y+1 = Z [=12];
Z+1 = 10 [=13], 10+1 = 11 [=14]…
Let a = 507, b = 105, c = 58Z. Then:
u = 507 + 105 – 58Z = 60Z – 58Z = 20.
k = 1.
u' = 7 + 5 – Z = 0.
u'' = u – u' = 500 + 100 – 580 = 20.
v = 5 + 1 – 5 = 1.(a_(k+1) + b_(k+1) – c_(k+1))_2 = (0 + 0 – 8)_2 = 0.
u''_(k+2)= [v + (a_(k+1) + b_(k+1) – c_(k+1))_2]_1 = 1 + 0 = 1 = (–1, 0 or 1).
OK?
vs

 

[learningphysics]

Let the digits in base 13: 1, 2, 3, 4, 5, 6, 7, 8, 9, 9+1 = X [=10 in base 10], X+1 = Y [=11], Y+1 = Z [=12];
Z+1 = 10 [=13], 10+1 = 11 [=14]…
Let a = 507, b = 105, c = 58Z. Then:
u = 507 + 105 – 58Z = 60Z – 58Z = 20.

In the above 60Z-58Z = 50.

k = 1.
u' = 7 + 5 – Z = 0.
u'' = u – u' = 500 + 100 – 580 = 20.

So if k=1, then u''_(k+2) = 0.

u''_(k+2)= [v + (a_(k+1) + b_(k+1) – c_(k+1))_2]_1 = 1 + 0 = 1 = (–1, 0 or 1).

u''_(k+2) = 0, but [v + (a_(k+1) + b_(k+1) – c_(k+1))_2]_1 = 1, so the equation is false it seems.

 

[robert Ihnot]

Victor Sorokine: 3. (a^n_1 + b^n_1 – c^n_1)_1 = (a_1 + b_1 – c-1)_1 = u_1.

What are you saying by a^n_1, a raised to the last digit in n? I don't think so. Everyone assumes that a^n+b^n-c^n = 0, though you have never bothered to even state as much. What I think you mean by a^n_1 is the last digit in a^n; do you ever read any mathematical literature? Why can't you just follow convention?:

a^n+b^n-c^n=0 \equiv a+b-c\equiv0 Mod (n)

As for the final u_1, what on Earth is that? Is it defined? I don't think so.

 

[Hurkyl]

I don't know of any particular convention for denoting the k-th digit of a number, so .

u was defined to be a + b - c, so I thought it was pretty clear u_1 meant the last digit of u, anyways.

 

[Victor Sorokine]

In the above 60Z-58Z = 50.

So if k=1, then u''_(k+2) = 0.

u''_(k+2) = 0, but [v + (a_(k+1) + b_(k+1) – c_(k+1))_2]_1 = 1, so the equation is false it seems.

Yes:
(9°) u''_{k+2} = [v + (a_(k+1) + b_(k+1) – c_(k+1))_k+2]1, where (a_(k+1) + b_(k+1) – c_(k+1))_ {k+2} = (–1, 0 or 1);

Let a = 507, b = 105, c = 58Z. Then:
u = 507 + 105 – 58Z = 60Z – 58Z = 50.
k = 1.
u' = a_1 + b_1 – c_1 = 7 + 5 – Z = 0.
u'' = u – u' = 500 + 100 – 580 = 50.
v = 5 + 1 – 5 = 1.
(a_(k+1) + b_(k+1) – c_(k+1))_3 = (07 + 05 – 8Z)_ 3 = (0Z – 8Z)_ 3 = 0.
u''_(k+2)= [v + (a_(k+1) + b_(k+1) – c_(k+1))_ 3]_1 = 1 + 0 = 1 = (–1, 0 or 1).
OK?
vs

 

[Victor Sorokine]

Victor Sorokine: 3. (a^n_1 + b^n_1 – c^n_1)_1 = (a_1 + b_1 – c-1)_1 = u_1.

Certainly: 3. (a^n_1 + b^n_1 – c^n_1)_1 = (a_1 + b_1 – c-1)_1 = 0.

vs

 

[Victor Sorokine]

I don't know of any particular convention for denoting the k-th digit of a number, so .

u was defined to be a + b - c, so I thought it was pretty clear u_1 meant the last digit of u, anyways.

Yes. u_1 = 0.
vs

 

[robert Ihnot]

Hurkly: u was defined to be a + b - c, so I thought it was pretty clear u_1 meant the last digit of u, anyways.

Well, if you want to guess about it, but more conventionally the writer could begin by introducing u as: Let u represent... A person might have thought it was defined somewhere else.

 

[learningphysics]

Yes:
(9°) u''_{k+2} = [v + (a_(k+1) + b_(k+1) – c_(k+1))_k+2]1, where (a_(k+1) + b_(k+1) – c_(k+1))_ {k+2} = (–1, 0 or 1);

Let a = 507, b = 105, c = 58Z. Then:
u = 507 + 105 – 58Z = 60Z – 58Z = 50.
k = 1.
u' = a_1 + b_1 – c_1 = 7 + 5 – Z = 0.
u'' = u – u' = 500 + 100 – 580 = 50.

So u''_{k+2} = 0, where k=1.

v = 5 + 1 – 5 = 1.
(a_(k+1) + b_(k+1) – c_(k+1))_3 = (07 + 05 – 8Z)_ 3 = (0Z – 8Z)_ 3 = 0.
u''_(k+2)= [v + (a_(k+1) + b_(k+1) – c_(k+1))_ 3]_1 = 1 + 0 = 1 = (–1, 0 or 1).

u''_(k+2) = 0... but [v + (a_(k+1) + b_(k+1) – c_(k+1))_ 3]_1 = 1 so the equation still seems false.

Is u''_(k+2) = 0 when k=1 and u''=50? I'm just looking at the third digit from the right of u'', which is 0.

 

[Hurkyl]

Well, if you want to guess about it, but more conventionally the writer could begin by introducing u as: Let u represent... A person might have thought it was defined somewhere else.

Statement 2° in his PDF:

Let u = a + b - c ...

 

[robert Ihnot]

Hurkyl: Statement 2° in his PDF:
Let u = a + b - c ...

I see that is correct.

 

[ramsey2879]

Yes:
(9°) u''_{k+2} = [v + (a_(k+1) + b_(k+1) – c_(k+1))_k+2]1, where (a_(k+1)
+ b_(k+1) – c_(k+1))_ {k+2} = (–1, 0 or 1);

But the pdf version reads
(9°) u''_{k+2} = [v + (a_(k+1) + b_(k+1) – c_(k+1))_2]_1, where (a_(k+1)
+ b_(k+1) – c_(k+1))_ 2 = (–1, 0 or 1);

Yes:
Let a = 507, b = 105, c = 58Z. Then:
u = 507 + 105 – 58Z = 60Z – 58Z = 50.
k = 1.
u' = a_1 + b_1 – c_1 = 7 + 5 – Z = 0.
u'' = u – u' = 500 + 100 – 580 = 50.
v = 5 + 1 – 5 = 1.
(a_(k+1) + b_(k+1) – c_(k+1))_3 = (07 + 05 – 8Z)_ 3 = (0Z – 8Z)_ 3 = 0.

But (a_(k+1) + b_(k+1) + c_(k+1))_2 = (a_2 + b_2 - c_2)_2 = (0+0-8)_2
= -8_2 = 0?

Thus
u''_{k+2} = 50_3 = 0 <> (1+0)_1 = 1?

This seems to be a contradiction at first, but -8 is equivalent to -10 + 5 thus this must be that occasion where (a_2 + b_2 - c_2)_2 = -1. I can think of no other logical reason that -1 could result.

Thus -8_2 in my mind should equal -1 which then gives u''_3 = 0 which is correct!

 

[ramsey2879]

Your PDF says that case 1 is that (bc)_1 \neq 0.

In introductory notes, Victor also rules out a_1 as equaling 0. This does not cause a problem as at least one of a_1 and b_1 <> 0.

Anyways, it doesn't matter: how about this one:

a = 507
b = 105
c = 58C (C in base 13 = 12 in base 10)

My calculations again show statement 9 to be in error.

But look at my analysis of this in my last post which I edited just previously. I think there is no contradiction of statement 9 here!

 

[Hurkyl]

There is by the way Victor defines his terms... the second digit of -8 is not -1, it's 0.

I think I'd prefer some n's complement type thing, where -80 would actually be the left-infinite triskadecimal¹ number ...CCCC50. (Adding 80 to this would yield 0)

In any case, if you would rather the second digit of -8 to be -1, then that should require that the first digit be a 5... does that mess anything up?

In any case, we'll have to wait and see if Victor likes one of these alterations, and sees if it works out the kinks in his logic. (And I still maintain it would be nice to see proofs of all steps)


1: I don't know if this is the standard term, or if there is even a standard.

 

[ramsey2879]

In any case, we'll have to wait and see if Victor likes one of these alterations, and sees if it works out the kinks in his logic. (And I still maintain it would be nice to see proofs of all steps)

I also would like Victor to explain this more as his last post is a contradiction. It seems that his statements are so brief, that it is difficult for even the author to fully understand what is exactly meant upon reading them. I think the failure to fully set out the logical steps that lead to each statement adds to the confusion here.

 

[Victor Sorokine]

For Hurkyl, robert Ihnot, learningphysics and ramsey2879

So u''_{k+2} = 0, where k=1.

u''_(k+2) = 0... but [v + (a_(k+1) + b_(k+1) – c_(k+1))_ 3]_1 = 1 so the equation still seems false.

Is u''_(k+2) = 0 when k=1 and u''=50? I'm just looking at the third digit from the right of u'', which is 0.

Dear Hurkyl, robert Ihnot, learningphysics and ramsey2879,
thank you very much for your help! Your criticism is polite and constructive.
The example of Hurkyl is exellent!

About u''_(k+2), or u''_(3).
Accurate difinition (for a = 507, b = 105, c = 58Z, Z = 7+5, k =1):
u' = 7 + 5 – Z = 0, u'' = u – u' = 500 + 100 – 580 = 50; u''_{3} = 0;
v = (a_{3} + b_{3} – c_{3})_1 = 1 (cf 4°);
(a_(2) + b_(2) – c_(2))_{3} = (–1, 0 or 1) = (00 + 00 – 80)_{3} = (–100 + 2)_{3} = –1;

From (9°):
u''_{3} = [v + (a_(2) + b_(2) – c_(2))_3]_1 = (1 – 1) = 0.

 

[ramsey2879]

Dear Hurkyl, robert Ihnot, learningphysics and ramsey2879,
thank you very much for your help! Your criticism is polite and constructive.
The example of Hurkyl is exellent!

About u''_(k+2), or u''_(3).
Accurate difinition (for a = 507, b = 105, c = 58Z, Z = 7+5, k =1):
u' = 7 + 5 – Z = 0, u'' = u – u' = 500 + 100 – 580 = 50; u''_{3} = 0;
v = (a_{3} + b_{3} – c_{3})_1 = 1 (cf 4°);
(a_(2) + b_(2) – c_(2))_{3} = (–1, 0 or 1) = (00 + 00 – 80)_{3} = (–100 + 2)_{3} = –1;

From (9°):
u''_{3} = [v + (a_(2) + b_(2) – c_(2))_3]_1 = (1 – 1) = 0.

First how do you obtain -100 + 2, should not it be -100 + 50? And isn't this a contradiction that negative numbers are negative at every digit?
I think you need to amend your statement or definitions to explain how exactly the sum in u'' (other than v) can equal either -1, 0 or 1, particularly when does it occur that 1 is substracted from v. Is not -8_{2} or -80_{3}in fact = 0 using your initial definitions?
I also think an explanation needs to be made how you got from [v + (a_{k+1} + b_{k+1} - c_{k+1})]_{2}, which is how your pdf version reads, to [v + (a(k+1) + b(k+1) - c(k+1))]_{3}. The former just uses the digits in the (k+1)th position, or (0+0-8) evaluated at the 2nd position of this sum, while the latter uses the numbers comprised of the last k+1 digits evaluated at the k+2 position, or (07+05-8Z)_{3}. I think it can be shown that this discrepancy does not make a difference since the digits in positions k or less cancel out with no borrow from the k+1 position since u(k) = 0 . There could be a carry over to the k+1 position but 1 substracted from 5 in the k+1 position does not affect any carry to or borrow from the k+2 position.

 

[robert Ihnot]

Victor Sorokine: Dear Hurkyl, robert Ihnot, learningphysics and ramsey2879,
thank you very much for your help! Your criticism is polite and constructive.

Thank you very much for the complement, and the edition of the proof with the heading: "Page Math Journal" is a much improved version.

 

[Victor Sorokine]

For ramsey2879, Hurkyl, robert Ihnot and learningphysics

First how do you obtain -100 + 2, should not it be -100 + 50? And isn't this a contradiction that negative numbers are negative at every digit?

Dear ramsey2879, Hurkyl, robert Ihnot and learningphysics,
Here are our acquisitions:

1. Correction of (9°) in the proof:
(9°) u''_{k+2} = [v + (a_(k+1) + b_(k+1) – c_(k+1))_ {k+2}]_1.
Cause of the error: erroneous copying.

2. First digit in the negative number (in the example n = [13], k = 1, w = –80, (–80)_{3}).
It is logically:
The augmentation of the number w by n^(k+2) the digit w_{3} increases by 1 (if w_{3} = n – 1, then new w_{3} = 0). ALWAIS!
(2 + 50)_{3} = (300 – 80)_{3} = 2,
(1 + 50)_{3} = (200 – 80)_{3} = 1,
(0 + 50)_{3} = (100 – 80)_{3} = 0,
(–100 + 50)_{3} = (– 80)_{3} = –1.
Therefore:
If w_(2) < 0 and !w! < n^2, then w_{3} = –1. (–80)_{3} = –1, but 0!

To be continued

 

[ramsey2879]

Dear ramsey2879, Hurkyl, robert Ihnot and learningphysics,

1. Correction of (9°) in the proof:
(9°) u''_{k+2} = [v + (a_(k+1) + b_(k+1) – c_(k+1))_ {k+2}]_1.
Cause of the error: erroneous copying.
>snip
To be continued

That is not the only error, see 13°, "3" should be --5--. Also is it necessary to the proof that a, b, and c can be either negative or positive? For instance if a, b and c are each positive then "5° u'_k+1 = -1, 0 or 1" should be --u' = 0 or 1--. I think the proof can be more concise if a, b, c are made to be each positive. The proof is also too difficult to read. I suggest you look at the link at the top of this page on How to write Proofs.

 

[Victor Sorokine]

For all

The proof is also too difficult to read.

I would like my readers to take into account the following:
1. Since I don't master English very well, my replies could be a little be late.
2. Because of lack of time and being a little bit distracted, please be tolerant to typos in my texts.
3. Altough my wording style is not too detailed, it makes the reader to THINK and to generate a number of non-standard ideas. It may be more interesting that a simple reading.
4. The beginning of the discussion gives useful results. However, no basic flaw in the proof has been found so far. It would be a mistake to end the discussion prematurely.
Thank you all,
Victor

Other equivalent version of (9°):
(9°) /u''_{k+2} - v/ <= 1

That is not the only error, see 13°, "3" should be --5--.

Yes. Thank you very much! The digit "3" was in a version of the proof for n > 5.

There is a version of the proof for n > 2, but with other multipliers in (3°) and in (1*°). This version is shorter in 2 times, but it contains 1 supplement lemma.

 

[Hurkyl]

3. Altough my wording style is not too detailed, it makes the reader to THINK and to generate a number of non-standard ideas. It may be more interesting that a simple reading.

Such a methodology certainly has merit, but I don't think the situation is appropriate for it, for two reasons:

(1) Claiming a proof of the FLT is a very serious thing, and people won't take you seriously if you leave much of the detail to the reader.
(2) Working with individual digits is generally an exercise in technical detail, not enlightened learning. Your proof doesn't seem to carry any insight into how numbers work: it looks more like a complicated series of calculations that (alledgedly) leads to a contradiction.

That's how I begin looking at the argument: I simply wrote a program to do the calculations, and checked to see if things behaved as advertised. I haven't updated it to handle your new definition of "k-th digit" yet... could you state it precisely?

4. The beginning of the discussion gives useful results. However, no basic flaw in the proof has been found so far. It would be a mistake to end the discussion prematurely.

Someone suggested to me today what might be a fatal flaw: he points out that your argument, if valid, would not only prove there are no integer solutions to the FLT, but that there are no p-adic solutions either... but he thinks there's good reason to expect that there are p-adic solutions to the FLT.

 

[ramsey2879]

Eight months have passed since the first time when an elementary proof of FLT was found, laid down and proposed to the attention of 70 mathematicians (they have asked for it; most of them are specialized in numbers theory). About 1,000 individuals visited internet sites with the text of the proof. However, no positive nor negative opinions have been expressed so far.

I would like my readers to take into account the following:
1. Since I don't master English very well, my replies could be a little be late.
2. Because of lack of time and being a little bit distracted, please be tolerant to typos in my texts.
3. Altough my wording style is not too detailed, it makes the reader to THINK and to generate a number of non-standard ideas. It may be more interesting that a simple reading.

Such a methodology certainly has merit, but I don't think the situation is appropriate for it, for two reasons:

(1) Claiming a proof of the FLT is a very serious thing, and people won't take you seriously if you leave much of the detail to the reader.
(2) Working with individual digits is generally an exercise in technical detail, not enlightened learning. Your proof doesn't seem to carry any insight into how numbers work: it looks more like a complicated series of calculations that (alledgedly) leads to a contradiction.

That's how I begin looking at the argument: I simply wrote a program to do the calculations, and checked to see if things behaved as advertised. I haven't updated it to handle your new definition of "k-th digit" yet... could you state it precisely?




Someone suggested to me today what might be a fatal flaw: he points out that your argument, if valid, would not only prove there are no integer solutions to the FLT, but that there are no p-adic solutions either... but he thinks there's good reason to expect that there are p-adic solutions to the FLT.

I think that Victor does not appreciate the lack of courtesy his methodology shows here. A proof is not supposed to make the reader full in the detail. It is assumed that the author has already covered all corners. It is the task of the author to communicate this to the reader. I again refer to the pages at http://www.artofproblemsolving.com/Resources/AoPS_R_A_HowWrite.php on how to write proofs. It begins with this opening paragraphs: "You've figured out the solution to the problem - fantastic! But you're not finished. Whether you are writing solutions for a competition, a journal, a message board, or just to show off for your friends, you must master the art of communicating your solution clearly. Brilliant ideas and innovative solutions to problems are pretty worthless if you can't communicate them. In this article, we explore many aspects of how to write a clear solution. Below is an index; each page of the article includes a sample 'How Not To' solution and 'How To' solution. One common theme you'll find throughout each point is that every time you make an experienced reader have to think to follow your solution, you lose.

As you read the 'How To' solutions, you may think some of them are overwritten. Indeed, some of them could be condensed. Some steps we chose to prove could probably be cited without proof. However, it is far better to prove too much too clearly than to prove too little. Rarely will a reader complain that a solution is too easy to understand or too easy on the eye"


As for my statement that to do otherwise demonstrates a lack of courtesy, I point out that if a simple proof of FLT is indeed at hand, it makes no sense to write the proof in such a manner that the reader has to redo much of the original work of the author. Most mathematicians already have a full schedule and don't appreciate having to give a lot of time to something that seems so dubious in the first place. If a simple proof is in fact known, then it should be simple to write it in a clear manner. As to the fact that English is a second language here, I don't see it as an excuse for omitting so much detail. While Victor's time may also be limited, he had the incentive to submit his proof to over seventy mathematicians for review and to post it in the internet. A little extra time spent on writing a clear proof in the first place thus could have saved untold volumns of time and would have been more rewarding for Victor's own interests, let alone the interests of others. This is not something that should be taken so lightly.

 

[Hurkyl]

Yes, there are (a great many) nontrivial solutions to x^p + y^p = z^p in the p-adics for all primes p.

If you can write down a solution to x^p + y^p = z^p modulo p^2, then that can be extended to a solution in the p-adics. (a great many, actually)


Here's an example for p = 13:

...3333^p + ...1217114C11^p = ...4444^p

which I generated from the base solution 3^13 + 1^13 = 4^13 (mod 13^2). (numbers here written base 10)

 

[ramsey2879]

Yes, there are (a great many) nontrivial solutions to x^p + y^p = z^p in the p-adics for all primes p.

If you can write down a solution to x^p + y^p = z^p modulo p^2, then that can be extended to a solution in the p-adics. (a great many, actually)


Here's an example for p = 13:

...3333^p + ...1217114C11^p = ...4444^p

which I generated from the base solution 3^13 + 1^13 = 4^13 (mod 13^2). (numbers here written base 10)

I am weak in this area so I did an internet search on p-adics. One site of interest was http://mathforum.org/library/drmath/view/65286.html Another is http://www.math.niu.edu/~rusin/known-math/93_back/aaargh The last site has a section that reads:
"FLT is Fermat's Last Theorem, and it is a falsity. Because when you accept that the Naturals are the Infinite Integers, or p-adics and that Finite Integer was a ill-defined concept, then , FLT has solutions in all exponents. FLT is a patently false conjecture.
FLT is easy to prove in p-adics as a false conjecture. However,
Pierre Fermat never had the p-adics in his day.
And all attempts at proving FLT were so difficult not because the
problem is a difficult one for anyone who knows p-adics can solve it in
an hour. FLT was difficult for 350 years because it was false in the
first place and those that tried to prove it were using a fake concept
of Finite Integer. Imagine today a Quantum Physicist trying to use
Newtonian spinning spheres for electrons. Electrons exist but they are
never little spinning Newtonian spheres. P-adics exist, but "finite" in
Finite Integers is a illusion, a mirage"
OK there are many solutions of FLT in p-adics. Now, how does that show Victor's proof to have a fatal error?

 

[Victor Sorokine]

A proof is not supposed to make the reader full in the detail. ... This is not something that should be taken so lightly.

About the difficult of the proof:
1. Pierre Fermat had had more difficulties: it was necessary to make all calculation mentally, in his mind.
2. In the September I chall illustrate all operation with numerical calculations. But I am ready to explain in every detail any statement in my proof.

 

[Hurkyl]

Victor's proof is all about looking at the digits of various numbers, and only the right-most few: it never looks at any of the digits to the left of the (k+3)-th digit.

So, there are two ways of looking at it...

(1) Anything Victor states ought to be true about p-adic values as well as integer values, because the proof doesn't care what's happening to the left of the (k+3)-th digit.

(2) A p-adic solution can be used to generate digits of a "problematic" set of integers a, b, and c, which do satisfy a^p + b^p = c^p, as long as you're only looking at the rightmost (k+3) digits.


If you look at the p-adic solution I generated, we could use it to generate the "problem" set:

a = 3333333
b = 7114C11
c = 4444444

for which u = 6003B00. So, we need to multiply everything by 4 so the rightmost digit is a 5...

a = 0CCCCCCC
b = 22454944
c = 144444443

and we have u = B010500, so that k = 2, and also that the rightmost 5 digits of a^13 + b^13 = c^13 agree, so all the assumptions Victor actually uses in his proof ought to be satisfied, so he shouldn't be able to derive a contradiction.

 

[ramsey2879]

Victor's proof is all about looking at the digits of various numbers, and only the right-most few: it never looks at any of the digits to the left of the (k+3)-th digit.

So, there are two ways of looking at it...

(1) Anything Victor states ought to be true about p-adic values as well as integer values, because the proof doesn't care what's happening to the left of the (k+3)-th digit.

(2) A p-adic solution can be used to generate digits of a "problematic" set of integers a, b, and c, which do satisfy a^p + b^p = c^p, as long as you're only looking at the rightmost (k+3) digits.


If you look at the p-adic solution I generated, we could use it to generate the "problem" set:

a = 3333333
b = 7114C11
c = 4444444

for which u = 6003B00. So, we need to multiply everything by 4 so the rightmost digit is a 5...

a = 0CCCCCCC
b = 22454944
c = 144444443

and we have u = B010500, so that k = 2, and also that the rightmost 5 digits of a^13 + b^13 = c^13 agree, so all the assumptions Victor actually uses in his proof ought to be satisfied, so he shouldn't be able to derive a contradiction.

I think you are assuming that if a^p + b^p -c^p = 0 , i.e. mod 13^7 where a b and c are p-adic numbers then (a*k)^p + (b*k)^p -(c*k)^p also equals zero mod 13^7. I did a check on this using your numbers and according to my calculations, it is not true. I must add that p-adic numbers are new to me, but my calculations cause me to wonder about your statement that "the rightmost 5 digits of the new p-adic numbers agree with the formula a^13+b^13=c^13. Am I off base here?

 

[Hurkyl]

I could have the wrong numbers: I generated them at work, and I haven't programmed a way to check them here at home.

I'll tell you how they were generated so you can play with them yourself:

You start with a (nonzero) solution to a^p + b^p = c^p (mod p^2)

For example, when p = 13, I believe that 1^p + 3^p = 4^p. In any case, you could always use 1^p + 0^p = 1^p, or (p-1)^p + 1^p = 0^p


Now, you add digits one at a time, allowing you to increase that exponent. Here's the derivation and proof:

You're given:

(1) a^p + b^p - c^p = 0 (mod p^n)

with at least 1 of the a, b, c nonzero, and you want to find digits x, y, z, such that:

a' = a + x p^(n-1)
b' = b + y p^(n-1)
c' = c + z p^(n-1)
(2) a'^p + b'^p - c'^p = 0 (mod p^(n+1))

We know from (1) that
a^p + b^p - c^p = q * p^n (mod p^(n+1))

So, we calculate (mod p^(n+1))
a'^p + b'^p - c'^p =
a^p + p a^(p-1) x p^(n-1) + 0 + b^p + p b^(p-1) y p^(n-1) + 0 - c^p - p c^(p-1) z p^(n-1) - 0
(The higher order terms from the binomial theorem all have a p^(2n-1) factor, and since n >= 2, 2n-1 >= n+1)
= (a^p + b^p - c^p) + p^n (a^(p-1) x + b^(p-1) y - c^(p-1) z)
= q p^n + p^n (a^(p-1) x + b^(p-1) y - c^(p-1) z)
= p^n (q + a^(p-1) x + b^(p-1) y - c^(p-1) z)

So all we need to do is to solve the equation
(q + a^(p-1) x + b^(p-1) y - c^(p-1) z) = 0 (mod p)

for x, y, and z. This is a severely underdetermined equation: it will have lots of solutions, unless the coefficients of x, y, and z are all zero... but that can't happen because at least one of a, b, and c are nonzero.

So, this yields an algorithm for generating a solution:

Set a = p - 1, b = 0, c = 1, n = 2. (Or, pick your favorite solution with nonzero a)

We see that a^p + b^p = c^p (mod p^n)

Now, find the q such that a^p + b^p - c^p = q p^n (mod p^(n+1))
Pick any value for y and z, then solve for x in

(q + a^(p-1) x + b^(p-1) y - c^(p-1) z) = 0 (mod p)

Now, set:
a <-- a + x p^(n-1)
b <-- b + y p^(n-1)
c <-- c + z p^(n-1)
n <-- n + 1

And we still have that a^p + b^p = c^p (mod n)

 

[ramsey2879]

Dear Hurkyl, Your calculations appear correct and work out, you just mistyped b = 22456944. I also would like to delete my two messages on this error. I tried to delete the first, without success as of now.

 

[Victor Sorokine]

All number-endings () and digits {},

Here are all number-endings () and digits {}, which are taken into account in the proof of the FLT (Case 1):

(1°) a^n + b^n – c^n :: [= 0]
(2°) u :: [= a + b – c > 0]
(2°) u_{k} :: [= 0; if a, b, c have not common factors then k >1]
(2°) u_{k+1} :: [=/ 0]
(4°) u' :: [= a_(k) + b_(k) – c_(k); !(a_(k) + b_(k) – c_(k))_ {k+1}! <= 1]
(3°, 3a°) u_{k+1} :: [= (u'_{k+1} + u''_{k+1})_1 = 5]
(5°) u'_{k+1} :: [!u'_{k+1}! <= 1]
(7°) u'_{k+2} :: [= 0 always]
(4°) u'' :: [= u – u' = a – a_(k) + b – b_(k) – c + c_(k)]
(6°) u''_{k+1} :: [= (u_{k+1} – u'_{k+1})_1 = (4, 5 or 6)]
(8°) u''_{k+2} :: [= u{k+2} always]
(4°) v :: [= (a_{k+2} + b_{k+2} – c_{k+2})_1; (vn^(k+1) + a_(k+1) + b_(k+1) – c_(k+1))_(k+2) = u_(k+2)]
(10°) a_(k+1) + b_(k+1) – c_(k+1) :: [!(a_(k+1) + b_(k+1) – c_(k+1))_{k+2}! <= 1]
(9°) u''_{k+2} :: [(v – 1)_1 <= u''_{k+2} <= (v + 1)_1]
(9°) v :: [(u_{k+2} – 1)_1 <= v <= (u_{k+2} + 1)_1]

(1*°) u* :: [= 11u = 11a + 11b – 11c = a* + b* – c*]
[Let u/n^k = pn + q; then (pn + q)11_(2) = ((pn + q)(n + 1))_(2) = (qn + pn + q)_(2);
from here (u/n^k)_1 = q, (u/n^k)_2 = (p + q)_1; example: 230 x 11 = …530]
(4°) u*' :: [= a*_(k) + b*_(k) – c*_(k); !u*'_{k+1}! <= 1]
(9°) a*_(k+1) + b*_(k+1) – c*_(k+1) :: [!(a*_(k+1) + b*_(k+1) – c*_(k+1))_{k+2}! <= 1]
(4°) u*'' :: [= u* – u*']
(4°) 11u'
(4°) 11u''
(11°) u*_{k+1} :: [= u_{k+1} = 5]
(12°) u*'_{k+1} :: [= u'_{k+1}]
(13°) u*''_{k+1} :: [= (u*_{k+1} – u*'_{k+1) }_1 = (5 – u*'_{k+1})_1 = (4, 5 or 6)]
(14°) (11u')_{k+2} :: [= (u'_{k+2} + u'_{k+1})_1; (11u')_{k+2} * 0]
(15°) (11u'')_{k+2} :: [= (u''_{k+2} + u''_{k+1})_1]
(16°) u*_{k+2} :: [= (u_{k+2} + u_{k+1})_1 = (u''_{k+2} + u_{k+1})_1 = (u''_{k+2} + 5)_1]
(16а°) u*'_{k+2} :: [= 0]
(17°) u*''_{k+2} :: [= (u*_{k+2} + (–1, 0 or 1))_1 = (u''_{k+2} + (4, 5 or 6))_1]
(18°) v* :: [= (u*_{k+2} – (–1, 0 or 1)) _{k+2})_1 = ((u_{k+2} + u_{k+1})_1 – (–1, 0 or 1))_1 ]

(19°) U' :: [= (a_(k+1))^n + (b_(k+1))^n – (c_(k+1))^n, U'' = (a^n + b^n – c^n) – U', U = U' + U'']
(19°) U*' :: [= (a*_(k+1)) ^n + (b*_(k+1)) ^n – (c*_(k+1)) ^n,
(19°) U*'' :: [= (a*^n + b*^n – c*^n) – U*'
(19°) U* :: [= U*' + U*'']
(19а°) U'_(k+1) :: [= U*'_(k+1) = 0]
(20°) U(k+2) = U'(k+2) = U''(k+2) = U*(k+2) = U*'(k+2) = U*''(k+2) = 0 [always!].
(20b°) U''_{k+3} :: [= (a_{k+2} + b_{k+2} – c_{k+2})_1 = v]
(22°) (11^nU') _{k+3} :: [= U'_{k+3} = (U*'_{k+3} + (11u') _{k+2})1 = (U*'_{k+3} + u'_{k+1})_1]
(23°) U*'_{k+3} :: [= U'_{k+3} – u'_{k+1}]
(24°) U*''_{k+3} :: [= v* = (u_{k+2} + u_{k+1})1 – (–1, 0 or 1)]

(25°) (U*'_{k+3} + U*''_{k+3})_1 :: [= (U*'_{k+3} + U*''_{k+3} – U'_{k+3} – U''_{k+3})_1 =
= (U*'_{k+3} – U'_{k+3} + U*''_{k+3} – U''_{k+3})1 = (– u'_{k+1} + v* – v) =
= (– u'_{k+1} + [u_{k+2} + u_{k+1} – (–1, 0 or 1)] – [u_{k+2} – (–1, 0 or 1)])_1 = (– u'_{k+1} + u_{k+1} + (–2, –1, 0, 1, or 2))_1 = ( u''_{k+1} + (–2, –1, 0, 1, or 2))_1 = (2, 3, 4, 5, 6, 7 or 8) =/ 0, therefore the expression 1° is an inequality]

 

[Hurkyl]

I don't understand your notation at all.