Dear Golden Gadzirayi Nyambuya, unfortunately, I need to inform you that your proof is erroneous. Like many other authors of the FLT "proofs", you used a statement which is claimed (or silently assumed) obvious, but really is wrong. In your case this is "Lemma §(3)": indeed, there is an infinite set of the pairs (c; d) whose sum equals a√b, but there are only (a-1) pairs of the positive integers (c1; d1) whose sum equals a. Therefore, it is easy to give a counterexample:

2√5 = (√5+1) + (√5-1).

As one can see, c/√b is irrational (c=(√5+1), b=5). Thus, the Lemma is not "a self-evident truth" as you wrote, but is obviously wrong. And the following Theorem, based on the Lemma, is, of course, wrong. According to the theorem, there is no solution of the equation z^3 = x^2 + y^2 "for any piecewise coprime triple of integers (x, y, z) each greater than unity such that z is not a perfect square". But x=2, y=11, z=5 is the solution.

Fermat very likely did not have the correct proof. By parsimony, we would expect that if there were a "simple proof" it very likely would have been found in the 350 years that mathematicians had considered this problem before Wiles' developed what we now accept as the definitive conclusion to the problem.

This is one of the most interest mystery of math...

Why Fermat did not give the proof of his theorem? He didi not die immediately, but a lot of time after his sentence...

The best mathematician of the math story was involved to solve this problem, but we waited a lot of years before the genius of Wiles gives the right proof.

Thanks a lot for this reflection...

When I send this to number theory journals, they simple say they don't publish such claims. I am still hunting for a journal to accept the paper, but it seems it may take some time. I will try to put a sketch of the proof.

I will try that. But ArXiv is very difficult to submit a paper because of their indorsement system. I have papers there, and I know submitting to a different section is very difficult.

To add to what Fausto said, history is filled with cases where famous mathematicians (i.e. not amateurs, posers, or cranks) were convinced they had a result, but instead it turned out to be completely wrong. One of the more famous instances was Hans Rademacher's (dis)proof of the Riemann Hypothesis, which was almost published before the mistake was found. There are cases where simpler proofs for known propositions are found later, but some problems just cannot be simplified substantially.  Again, given the 350 years that mathematicians have looked at the Fermat problem (and found no simpler proof than Wiles'), I think it's safe to say that Fermant himself did not have the solution he thought he had.

Dear Michael Wendl,

I hear you. But, if someone claims a proof, it must be evaluated.

I am well aware of the plethora of failures along the FLT Proof High Road.

The proof I pause needs to be checked by someone competent.

That's all I am saying.

Journals are shying away from doing so.

I will be very happy if a flaw is found because it would rest my mind, I will stop thinking about this proof.

I will never beat my chest and say, I have found a simpler proof than that of Professor Wiles. No! Until someone competent looks at the paper and gives it a KUDO. I know the great many embarrassments along this highway. But one has to be confident and this should not be mistaken as beating of one's own chest!

Einstein was very confident, even when he was wrong. However, he was lucky man because he always managed to get time to correct himself before the experiments are done.

@Peter T Breuer, the for the correction Euler was indeed not Italian, but Swiss. We have taken that into account.

@Golden: I don't recall commenting on your manuscript. Rather, I was addressing your original question, whether Fermat had the proof he thought he had. If you have your own manuscript on this subject, I would follow Peter's advice and go to arxiv. You appear to be a member of an academic institution, meaning that you should be automatically endorsed at arxiv (see https://en.wikipedia.org/wiki/ArXiv#Peer_review). If your manuscript starts to gather some momentum there, you might then think about journal submission. Good luck!

My take on Fermat's short solution is that each equation resolves to ax2+by2=cz2where a,b and c don't have any factors in common. Then it comes down to proving that an ellipse can't be the same as a circle.

Hi Richard. That is a mis-statement of the problem. It is only defined for a=b=c=1. The proof turns out to be so much more than showing an ellipse is not the same as a circle (actually a circle is a special case of an ellipse having the foci at the same x,y point)

https://en.wikipedia.org/wiki/Fermat%27s_Last_Theorem

Hello Michael,

I may have missed some steps in my previous explanation. Xn+Yn=Zn

can be written as X(a2)+Y(b2)=Z(c2)

where X,Y and Z can't have any factors in common and this can be written as X/Za2+Y/Zb2=c2

The function on the left is an ellipse and that on the right a circle, as you say, it is a matter of proving that no sets of integers satisfy this equation.

Just to clarify, I don't see this as necessarily provable, but I can see Fermat in his carriage thinking along lines like this.

Dear Ellis D. Cooper, I will be happy if you download my paper and went through it and thereafter ... ask if Fermat really had a proof. I think he did!

@  Geoffrey ...... link me to the Linkedin group

I have completed a proof of Beal using this method. I am checking the proof and giving it to other to check for error. There is another paper I wrote on beal, check:

http://www.scirp.org/journal/PaperInformation.aspx?PaperID=49935#.VEpX1cngVY8

Thanks,  I have joined the group, looks like an active group. I will post my paper there. I will make some adjustments based on my private communication with Professor Andre C. M. Ran.

I have no intention to read your paper. Assume it is fully correct. How do you explain you are the first in three centuries to find it? Its not for lack of candidates. Already in the 18th century the french academy science refused to consider for publication any proof of FLT.

I hear you Patrick. The is simple trivial Lemma to it. I have worked in private on this for 20 years with many wrong proofs. But when this one came, I was shocked that it required this simple lemma which is found in the paper. I am hoping some mathematician of note will read this and try and find the flaw in it. For all I know, an arithmetic proof is easy to know if it is correct or not. Thus far, the only eminent person that has tried to find flaws is Professor Andre C. M. Ran who is on researchgate. Thus far he has only pointed out some areas of improved not flaws.

Dear Golden Gadzirayi Nyambuya, unfortunately, I need to inform you that your proof is erroneous. Like many other authors of the FLT "proofs", you used a statement which is claimed (or silently assumed) obvious, but really is wrong. In your case this is "Lemma §(3)": indeed, there is an infinite set of the pairs (c; d) whose sum equals a√b, but there are only (a-1) pairs of the positive integers (c1; d1) whose sum equals a. Therefore, it is easy to give a counterexample:

2√5 = (√5+1) + (√5-1).

As one can see, c/√b is irrational (c=(√5+1), b=5). Thus, the Lemma is not "a self-evident truth" as you wrote, but is obviously wrong. And the following Theorem, based on the Lemma, is, of course, wrong. According to the theorem, there is no solution of the equation z^3 = x^2 + y^2 "for any piecewise coprime triple of integers (x, y, z) each greater than unity such that z is not a perfect square". But x=2, y=11, z=5 is the solution.

Dear Vadim, thanks for you input, will study it and come back to you!

As a general observation (one that I believe many active mathematician colleagues would agree with), it's usually not profitable to spend too much time on problems like this because (1) there is already a conclusive verdict, in this case Wiles' solution, and (2) so many feet have already trodden this path that it is incredibly unlikely that some simpler solution can be found. The world of unsolved mathematics problems is so vast, especially in rapidly emerging application areas like mathematical genomics or financial mathematics, that one's time might probably be better spent blazing trails rather than retracing very old footsteps.

Correct. There is no real debate today about Wiles' proof. Many specialists have examined it over the last (almost) 20 years and consensus holds that it is valid. The wikipedia page has a brief history for those interested in this:

https://en.wikipedia.org/wiki/Wiles%27_proof_of_Fermat%27s_Last_Theorem

PS: for those who are *really* interested, I recommend the book by Cornell et al that goes into much more detail:

G. Cornell, J. H. Silverman and G. Stevens, Modular forms and Fermat's Last Theorem, ISBN 0-387-94609-8

@Michael: there is a more elementary reference  than Cornell et al.

http://www.amazon.com/Invitation-Mathematics-Fermat-Wiles-Yves-Hellegouarch/dp/0123392519

At a philosophical level there are many many Theorems that require a book just by recalling the context.

Dear Golden,

I happened to miss your question when you posted it first, but it’s still time to give an answer, isn’t it ? In order to avoid any ambiguity, let me fix the terminology , because you sometimes use an unorthodox vocabulary. So : N is the set of natural integers, Z the set of rational integers ( = integers with a sign), Q the set of rational numbers ( = fractions with rational integers as numerators and denominators), R the set of real numbers. In your three papers you often say « rationals » for N, and « numbers» for R  (I think). This is not important, but somewhat irritating for number theorists. More important: in the Fermat equation, you seem to restrict the variables to N, whereas they should be in Z. This bears no consequence for even exponents, but modifies dramatically the problem for odd exponents (*). Summarizing: even if your arguments were correct, they would not prove Fermat’s last theorem (FLT for short).

Let us come now to the heart of the discussion. You complain that « official » journals are « shying away » from « elementary » proofs of FLT. This could mean that the « establishment » is narrow minded, but actually no. The referees know a « trick » (due to Marc Krasner) which allows them to affirm that no elementary proof of FLT can exist (**). Let me explain. We must of course define « elementary ».  It is widely accepted that « elementary arithmetic » makes use of only three tools: the laws of addition and multiplication (and their interrelated properties), and the existence and unicity of factorization into powers of primes. These three ingredients actually express structural properties of Z : it’s not merely a set, it’s a ring (properties of + and x), and this ring is factorial (existence and uniqueness of factorization). Summarizing: to do « elementary arithmetic » is to work inside Z using only the structure of a factorial ring. But in algebraic number theory, many equally interesting factorial rings are known to exist. One of the most fundamental is the ring  of p-adic integers, p being a given prime. One can define on Z a « p-adic metric » (a good exercise !) , quite different than the usual metric coming from the absolute value. Then  is just the completion of Z with respect to that p-adic metric. The point is that  is also naturally  a factorial ring, and the key point for us is that the Fermat equation with exponent p indeed admits non trivial solutions in . Summarizing: if one could prove FLT in Z by  elementary arithmetic arguments, then one could prove it also in  by exactly the same arguments, a contradiction.

Let us apply this to your three papers (let us call them Fermat 1, Fermat 2 and Beal for short) :

1)The Krasner trick clearly applies to Fermat 1, so we can say undoubtfully that Fermat 1 is not correct

2)In Fermat 2, things could seem more obscure because you introduce a parameter  , which belongs a priori to an algebraic closure of Q. But if z belongs to , then  also exists in an algebraic closure of , the field of fractions of , and your arguments go on when replacing Z by . So the Krasner trick also applies, and Fermat 2 is wrong. Perhaps I could convince you more easily  by staying more down to earth and saying that your crucial lemma 3 is not correct. The proof at the end of p.5 is hardly understandable, but I interpret it as follows (tell me if I’m mistaken) :  you assert - without any justification - that if two real numbers c, d are such that (c+d)/  (where b is an integer which is not a perfect square) is an integer, then c/  and d/  separately are also integers. This is obviously false, just take c =    and d = (1 – )  . See also  Vadim Khodorovskiy’s counter-example.

3)I haven’t had the courage to examine Beal thoroughly, but since it’s the same ship sailing on the same waters, it’s condemned to the same fate

Good luck all the same,   Thong NQD

(*) FLT for even exponents has been given a simple « almost elementary » proof by Guy Terjanian in Comptes Rendus Académie Sciences Paris, 285, 16 (1977), 973-975. « Almost» means that Terjanian makes use of a non elementary (in the above sense) tool, the Legendre symbol, behind which is lurking the definitely non elementary « quadratic reciprocity law » of Gauss

(**) This could also answer the questions posed by Victor Christiano and Vinay Kumar

@Thong: can you repost with the symbol \overline Z which is missing. Krasner trick is interesting but incomprehnesible with the current notation.

Dear Golden,

I happened to miss your question when you posted it first, but it’s still time to give an answer, isn’t it ? In order to avoid any ambiguity, let me fix the terminology , because you sometimes use an unorthodox vocabulary. So : N is the set of natural integers, Z the set of rational integers ( = integers with a sign), Q the set of rational numbers ( = fractions with rational integers as numerators and denominators), R the set of real numbers. In your three papers you often say « rationals » for N, and « numbers» for R  (I think). This is not important, but somewhat irritating for number theorists. More important: in the Fermat equation, you seem to restrict the variables to N, whereas they should be in Z. This bears no consequence for even exponents, but modifies dramatically the problem for odd exponents (*). Summarizing: even if your arguments were correct, they would not prove Fermat’s last theorem (FLT for short).

Let us come now to the heart of the discussion. You complain that « official » journals are « shying away » from « elementary » proofs of FLT. This could mean that the « establishment » is narrow minded, but actually no. The referees know a « trick » (due to Marc Krasner) which allows them to affirm that no elementary proof of FLT can exist (**). Let me explain. We must of course define « elementary ».  It is widely accepted that « elementary arithmetic » makes use of only three tools: the laws of addition and multiplication (and their interrelated properties), and the existence and unicity of factorization into powers of primes. These three ingredients actually express structural properties of Z : it’s not merely a set, it’s a ring (properties of + and x), and this ring is factorial (existence and uniqueness of factorization). Summarizing: to do « elementary arithmetic » is to work inside Z using only the structure of a factorial ring. But in algebraic number theory, many equally interesting factorial rings are known to exist. One of the most fundamental is the ring Z_p of p-adic integers, p being a given prime. One can define on Z a « p-adic metric » (a good exercise !) , quite different than the usual metric coming from the absolute value. Then Z_p is just the completion of Z with respect to that p-adic metric. The point is that Z_p is also naturally  a factorial ring, and the key point for us is that the Fermat equation with exponent p indeed admits non trivial solutions in Z_p. Summarizing: if one could prove FLT in Z by  elementary arithmetic arguments, then one could prove it also in  Z_p by exactly the same arguments, a contradiction.

Let us apply this to your three papers (let us call them Fermat 1, Fermat 2 and Beal for short) :

1)The Krasner trick clearly applies to Fermat 1, so we can say undoubtfully that Fermat 1 is not correct

2)In Fermat 2, things could seem more obscure because you introduce a parameter square root of z , which belongs a priori to an algebraic closure of Q. But if z belongs to Z_p, then s.r. z   also exists in an algebraic closure of the field of fractions of Z_p, and your arguments go on when replacing Z by Z_p. So the Krasner trick also applies, and Fermat 2 is wrong. Perhaps I could convince you more easily  by staying more down to earth and saying that your crucial lemma 3 is not correct. The proof at the end of p.5 is hardly understandable, but I interpret it as follows (tell me if I’m mistaken) :  you assert - without any justification - that if two real numbers c, d are such that (c+d)/s.r. b  (where b is an integer which is not a perfect square) is an integer, then c/s.r. b and d/s.r. b separately are also integers. This is obviously false, just take c = pi s.r. b and d = (1 – pi)s.r. b  . See also  Vadim Khodorovskiy’s counter-example.

3)I haven’t had the courage to examine Beal thoroughly, but since it’s the same ship sailing on the same waters, it’s condemned to the same fate

Good luck all the same,   Thong NQD

(*) FLT for even exponents has been given a simple « almost elementary » proof by Guy Terjanian in Comptes Rendus Académie Sciences Paris, 285, 16 (1977), 973-975. « Almost» means that Terjanian makes use of a non elementary (in the above sense) tool, the Legendre symbol, behind which is lurking the definitely non elementary « quadratic reciprocity law » of Gauss

(**) This could also answer the questions posed by Victor Christiano and Vinay Kumar

@luiz

I personally "buy it", as you say. I remember we have already had a discussion on the Shimura-Taniyama-Weil-Wiles stuff in the Algebraic Number Theory section - curiously starting from the Riemann hypothesis ! I have nothing to add to the lengthy development which I wrote then to your intention. Except perhaps a literary quotation: "There are more things in heaven and earth, Luiz, than are dreamt on in your philosophy" (Hamlet to Horatio)

Dear Luiz,

let me ask you some simple questions.

1) Why do you try to verify a proof that is not in your domain?

2) How can you expect a problem open for three centuries to admit a proof that is elementary or easily verifiable?

Let me observe that Wiles proof has been checked by the referees of Annals of Math plus by the many number theorists that have run seminars written books on the topic, etc

I believe the interest for any of the members of the small group of people you describe to find a mistake is much larger than to promote the proof of another member of the group.

Eventually the likelyhood of a tacit conspiracy between all the members of this group (easily a thousand people in the world) to promote Wiles proof for another reason than the belief its true is very very low!

Sincerely Yours

PS

If you want a almost a margin proof, please read my work

https://www.researchgate.net/publication/301674377_Fermat%27s_Last_Theorem_Proof_fixed_edition?ev=prf_pub

Data Fermat's Last Theorem (Proof) (fixed edition)

Luiz C. L. Botelho, please substantiate your statement!

edition error

https://www.researchgate.net/publication/302602715_Fermat%27s_Last_Theorem_Proof_fixed_edition_2?ev=prf_pub

Data Fermat's Last Theorem (Proof) (fixed edition) 2

Luis "Mr Idioces"

Are you high? by your comments you are that kind of people with pathologies mutiples (if you want I recommend a therapist) . please avoid that kind of comments that contribute nothing better explains your point of view here

To Carlos Mauricio Santacruz who wrote

"If you want a almost a margin proof, please read my work".

Dear Carlos Mauricio Santacruz, in your work you made some simple manipulations that yielded your "Equation No. 1" and "Equation No. 2". After this you stated (in inaccurate words) that there is no solution of the system of the Eq. No.1 and Eq. No.2 in Rational Numbers for the cases n=3 and more. This statement is equivalent to the FLT.

But you did not prove this statement (your figures and examples do not contain a proof).

So, you just reformulated the FLT, but you did not prove it.

Unfortunately, it is an often situation in many the FLT "proofs".

Like everything in life, sadly I have encountered with these comments that do not seek to understand my work only discredit him, but I've also had good cometary. At first I thought that for this web would find better answers, answers with fundamentals not only deny that is not understood. I think this is my last contribution to the question of Golden Gadzirayi Nyambuya. if studying document found that if n = 2 always find the values of x, y, z. For this conjecture and the formulation is correct (call manipulation of equations understand as if performing some trick) in the case of n> 2, equation 2 resulting not have rational points (yes, you have to work in each equation before of discrediting) that meet the condition of intersect or meet also to equation 1, due to the nature of problem where each case is an equation with a greater degree, it is impossible to prove with a simple "x ≠ y", so the use of graphics to review cases is the only way to see it in the simplest way, due to other questions, I work in a formulation that can review on my list ...... (( x ^ (1 / n) +1)) ^ (1 / n) .............). Because this is that the method of testing FLT becomes so important to math.

After this. if you or someone else have a specific question, please send me a personal msg

Regards

Carlos never ever be discouraged by negative comment .. reading in-between the lines of these negative comments and try to clean the new proof so that this negative comment no longer applies, in the end, you proof gets better ... Continue pedal on man.

To Carlos Mauricio Santacruz

Dear Carlos, it seems to me, your problems are your unwillingness to hear and comprehend your opponents and, excuse me, your English. Please try to understand:

1) "manipulations of equations" does not mean "some trick" but this is the standard phrase which means usual mathematical transformations of equations. Therefore, this phrase is not offensive. By "simple manipulations" I only meant that you made some trivial transformations of equations.

2) As a result of these transformations, you obtained the statement which has the same complexity as the Fermat's Last Theorem. Your statement "in the case of n> 2, equation 2 resulting not have rational points…" is needed to be proved (it is equivalent to the FLT). To say (or silently assume) that it is right is not enough. And several (or even millions) examples is not a proof.

Please prove it! And by this you will prove the FLT.

Dear Golden,

I haven't read your paper about FLT but I read the one about Beal's conjecture. Is this a peer reviewed paper?

I have a few questions about your demonstration:

1) In the lemma, you mention gcd(p,q) different from 1 but when you use it in the decomposition of A^x, B^y, C^z into a sum of numbers p,q, you say gcd(p,q)=1. Is this a typo?

2) You wrote B^y = 2q, which is always even. You may mention that you could choose any number among A^x, B^y and C^z to be set equal to 2q.

3) The lemma holds only for n>2?

4) You mention that if gcd(A^x,B^y,C^z) different from 1, then gcd(A,B,C) is also different from 1. The opposite is certainly true, but not this I think?

Thanks.