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proposed elementary proof of Fermat's last theorem

Major Section: 
Reference
Type of Math Object: 
Proof

Mathematics Subject Classification

11D41 no label found

Comments

In the third displayed equation, your've written a^b instead of a^n.

In the definition of beta, I think you mean gcd(k,t) = n^beta (not gcd(k,b)).

Also, it is clear that at least one of alpha and beta must be zero; otherwise n divides f and k which contradicts gcd(f, k) = 1.

#1. Usually, in work on FLT using elementary methods, there is a Case 1 in which n does not divide the product abc, and a Case 2 in which n does divide this product. These cases clearly cover all possible triples of coprime natural numbers for given n. My impression after some reading and studying is that this attempted proof may produce a reasonable stab at Case 1, but is incomplete with respect to Case 2. This would not be surprising since there are many more results for Case 1 than for Case 2, and Case 2 is considered to be much more difficult (according to Ribenboim in Fermat's Last Theorem for Amateurs). It may be that the original author (McPogor) really did nothing essentially wrong, but merely that his Version A and Version B do not cover all the possibilities.

#2. A related problem occurs when the substitutions (or transformations) used in a proof cannot be reversed in a 1-1 fashion (e.g., even n-th power and n-th root). This also has the effect of proving results for fewer than all the possibilities. In a proof such as this in which many substitutions are made, this might be helpful to investigate. A very brief article that addresses this common shortcoming is:

A Classic Roadblock in Efforts to Prove Fermat's Last Theorem
Glenn James
Mathematics Magazine, Vol. 32, No. 2. (Nov. - Dec., 1958), pp. 101-102.

#3. I was wondering if it would simplify or complicate the situation to let a, b, c be integers rather than natural numbers. I have typically seen FLT phrased this way. The two formulations are equivalent but it seems to me easier to work with integers and not worry all the time if you are subtracting too much (as in the two holes pointed out by MathProf).

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