A Suggested Proof of Fermat’s Last Theorem (Revised 3/22/2021)

Pierre de Fermat

Introduction.

“After his death in 1665, Pierre de Fermat’s son Clement-Samuel discovered a copy of Arithmetic, a third-century math book by Diophantus, in which Fermat had written on one page, “It is impossible…for any number which is a power greater than the second to be written as the sum of two like powers [xⁿ + yⁿ = zⁿ for n > 2]. I have a truly marvelous demonstration of this proposition which this margin is too narrow to contain.” His son published the note, and “Fermat’s Last Theorem” beguiled mathematicians for over 350 years, until it was proved by Andrew Wiles, with Richard Taylor, in 1995.”⁽¹⁾

Axiom 1: Every number >0 is integer or rational or irrational. 

Axiom 2: integer is not rational

Axiom 3: integer+rational= rational.

Axiom 4: integer or rational is not irrational.

Proof of Fermat’s Last Theorem.

In the following proof, n is an integer >2.

According to Fermat, the equation

xⁿ+yⁿ=zⁿ                                            (1)

has no integer solution, z, if z>y>x>0 and n is an integer>2. We shall call Eq 1 “Fermat’s Equation”. And it has become customary to refer to his claim as “Fermat’s Last Theorem”. Following is a suggested proof of Fermat’s Last Theorem.

We begin by reordering Fermat’s equation:

zⁿ=xⁿ+yⁿ.                                                (2)

Let us define the positive integer p to be such that

y=x+p.                                                    (3)

Then

zⁿ=xⁿ+(x+p)ⁿ                                            (4a)

    = xⁿ+(xⁿ+nx^n-1p+…+pⁿ)                        (4b)

    =2xⁿ+(nx^n-1p+…+pⁿ)                            (4c)

   =2[xⁿ+(nx^n-1p+…+pⁿ)/2]                        (4d)

where (nx^n-1p+…+pⁿ)/2] is integer or rational.

xⁿ is of course integer. It follows that xⁿ+(nx^n-1p+…+pⁿ)/2) is integer or rational,  say  

xⁿ+(nx^n-1p+…+pⁿ)/2=N.                        (5)

Since (nx^n-1p+…+pⁿ)/2 is >1, N>2 and N is integer or rational.

We have from Eq. 4d

zⁿ=2N,                                                    (6)

and N is always integer>2 or rational>2.

Hence

z=2^1/nN^1/n.                                                (7)

Now since 2^1/n is irrational, z is irrational except if N^1/n=1/2^1/n<1. But N^1/n is, by definition, >1 (see Eq. 5), and hence N^1/n=1/2^1/n is a contradiction. It follows that z in Eq. 7 is always irrational and therefore non-integer.

One final caveat: 1/2^1/2 is also less than 1. It might be argued, therefore, that z should also be irrational in the “Pythagorean equation”, z²=x²+y². But a much older theorem, the Pythagorean theorem, proves that, for selected values of x and p, z has some integer solutions. Hence the argument is without merit. In brief, we cannot use the proof herein to prove that z is always irrational when n<3. 

QED. For all n>2, solutions for z are always irrational. 

Fermat was right!

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1. From Lapham’s Quarterly, https://www.laphamsquarterly.org/miscellany/fermats-last-margin-note. The Wiles proof is much longer than the proof suggested in this article, and it includes concepts unknown in Fermat’s time.