proof of equivalence of Fermat’s Last Theorem to its analytic form

Consider the Taylor expansion of the cosine function. We have

${\rm lim}_{s\to\infty}(A_{s})=2-{\rm cos}\,x-{\rm cos}\,y$

and

${\rm lim}_{s\to\infty}(B_{s})=1-{\rm cos}\,z$ .

For $r>x,y$ the sequence $a_{r}$ is decreasing as the denominator grows faster than the numerator. Hence for $s>x,y$ the sequence $A_{s}$ is increasing as $A_{s+4}=A_{s}+a_{s+2}-a_{s+4}$ and $a_{s+2}>a_{s+4}$ . So if $A_{N}>0$ for some $N>x,y$ , we have $2-{\rm cos}\,x-{\rm cos}\,y>0$ . Conversely if no such $N$ exists then $A_{s}\leq 0$ for $s>x,y$ , so its limit, $2-{\rm cos}\,x-{\rm cos}\,y$ , is also less than or equal to $0$ . However as this expression cannot be negative we would have $2-{\rm cos}\,x-{\rm cos}\,y=0$ .

Similarly for $r>z$ the sequence $b_{r}$ is decreasing, and for $s>z$ the sequence $B_{s}$ is increasing. So if $B_{M}>0$ for some $M>z$ we have $1-{\rm cos}\,z>0$ . Conversely if no such $M$ exists then $1-{\rm cos}\,z\leq 0$ . However as this expression cannot be negative we would have $1-{\rm cos}\,z=0$ .

Note that $2-{\rm cos}\,x-{\rm cos}\,y=0$ precisely when $x,y\in 2\pi\mathbb{Z}$ . Also $1-{\rm cos}\,z=0$ precisely when $z\in 2\pi\mathbb{Z}$ .

So the form of the theorem may be read:

If for positive reals $x, y, z$ we have $x^{n}+y^{n}=z^{n}$ for some odd integer $n>2$ , then either $x$ or $y$ not in $2\pi\mathbb{Z}$ or $z$ not in $2\pi\mathbb{Z}$ .

Clearly this only fails if for positive integers $a, b, c$ and some odd $n>2$ , we have

$(2\pi a)^{n}+(2\pi b)^{n}=(2\pi c)^{n}$ .

Dividing through by $(2\pi)^{n}$ we see that $a^{n}+b^{n}=c^{n}$ .

Conversely suppose we have non-zero integers satisfying $a^{n}+b^{n}=c^{n}$ for some $n>2$ . If $n=4k$ we have $(a^{k})^{4}+(b^{k})^{4}=(c^{k})^{4}$ , contradicting example of Fermat’s last theorem. Hence if $n$ is even we may replace $a, b, c$ with $a^{2},b^{2},c^{2}$ and $n$ with $n/2$ , which will be odd and greater than 1 (and hence greater than 2 as it is odd). So without loss of generality we may assume $n$ odd.

Finally replace $a, b, c$ with their absolute values and if reorder to obtain a positive integer solution. This would be a counterexample to the form of the theorem as stated above.

Title	proof of equivalence of Fermat’s Last Theorem to its analytic form
Canonical name	ProofOfEquivalenceOfFermatsLastTheoremToItsAnalyticForm
Date of creation	2013-03-22 16:19:04
Last modified on	2013-03-22 16:19:04
Owner	whm22 (2009)
Last modified by	whm22 (2009)
Numerical id	7
Author	whm22 (2009)
Entry type	Proof
Classification	msc 11D41