# FLT for rational functions

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Following is the problem 2.16 in The Math Problems Notebook:

Prove that if $n>2$, then we do not have any nontrivial solutions of the equation $F^n + G^n = H^n$ where $F,G,H$ are rational functions. Solutions of the form $F = aJ, G=bJ, H=cJ$ where $J$ is a rational function and $a,b,c$ are complex numbers satisfying $a^n + b^n = c^n$, are called trivial.

This problem is analogous to the Fermat’s Last Theorem (FLT) which states that for $n> 2$, $x^n + y^n = z^n$ has no nontrivial integer solutions.

The solution of this problem involves proof by contradiction:

Since any rational solution yields a complex polynomial solution, by clearing the denominators, it is sufficient to assume that $(f,g,h)$ is a polynomial solution such that $r=\max(\deg(f),\deg(g),\deg(h))$ is minimal among all polynomial solutions, where $r>0$.

Assume also that $f,g,h$ are relatively prime.  Hence we have $f^n+g^n = h^n$, i.e. $f^n-h^n = g^n$. Now using the simple factorization identity involving the roots of unity, we get:

$\displaystyle{\prod_{\ell = 0}^{n-1}\left(f-\zeta^\ell h\right) = g^n}$

where $\zeta = e^{\frac{2\pi i}{n}}$ with $i = \sqrt{-1}$.

Since $\gcd(f,g) = \gcd(f,h) = 1$, we have $\gcd(f-\zeta^\ell h, f-\zeta^k h)=1$ for $\ell\neq k$. Since the ring of complex polynomials has unique facotrization property, we must have $g = g_1\cdots g_{n-1}$, where $g_j$ are polynomials satisfying $\boxed{g_\ell^n = f-\zeta^\ell h}$.

Now consider the factors $f-h, f-\zeta h, f-\zeta^2 h$. Note that, since $n>2$, these elements belong to the 2-dimensional vector space generated by $f,h$ over $\mathbb{C}$.  Hence these three elements are linearly dependent, i.e. there exists a vanishing linear combination with complex coefficients (not all zero) in these three elements. Thus there exist $a_\ell\in\mathbb{C}$ so that $a_0g_0^n + a_1g_1^n = a_2g_2^n$. We then set $h_\ell = \sqrt[n]{a_\ell}g_\ell$, and observe that $\boxed{h_0^n+h_1^n = h_2^n}$.

Moreover, the polynomials $\gcd(h_\ell,h_k)=1$ for $\ell \neq k$ and $\max_{\ell}(h_\ell) = \max_{\ell} (g_\ell) < r$ since $g_\ell^n \mid f^n-h^n$. Thus contradicting the minimality of $r$, i.e. the minimal (degree) solution $f,g,h$ didn’t exist. Hence no solution exists.

The above argument fails for proving the non-existence of integer solutions since two coprime integers don’t form a 2-dimensional vector space over $\mathbb{C}$.

# FLT proof fits on a shirt!

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Around 1637, Pierre de Fermat wrote his Last Theorem in the margin of his copy of the Arithmetica next to Diophantus’ sum-of-squares problem:

It is impossible to separate a cube into two cubes, or a fourth power into two fourth powers, or in general, any power higher than the second, into two like powers. I have discovered a truly marvelous proof of this, which this margin is too narrow to contain.

In May 1995, Andrew Wiles proved “Fermat’s Last Theorem” (FLT) ! To celebrate his achievement various conferences and meetings were organized. A Fermat’s Last Theorem T-shirt was designed for the Boston University meeting on FLT, August 9-18, 1995.  The T-shirt was designed by members of the 1995 PROMYS counselor staff who attended ” A Conference On Number Theory And Fermat’s Last Theorem ” . The conference was intended to be as accessible as possible to a general mathematical audience.  The conference focused on two major topics: (1) Andrew Wiles’ proof of the Taniyama-Shimura-Weil conjecture for semistable elliptic curves; and (2) the earlier works of Frey, Serre, and Ribet showing that Wiles’ Theorem would complete the proof of Fermat’s Last Theorem.

PROMYS T-shirt which summarize the proof of FLT, with complete references on the back.

Remarking on information printed on these T-shirts, Fernando Q. Gouvêa wrote following poem:

They said the proof was long and hard,
and painful to behold,
But at the conference at BU,
we got the real dirt.
The proof, it sure is tricky,
but its length isn’t so bold–
It doesn’t fit the margin,
but it does fit on a shirt.

There are many more poems on FLT: Fermat’s Last Theorem and Poetry (Lecturas Matem´aticas
Volumen 22 (2001), p´aginas 137–147)