The below proof can be seen as an example of a pathological proof. It gives no information to ``why" the result holds, or how non-trivial the result is. Yet, assuming Wiles' proof does not use the above theorem anywhere, it proves the statement. Otherwise, the below proof would be an example of a circular argument.

Proof. Suppose $\sqrt[n]{2}=a/b$ for some positive integers $a,b$ . It follows that $2=a^n/b^n$ , or \begin{eqnarray} \label{fermat} b^n + b^n &=& a^n. \end{eqnarray}We can now apply a recent result of Andrew Wiles [1], which states that there are no non-zero integers $a$ , $b$ satisfying equation (). Thus $\sqrt[n]{2}$ is irrational.

The above proof is given in [2], where it is attributed to W.H. Schultz.

Bibliography

1

A. Wiles, Modular elliptic curves and Fermat's last theorem, Annals of Mathematics, Volume 141, No. 3 May, 1995, 443-551.

2

W.H. Schultz, An observation, American Mathematical Monthly, Vol. 110, Nr. 5, May 2003. (submitted by R. Ehrenborg).