Feynman on Fermat's Last Theorem
This comment isn&#x27;t meant to be dismissive. I think it&#x27;s interesting to explore arguments like this. Math progresses by exploring thoughts. But...<p>Couldn&#x27;t we argue in the same way that there are unlikely to be any solutions to the simpler equation x^n = z^n?<p>After all, taking p(x) = x^(1&#x2F;n - 1)&#x2F;n to be the probability that x is an n-th power, as Feynman does, and then integrating p(x^n) dx from x = x_0 to infinity to find the expected number of n-th powers of the form x^n for x &gt; x_0, as Feynman does with p(x^n + y^n), we find for n &gt; 2 that this comes out to 1&#x2F;(x_0^(n - 2) * n * (n - 2)), which is quite small for sizable x_0 and n.<p>This yields, for example, that the expected number of solutions (and thus an upper-bound for the probability of the existence of any solutions) to the equation x^100 = z^100 with x &gt; 10 should be 1 out of 98 googol. This is about as certain as certain gets that there are no solutions... and yet solutions are as ubiquitous as ubiquitous gets!