Feynman on Fermat’s Last Theorem (2016)

If \(N\) is a perfect power \(N=z^n\), there exists at least one integer ( \(\sqrt[n]{N} = z \)) in the interval Since the distance between all consecutive integers is \(1\) the probability that contains an integer is the ratio of the length of the intervals between two integers and the distance between \(\sqrt[n]{N}\) and \(\sqrt[n]{N+1}\): \(\frac{d}{1}\). Now in the case of FLT, \( N=x^n + y^n \) and so the probability that \( x^n + y^n \) is a perfect perfect \(n^{th}\) power is \(\frac{\sqrt[n]{x^n + y^n}}{n(x^n + y^n )}\). Setting \(\mathsf{x}_{0}=2\) we can see that the probability of there being integer solutions to \( z^n=x^n + y^n \) \mathsf{\int}_{1}^{\infty} \mathsf{\int}_{1}^{\infty} (u^n + v^n)^{-1+\frac{1}{n}} d u \ d v\)) does decrease with increasing \(n\).

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