Previous HN discussion around the time of publication just over a year ago:<p><a href="https://news.ycombinator.com/item?id=12018221" rel="nofollow">https://news.ycombinator.com/item?id=12018221</a>
Sigh, this proof bugs me so much.<p>Compute an approximate distance between nth powers, interpret this as the probability of an integer being an nth power, integrate this probability over the sum x^n + y^n, see that the probability of this being an nth power is also very low.<p>I guess this is close enough for government work, but it's so utterly fallacious. For example, the distance between n^2 and (n-1)^2 is 2n - 1. That "means" that the "probability" of N being a perfect square is about 1/(sqrt(2N - 1)). This probability also goes to zero in the limit as N goes to infinity. Not very quickly, but it does.<p>Does that mean that square numbers don't exist?<p>We have many examples of conjectures being disproved by very large counterexamples:<p><a href="https://www.quora.com/What-is-an-example-of-a-conjecture-that-was-proven-wrong-for-very-large-numbers" rel="nofollow">https://www.quora.com/What-is-an-example-of-a-conjecture-tha...</a>
Feynman's approach seems very reminiscent of Jake Vanderplas's Statistics for Hackers talk[0] as opposed to the purely theoretical physicist approach that the author notes at the end.<p>[0]: <a href="https://www.youtube.com/watch?v=Iq9DzN6mvYA" rel="nofollow">https://www.youtube.com/watch?v=Iq9DzN6mvYA</a>
Here's a more straight-forward (but equally hand-wavy) way to calculate the probability that N is a perfect power:<p>P(N) ≈ (number of perfect powers near N) / (size of the neighborhood)<p>≈ (ⁿ√(N + r) − ⁿ√N) / r, for some smallish r<p>≈ d/dN (ⁿ√N)<p>= ⁿ√N / nN
Apparently the author also hosts this interesting website: <a href="http://www.fermatslibrary.com" rel="nofollow">http://www.fermatslibrary.com</a>