Bethe, Fermi and Feynman and a simple math proof

 The Los Alamos Labs in New Mexico housed some of the greatest collections of human intellect on the planet ever in the history of the world. Hans Beth and Enrico Fermi were the superstars but Feynman  was not to be outdone. He had just graduated from Princeton with a Ph.D and had joined Hans Bethe team. He impressed everyone so much with his uncompromising approach to Physics and Math that Niels Bohr would come to him to brainstorm ideas and ask Feynman to find the flaw. He was the sounding board. Less known about Feynman than his famous diagrams were his numerical calculation abilities. Interestingly Hans Bethe was no slouch either in this area. They would have compute-offs and once the answer came down to calculating the difference between two consecutive squares. Hans Bethe answered it as an even number and Feynman jumped and said that it was not possible. Hans Bethe grinned and corrected himself to an odd number.  Both of them had quickly done the below proof in their heads in a flash. Hans Bethe then recruited Feynman to Cornell were he worked after the war. Feynman of course tired of living all his life in the North East, moved to Caltech.

Going back to the problem of why the difference between two consecutive squares cannot be an even number, it is one of the more simpler and elegant proofs in mathematics.

Consider a number n and a consecutive number n-1. (n)square - (n-1)square is (n) square - (n(square)-2n+1) whhc is n(square)-n(square)+2n-1 which is 2n-1. 2n is always even and 2n-1 will always be odd.

Or in other words, consider a number n+1 and a consecutive number n. (n+1)square -(n)square is ((n) square+2n+1)- n(square) which is 2n+1. 2n is always even and 2n+1 will always be odd.

Simple and elegant and Feynman and Bethe did this really fast in their heads.