I went to a math graduate student talk yesterday about regular primes and their relations to Fermat’s Last Theorem, class numbers of fields, zeta functions, and the like. The thing that struck me most about the talk was how many “proofs” due to Euler were used that really did nothing like what a proof is supposed to do.
Here’s a simple example of the sorts of “proof” involved in the lecture – we know that if a geometric series 1+r+r^2+r^3+… converges, then a simple calculation shows that it converges to 1/(1-r). (If we just multiply through by 1-r, we see that every term cancels except for the 1 – more rigorously, if we multiply the partial sums by 1-r, we get 1-r^(n+1), and if |r|p-adic distance for some prime p, we get the p-adic numbers as the completion. Amazingly enough, in the 2-adics, the series 1+2+4+8+16+… really does converge to -1. And in the 5-adics, the series 1+5+25+125+… really does converge to -1/4. (The argument from above actually works completely unchanged, except that |r|Here’s another blogospheric discussion of this phenomenon.