To recap: we know that 3²+4²=5² and 3³+4³+5³=6³. Cyprian’s Last Theorem states that these are the only cases of n consecutive nth powers adding up to the next nth power.
I’ve nearly proved it.
Specifically, I’ve proved it for all values of n that are not of the form 16m+2 or 8m+3, and, in addition, for all values of n (of whatever form) that are less than . This is not the same as proving it for absolutely all values of n, but it’s enough to be going on with.
The final gap in that proof was proving Part B of the problem described here. I floundered around for a while, getting nowhere in particular, and then submitted it to the American Mathematical Monthly for publication in its problems page. In accepting it, the editor sent me a proof by one of his colleagues of a weaker version of Part B, and I was able to strengthen it to prove the result I needed.
So now I have a 45-page paper describing the whole adventure in a chatty and discursive style: I think it’s important that mathematics should work as a spectator sport as long as the spectators can be helped to participate a little bit in what’s going on. I’m letting the paper infuse for a few weeks and then I’ll tighten it up a bit and smooth it out.