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 *n*th powers adding up to the next *n*th power.

I’ve *nearly* proved it.

Specifically, I’ve proved it for all values of *n* that are **not** of the form 16*m*+2 or 8*m*+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.