When you see a proof in print it’s like an Old Master painting – everything in its place and polished and perfect. You get the feeling that no human being could produce such a thing, and you despair. This is why Old Master drawings are so encouraging. Repeated attempts at getting the same thing. There is a lovely Guercino drawing where the hero has three left arms: one because Guercino got the proportions wrong, one because he changed his mind about where to put it.
The description of how something comes about is inevitably far longer than the act of doing it. Try describing a BJJ move in terms of what happens, why it happens, and what would have happened if you’d done something else. Try describing walking, for that matter.
So this description of how to prove the Fundamental Theorem of Arithmetic (that every number has a unique factorization into prime numbers) is invaluable. It describes how one actually thinks when proving things, not merely the result of doing the thinking.
Here is a video clip demonstrating the mathematical process of turning a sphere inside out. It’s very well scripted, with the dialogue form fitting the subject-matter perfectly.
(From John Baez via Scott Aaronson‘s blog)
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.
I submitted the problem outlined here to the American Mathematical Monthly and they seem willing to publish it in their monthly problems page. One of the collaborating editors has provided a proof of part (b), which is delightful news. It takes a route I considered briefly but then discarded as being impassable!
The only thing missing in the editor’s proof is that the error in the approximation is proved to be , whereas I really need and a good idea of what multiple of 1/n is involved. Perhaps one of the solvers will improve the error term once the problem appears in the magazine.
An entry in Ars Mathematica has alerted me to John Cramer’s Transactional Interpretation of Quantum Mechanics [see also Wikipedia]. It feels exactly right.
The trouble with quantum mechanics has always been that it makes accurate predictions but doesn’t make sense. People make a virtue of this. It shows how far above our heads the whole theory is. “My thoughts are not your thoughts, my ways are not your ways, says the Lord”.
This is self-indulgent obscurantism and it leads to such New Age loopiness as The Dancing Wu Li Masters (in which, among other delights, every chapter is called Chapter One).
The transactional interpretation is solidly and sensibly based on mathematics – specifically, on a bit of mathematics that has mostly been ignored because it’s embarrassing.
This problem is in two parts. Part A is A-level standard; part B is more advanced.
A. Given that k > 1.0, prove that, for large n, the largest real root of
B. Given that k > 1.0, prove that, for large n, the largest real root of