#mathematics question: is there any good collection of all the knowledge about Conway's "nimbers"? I've found any number of beginners' explanations, but nothing further.
(I mean the thing that makes a field of characteristic 2 out of ℕ₀, with addition corresponding to bitwise XOR, and multiplication a really weird inductively defined thing, such that every initial segment of ℕ₀ of size 2^{2^n} forms a subfield.)
I was idly playing around with them recently – wrote a tiny Python thing to implement their multiplication. I happened across the fact that in each of the first nine of those finite subfields, the _largest_ integer in that subfield – 2^{2^n}-1 – generates its whole multiplicative group. That is, 3 has order 3, 15 has order 15, 255 has order 255, … up to and including 2^512−1.
Is that true for all n? I'm curious to know if this is a known result, or at least a known conjecture. Seems to me it would be pretty neat if it were true, because it would give you a "canonical" choice of primitive polynomial over GF(2) of degree 2^n – just take the min poly of the corresponding nimber.
By my calculations, the first few of those polynomials (in the usual binary representation where 2^n stands in for x^n) are 3, 7, 25, 425, 101021, 7158330089, 27971386341277386797. But OEIS doesn't know that sequence, so maybe this isn't known?
