## Friday, August 22, 2014

Hm... 32, eh? That's $2\times 2 \times 2 \times 2\times 2$. Just for amusement, $3 \times 3 \times 3 \times 3 \times 3$ is 243, whereas Abraham himself only lived to 170. Two is a very odd prime, you know, the way its powers pack so closely together that way and other things...

There's a nifty thing about primes — from Fermat's little theorem $$n^p \equiv n \pmod{p} \tag{Fermat}$$ we have a factorization $$x^p - x \equiv x (x+1) (x+2) \cdots (x+p-1) \pmod{p}$$ which in particular gives $$(p-1)! \equiv -1 \pmod{p} ;$$ on the other hand, if $q$ is a composite number, then $q = p N$ for some minimal prime $p$ and some $N$ which is not smaller than $p$. That is, either $p = 2$ and $N =2$, or $2 \lt N$ ; in the first case, $p N = 2 \times 2 = 4$ and $3 ! = 6 \equiv 2 \pmod{4}$; in all other cases, we have one of $p \lt N$ or $2 \lt p = N \lt 2 p \lt 3 p \leq q$, both of which lead to $q = p N | ( q - 1 ) !$, so that the full repertoire of $( q-1) ! \pmod{q}$ is : $-1$, if $q$ is prime ; $2$, if $q = 4$; $0$ otherwise. The odd case out, $q=4$, highlights in a number-theory way just how odd the thickness of powers of $2$ really is. It also arises as a thing in my research, the natural operations in homotopy... but never mind that for now! It's my Birthday, and I think I'll have a sleep.