*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.

## 2 comments:

happy belated birthday! with many more to come!

And a belated thank-you! I had a lovely time with family this past week; am now safely in my academic-time private residence again. Since you must have had a birthday since the last time I wrote one of these, a Happy Birthday to you, too! (And the same to anyone else who drops by! I'm feeling silly)

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