If we had two extra thumbs, how would we check if “2024” is divisible by eleven? Or by “11”? We will see a simple test in any base
$B$, i.e. usable by species having any number of fingers (whether shaped like hot-dogs or not); and for any divisor $d$.
That is, the test works for everything ($d$), everywhere ($B$), all at once.
We will then move to recurring decimals. Note that 1/3 = 0.3333… and 1/3x3 = 0.1111… have the same number of digits - one -
in their recurring parts. (Is 3 the only prime with this property in base 10?) More generally, we will see how many digits $1/d$
has in its recurring “decimal” expansion, for us or for any species as above.
Finally, for a species with a given number of fingers (= digits!), are there infinitely many primes $p$ for which the recurring part
of $1/p$ has $p-1$ digits? (E.g. for us, 1/7 has the decimal recurring string (142857).) And what does this have to do with Gauss,
Fermat, and one of the Bernoullis? Or with Artin and a decimal number starting with 0.3739558136… ? I will end by mentioning why
this infinitude of primes holds for at least one species among humans (10), emus (6), ichthyostega (14), and computers (2) - but,
we don’t know which one!