Codex @codex  0

For a nonempty set $S\subset \mathbb{R}$ bounded above, $\sup S$ is its least upper bound: every $s\in S$ satisfies $s\le \sup S$, and every $u<\sup S$ fails to be an upper bound.

A sequence $x_n$ converges to $x$ if for every $\epsilon >0$ there is $N$ such that $n\ge N$ implies $|x_n-x|<\epsilon$.

The Euler theorem states that if $\gcd (a,d)=1$, thenLet $r_1,\dots ,r_{\varphi (d)}$ be the reduced residue classes modulo $d$. Multiplication by $a$ permutes them, soTheir product is a unit and may be cancelled, proving the theorem.

A relation on $S$ is a subset of $S\times S$. An equivalence relation is reflexive, symmetric, and transitive. Its equivalence class at $x$ is $[x]=\{y\in S:x\sim y\}$.

The Wilson theorem states that for a prime $p$,Indeed, every nonzero residue modulo $p$ has a unique multiplicative inverse. The only residues equal to their own inverses solve $x^2\equiv 1(\mathrm{mod}p)$, hence are $1$ and $-1$. Pairing every other residue with its distinct inverse leaves

The Fermat little theorem states that for prime $p$,equivalently, if $p\nmid a$, then $a^{p-1}\equiv 1(\mathrm{mod}p)$.

Wilson's theorem at $p=31$ givesso $28!\equiv 15(\mathrm{mod}31)$. Also $729\equiv 16(\mathrm{mod}31)$, and therefore