1911 Encyclopædia Britannica/Number/The Theorems of Fermat and Wilson

28. The Theorems of Fermat and Wilson.—Let $r_{1},r_{2},\dots r_{t}$ where $t=\phi (m)$, be a complete set of residues prime to the modulus $m$. Then if $x$ is any number prime to $m$, the residues $xr_{1},xr_{2},\dots xr_{t}$ also form a complete set prime to $m$ (§ 27). Consequently $xr_{1}\cdot xr_{2},\dots xr_{t}\equiv r_{1}r_{2}\dots r_{t}$, and dividing by $r_{1}r_{2}\dots r_{t}$, which is prime to the modulus, we infer that

$x^{\phi (m)}\equiv 1{\pmod {m}}$

which is the general statement of Fermat's theorem. If $m$ is a prime $p$, it becomes $x^{p-1}\equiv 1{\pmod {p}}$.

For a prime modulus $p$ there will be among the set $x,2x,3x,\dots (p-l)x$ just one and no more that is congruent to $1$: let this be $xy$. If $y\equiv x$, we must have $x^{2}-1=(x-1)(x+1)\equiv 0$, and hence $x\equiv \pm 1$: consequently the residues $2,3,4,...(p-2)$ can be arranged in ${\tfrac {1}{2}}(p-3)$ pairs $(x,y)$ such that $xy\equiv 1$. Multiplying them all together, we conclude that $2.3.4.\dots (p-2)\equiv l$ and hence, since $1.(p-1)\equiv -1$,

$(p-1)!\equiv -1{\pmod {p}}$.

which is Wilson's theorem. It may be generalized, like that of Fermat, but the result is not very interesting. If $m$ is composite $(m-1)!+1$ cannot be a multiple of $m$: because $m$ will have a prime factor $p$ which is less than $m$, so that $(m-1)!\equiv 0{\pmod {p}}$. Hence Wilson’s theorem is invertible: but it does not supply any practical test to decide whether a given number is prime.