Table of congruences

 $2^{n-2} \equiv 1 \pmod{n}\,\!$ satisfied by infinitely many positive integers n[1][2][3][4] $2^{p-1} \equiv 1 \pmod{p}\,\!$ special case of Fermat's little theorem, satisfied by all odd prime numbers $2^{p-1} \equiv 1 \pmod{p^2}\,\!$ solutions are called Wieferich primes $(n-1)!\ \equiv\ -1 \pmod n$ by Wilson's theorem a natural number n is prime if and only if it satisfies this congruence $(p-1)!\ \equiv\ -1 \pmod{p^2}$ solutions are called Wilson primes ${2n-1 \choose n-1} \equiv 1 \pmod{n^3}$ by Wolstenholme's theorem satisfied by all prime numbers greater than 3 ${2p-1 \choose p-1} \equiv 1 \pmod{p^4},$ solutions are called Wolstenholme primes ${2p-1 \choose p-1} \equiv 1 \pmod{p^6},$ must be satisfied by a counterexample to the converse of Wolstenholme's theorem[5] $F_{p - \left(\frac{{p}}{{5}}\right)} \equiv 0 \pmod{p}$ satisfied by all odd prime numbers $F_{p - \left(\frac{{p}}{{5}}\right)} \equiv 0 \pmod{p^2}$ solutions are called Wall–Sun–Sun primes