Solution of Nathanson’s exponential congruence

Abstract

The exponential congruence 5 n ≡ 2 ( mod 3 n ) {5^n} \equiv 2\;\pmod {3^n} has no solution n > 1 n > 1 . This result is proved by using a theorem of van der Poorten to produce an upper bound for the size of such solutions n which is within range of machine verification, and then checking that no n below this bound satisfies the congruence.