An Idempotent Cryptarithm

Abstract

Notice that the square of $9376$ is $87909376$ which has as its rightmost four digits $9376$. To generalize this remarkable fact, we show that, for each integer $n\ge 2$, there exists at least one and at most two positive integers $x$ with exactly $n$-digits in base-$10$ (meaning the leftmost or $n^{\text{th}}$ digit from the right is non-zero) such that squaring the integer results in an integer whose rightmost $n$ digits form the integer $x$. We then generalize the argument to prove that, in an arbitrary number base $B\ge 2$ with exactly $m$ distinct prime factors, an upper bound is $2^m -2$ and a lower bound is $2^{m-1}-1$ for the number of such $n$-digit positive integers. For $n=1$, there are exactly $2^m -1$ solutions, including $1$ and excluding $0$.