An Idempotent Cryptarithm

Abstract

Summary 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 , there exists at least one and at most two positive integers x with exactly n-digits in base 10 (meaning the leftmost or nth digit from the right is nonzero) 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 with exactly m distinct prime factors, an upper bound is and a lower bound is for the number of such n-digit positive integers. For , there are exactly solutions, including 1 and excluding 0.