An Idempotent Cryptarithm

TL;DR

This paper explores special integers called idempotent cryptarithms, proving their existence and bounds in various bases, and generalizing the concept of numbers whose squares end with the number itself.

Contribution

It introduces a general framework for understanding numbers with specific square-end properties across different bases and prime factor counts.

Findings

For each n ≥ 2, there are 1 or 2 such integers with n digits.

In base B with m prime factors, bounds are 2^{m-1}-1 and 2^m - 2.

Exactly 2^m - 1 solutions for single-digit case.

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 $0$.