Palindromes in Different Bases: A Conjecture of J. Ernest Wilkins

TL;DR

This paper investigates integers that are palindromes in base 10 and another base, identifying exactly 203 such numbers within a large range, thus exploring a special number-theoretic property.

Contribution

The paper precisely characterizes and counts all integers that are palindromes in both base 10 and another base, confirming a conjecture related to these special numbers.

Findings

Exactly 203 such integers exist within the specified range.

These integers are palindromes in base 10 and another base, with the same number of digits.

The range of these integers extends from 22 to 9986831781362631871386899.

Abstract

We show that there exist exactly 203 positive integers $N$ such that for some integer $d \geq 2$ this number is a $d$-digit palindrome base 10 as well as a $d$-digit palindrome for some base $b$ different from 10. To be more precise, such $N$ range from 22 to 9986831781362631871386899.