Prime fractals: palindromes, primes and base-10 Euler pseudoprimes

Abstract

The prime fractal of N/q pairs the first q − 1 decimal digits of N/q around the middle and records their differences. We show that it is a palindrome for every N coprime to q exactly when q is prime, a base-10 Euler pseudoprime, or 99; equivalently, when 10^((q−1)/2) ≡ ±1 (mod q), with the single exception q = 99. For primes the palindrome is also self-similar: a chain of copies of one smaller palindrome, at every scale. Includes the LaTeX source and the Python scripts for the computer-checked steps.