Reciprocal Periods and Primality

Abstract

The paper develops a period-primality toolbox for an odd integer whose exact reciprocal period is known. It connects multiplicative order with Lucas's criterion, Fermat and strong pseudoprimality, Midy's property, cyclotomic factors, and prime-only tests.

It studies the rarity of small reciprocal-period indices, certifies the complete inclusive spectrum through index 33, examines the fixed-base prime spectrum, and supplies reproducible computational certificates, source, figures, data, and execution logs.

Acknowledgements

I must acknowledge the assistance and support of my colleagues: Aliaa Burqan, Samir Brahim Belhaouari, Yunis Carreon Kahalan, and Hatem Bishtawi. They were essential in creating and inspiring many parts of this work. As I come from physics, they were the ones who introduced me to this fascinating topic. The proof of the exceptional pseudoprimes with indices d ≤ 8 is all theirs, and was included with their permission. I am very grateful to all of them, but am especially grateful to Hatem Bishtawi, who I consider to be “William Shanks” of Jordan. He is an engineer and not a professional mathematician, but nevertheless spent 60 years of his life calculating L and d for very big numbers, and recognizing patterns and rediscovering repunit relations and Midy theory, without ever being introduced to modern number theory.