Reciprocal Periods and Primality: A period-primality toolbox and the rarity of small-index pseudoprimes

Abstract

Suppose the exact reciprocal period L_b(n) is known for an odd integer n and a base b coprime to n. How much primality information is already contained in that period? This work assembles a period–primality toolbox connecting multiplicative order with Lucas’ criterion, Fermat and strong pseudoprimality, Midy’s property, overpseudoprimes, cyclotomic factors, and N±1 certificates. For the important index d=(n−1)/L, it gives certified small-index consequences: for d≤33, an odd integer with L=(n−1)/d is prime except for an explicit table containing 80 distinct composites, together with an explicit upper bound for every composite spectral point and an asymptotic primality consequence. This deposit contains the 59-page manuscript, LaTeX source, figures, a proof-critical Julia supplement with data, logs and execution report, an exploratory Python script and log, and a manifest.