When the Divisor Sum Meets the Fibonacci Clock: The Solutions of sigma(n) = pi(n)

Abstract

Let sigma(n) denote the sum of divisors of n and pi(n) the Pisano period of n (the period of the Fibonacci sequence modulo n). We determine the solutions of sigma(n) = pi(n) below 10^7 and prove the mechanism behind them. The solutions are exactly n = 1, n = 2, the semiprimes pq whose prime factors both attain the maximal Pisano period 2(p+1) with gcd(p+1, q+1) = 2, and the numbers 2pq satisfying the same conditions together with 3 not dividing (p+1)(q+1). We prove the semiprime criterion in both directions unconditionally (Appendix A2), with exhaustive census corroboration below 10^7, classify when the common value sigma(pq) = pi(pq) is a perfect square (precisely when p = 2a^2 - 1 and q = 2b^2 - 1 with gcd(a, b) = 1, the value then being (2ab)^2), and prove that for primes 2k^2 - 1 in the class p == +-2 (mod 5) the observed entry-point parity law is equivalent to period maximality, reducing an apparent empirical mystery to a single open question about the density of maximal-period primes. All computational claims are certified by independently written engines, pairwise sharing no code (five engine lines at the classification's largest bound; four at the 10^9 pool census). A dated novelty search (Appendix C), consisting of exact-term probes of the complete OEIS database and bibliographic sweeps of zbMATH, arXiv, and the open web, found no prior statement or study of the equation or of any of its solution families (one adjacent object, the doubled-prime wing of the reverse-direction spectrum, matches a 2002 OEIS entry, recorded in Section 8). MSC 2020: 11B39 (primary); 11A25, 11B50 (secondary). All computational claims are certified by independently written engines included in the deposit; the evidence manifest (evidence-manifest.txt) lists every file's SHA-256 hash. This work was carried out in partnership with AI research systems (Kimi, Claude, Grok); the discovery of the mechanism is due to Kimi, and the full division of contributions is disclosed in Section 10 of the paper.