The Jacobi-positive squarefree odd semiprimes: a complete parametrisation

Abstract

Let N = ℓM with 3 ≤ ℓ < M distinct odd primes and let 𝒥(N) be the set of denominators Q_k in the period of the continued fraction of √N whose Jacobi symbol (Q_k/N) equals +1. A previous paper (DOI 10.5281/zenodo.21285697) proved that 𝒥(N) = {1} forces the period length L to lie in {1,2,4} for ℓ ≥ 5 and in {2,4,6} for ℓ = 3, and deferred the exact converse to a companion note. This is that note. We show that the set of N with 𝒥(N) = {1} is the disjoint union of five explicitly parametrised families: N = a₀²+1, N = a₀²+2, N = ℓ(ℓr²+c) with L = 2, N = ℓ(ℓr²−c)/y² with L = 4, and the exceptional family N = 81v²−30v+3 at ℓ = 3, L = 6, where c ∈ {1,2}. The parameters are read off the expansion, so the parametrisation is a bijection. The new ingredient is a trichotomy Q₁ ∈ {2, ℓ, 2ℓ} at the first step of the expansion whenever L = 2, an exact analogue of the central trichotomy of the previous paper. Two theorems of that paper become corollaries, the products of twin primes are identified as one member of the L = 2 family, and the classification yields a decision procedure for 𝒥(N) = {1} that is independent of the (unbounded) period length. All proofs are elementary. The classification has been verified exhaustively against direct computation for all N ≤ 5×10⁶.