Shaped Heights on Pell Unit Orbits: Monotone Dynamics, Depth Recovery, and Sharp Small-Height Bounds

Abstract

Let N = ℓM ≡ 3 (mod 4) be the product of distinct odd primes 3 ≤ ℓ < M, and put a₀ = ⌊√N⌋. We study positive integral solutions of X² − Ny² = σcℓ, X = a₀y + t, where c ∈ {1,2}. A description by the height t alone is not valid: multiplication by the positive fundamental unit preserves the norm and produces infinitely many further positive solutions. Mollin’s work on aX² − bY² = ±1, ±2 supplies the fundamental solution and its unit orbit. We specialize that structure to the half-period generator and show that the shaped height along its orbit is strictly increasing and satisfies a second-order Pell-type recurrence. In the semiprime case, the solutions are thereby coordinatized by an admissible half-period continuant cell and a nonnegative unit depth. We give an exact inverse procedure, prove sharp first-translate bounds 6 and 59, and classify all nonminimal translates of height at most 58.