Divisibility in Two-Cycles of Generalized Subprime Recurrences

Ponomarenko, Rhinehart, Schwarz, and Tong introduced generalized subprime recurrences and conjectured that the two entries of every two-cycle are comparable by divisibility. We prove this divisibility assertion whenever the first recurrence coefficient is nonzero. The argument also shows that the second coefficient need not be nonzero, slightly extending the coefficient range of the standing convention used in the original two-cycle analysis. This is Version 1.0 of the preprint. The manuscript includes a detailed generative-AI disclosure. It has not undergone independent peer review.