Distribution of the period-8 prime-valued survivor sequence

Prove or disprove the conjecture that, for the pure periodic profile F^(8), the counting function A_8(x) of survivor values up to x satisfies A_8(x) asymptotic to C_8 x/(log x)^2 for some constant C_8>0.

Background

The paper constructs a pure periodic residue profile F8 whose survivors, apart from finitely many initial terms, are primes q satisfying q−1=2αp for α in {1,2}, with p also prime and p congruent to 5 modulo 8. This is an exact structural characterization, not an asymptotic result.

The authors define A_8(x) to count these survivor values up to x and explicitly formulate a distributional conjecture predicting an asymptotic of order x/(log x)2 with a positive constant. They state immediately afterward that the conjecture is not proved in the paper.

References

There exists a constant $C_8>0$ such that

A_8(x)\sim C_8\frac{x}{(\log x)2} \qquad (x\to\infty).

This conjecture is not proved in the present paper.

— Counting Survivor Sets: Exponential Equivalence with Prime-Admissible Sets  (2609.08528 - Raso et al., 8 Sep 2026) in Appendix, Section “Distributional conjectures,” Conjecture 10.1 (labelled conj:mod8-density)