Infinitely many cyclotomic nearly-doubly-regular tournaments

Establish that there exist infinitely many primes p congruent to 5 modulo 8 for which the cyclotomic tournament CT_p is nearly-doubly-regular.

Background

For primes p congruent to 5 modulo 8, the cyclotomic tournament CT_p is defined using the index-four cyclotomic classes of the finite field F_p. Nearly-doubly-regularity is a strong local regularity property that is known for only a small number of such tournaments, including orders 5, 13, 29, 53, 173, 229, 293, and 733.

The paper proves that CT_q is nearly-doubly-regular precisely when q=s²+4 for an odd integer s. Consequently, the infinitude assertion would follow from the existence of infinitely many primes of the form s²+4; the paper obtains this implication conditionally through the Hardy–Littlewood conjecture F. The underlying infinitude statement remains conjectural.

References

On the other hand, for the case that ℓ = 2, it has been conjectured ( [3], [19]) that there exist infinitely many primes p ≡ 5 (mod 8) such that CTp is nearly-doubly-regular while the authors of [3] and [24] verified that CTp is nearly-doubly-regular when p = 5, 13, 29, 53, 173, 229, 293, 733 by computer search.

— On cyclotomic nearly-doubly-regular tournaments  (2502.12090 - Satake, 17 Feb 2025) in Section 1, p. 1