Infinitely many primes represented by s² + 4

Prove that there exist infinitely many positive integers s such that s²+4 is prime, equivalently establish the relevant instance of the Hardy–Littlewood conjecture F for the quadratic polynomial s²+4.

Background

The paper invokes the Hardy–Littlewood conjecture F, which predicts infinitely many prime values for quadratic polynomials satisfying specified local and nonsquare-discriminant conditions. The polynomial s²+4 satisfies those hypotheses.

Applying this conjecture to s²+4 would yield infinitely many primes p congruent to 5 modulo 8 with p=s²+4. By Theorem 1, each such prime produces a nearly-doubly-regular cyclotomic tournament CT_p, and by Theorem 2 these tournaments have the canonical spectrum. Thus this number-theoretic conjecture supplies the unresolved infinitude needed for the corresponding tournament construction.

References

Conjecture 3 (The Hardy-Littlewood conjecture F, [13]). Suppose that a > 0, b, c are integers such that (a, b, c) = 1, either a + b or c is odd, and b2 − 4ac is not a square. Then there exist infinitely many positive integers m such that am2 + bm + c is a prime.

— On cyclotomic nearly-doubly-regular tournaments  (2502.12090 - Satake, 17 Feb 2025) in Section 1, p. 2