Exceptional prime in the unresolved p ≡ 3 mod 8 case

Determine whether an exceptional odd prime exists with p ≡ 3 (mod 8), p=a^2+2b^2 for a positive odd integer a and a prime b=2a±1, such that p-x^2 has at most two different prime factors for every odd integer x with x^2<p.

Background

The paper completely classifies the primes satisfying the stated additive condition in the congruence classes p ≡ 7, 5, and 1 (mod 8), and gives a classification in the class p ≡ 3 (mod 8) subject to one possible exception. For p ≡ 3 (mod 8), the relevant representation is p=a2+2b2, where b is either 1 or a prime equal to 2a±1. The b=1 subcase is solved, while the prime-b subcase is reduced using class-number and continued-fraction arguments to the listed values, with the possibility that one additional value remains.

References

For p\equiv 3 \pmod 8 the class number of $\mathbb{Q}(\sqrt {p})$ is equal to 1. There are two possibilities. For $p=a2+2$, we solve completely the problem (using again the result of Lee). The other possibility is $p=a2+2b2$, where $b$ is a prime number which equals $2a\pm 1$. This case is not completely solved.

— An additive problem in connection with some quadratic real fields with class number one  (2609.16960 - Gica, 15 Sep 2026) in Section 1, Introduction