Let k (n) denote the smallest positive integer such that (n+1) ² − k (n) is prime. We introduce and numerically investigate two conjectures concerning the arithmetic structure of k (n). The first conjecture states that consecutive gaps differ by an odd integer: k (n+1) − k (n) ≡ 1 (mod 2) for all n ≥ 2. The second, and main conjecture, asserts a logarithmic upper bound: k (n) < 2ln (n+1) ² for all n ≥ 2. Both conjectures are verified for all n ≤ 10⁶. We additionally prove a structural theorem fully characterizing which residue classes modulo 6 are admissible for k (n), depending only on (n+1) mod 6. Unlike Legendre's and Oppermann's conjectures, which assert the existence of primes near perfect squares without quantifying the gap, the main conjecture provides an explicit window of size 2ln (n+1) ² below each perfect square that is guaranteed to contain a prime. Numerical evidence suggests that sup₍≥₂ k (n) /2ln (n+1) ² ≈ 0. 9353, attained at n = 28, 633
Judicael Brindel (Sun,) studied this question.