On the Gap Between Perfect Squares and the Nearest Prime Below: A Logarithmic Bound Conjecture

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 supn≥2 k(n)/2ln(n+1)² ≈ 0.9353, attained at n = 28,633