A Short-Interval Refinement of Brocard's Conjecture at Products of Consecutive Primes

Abstract

Let $p < q$ be consecutive primes with $p \ge 3$. Brocard’s conjecture asserts that at least four primes lie between $p^2$ and $q^2$. We propose a more structured conjectural distribution of these primes by introducing the intermediate points $p(q - 1)$ and $pq$. Our principal conjecture asserts that there is at least one prime in each of $(p^2, p(q - 1))$, $(p(q - 1), pq)$, and at least two primes in $(pq, q^2)$. Thus the four-prime lower bound of Brocard’s conjecture is decomposed as a $1 + 1 + 2$ lower bound across three subintervals. The middle interval has the particularly short length $p$, while $pq \asymp p^2$. We formulate the conjecture in terms of the prime-counting function, compare it with Bertrand’s, Legendre’s, Oppermann’s, Andrica’s, and Brocard’s conjectures, and give elementary implications between them. An exhaustive computation is reported for all relevant consecutive-prime pairs up to $q^2 \lesssim 1.05 \times 10^9$ (extending an initial check at $q^2 \le 10^6$); no counterexample was found in either range. This paper is intended as a conjectural note: no proof of the proposed statements is claimed.