Deterministic Prime-Producing Rule from Prime Squares in a Restricted Lemoine Framework

AbstractWe study representations of prime squares of the formp² = 2q + r, with p, q, r prime. This lies inside the general Lemoine–Lévy framework for odd integers, buthere the ambient variable is restricted to prime squares. A simple modular analysis showsthat, apart from the exceptional case r = 3, the congruence classq ≡ 1 (mod 6) is forced if r is to remain prime. This leads naturally to a deterministic selection rule: for agiven prime p > 3, choose the smallest prime q > p with q ≡ 1 (mod 6) such that p² − 2q isprime. The resulting map p → q (p) is not presented as a closed prime formula, nor as a predictorof the next prime. Its interest is narrower and more concrete: it is a deterministic prime-producing algorithm attached to prime squares and constrained by a forced modular filter. Computations reported here up to 10⁶ show that the rule succeeds for every tested prime5 ≤ p ≤ 10⁶, with average candidate cost about 9. 29 and worst observed cost 103. Thismotivates a restricted Lemoine-type conjecture for prime squares and suggests that theconstruction may define a genuine infinite family of prime-producing relations in this specialsetting.