This paper formulates an open problem regarding the local distribution of prime numbers constrained by the left-edge geometry of the Ulam spiral. By establishing a 0-indexed bijective mapping from the integer lattice Z² to the non-negative integers N₀, we isolate the arithmetic sequence governing this boundary. The proposed hypothesis asserts that for every spiral level n 2, the open interval (4n², 4n²+n) contains at least one prime. We demonstrate that this formulation corresponds to an exact algebraic bisection of Oppermann's interval for even perfect squares. Assuming the conjecture yields immediate structural corollaries, including an elementary localized resolution of Legendre's conjecture, a linear lower bound for prime counts within Brocard's intervals, and the geometric confinement of prime square root fractional parts strictly below 0. 25. We contextualize this problem within current analytic limitations, noting that an unconditional proof is presently obstructed by the O (X^0. 525) zero-density barrier and the combinatorial parity problem. However, by transitioning to an algebraic evaluation of the boundary, we identify a deterministic micro-segmentation of the interval. We show via polynomial discriminants that the boundary inherently factorizes at coordinates y = 4t² t, constraining potential primes into micro-bands of maximum width O (X^1/4). Furthermore, applying the Law of Quadratic Reciprocity reveals that the extrema of the left boundary strictly avoid divisibility by the first four odd primes (3, 5, 7, 11). Supported by these local algebraic constraints, heuristic variance expectations, and exhaustive empirical verification up to n = 2 10⁹, we present this geometry-derived bound as an open problem.
Huynh Hai Dang Vo (Tue,) studied this question.