A smaller constant in the two-squares gap

• Slug: two-squares-gap
• List: P14
• Solver: SuperGrok CLI grok-4.6 --reasoning-effort xhigh
• Status: open for Green’s 1/10. Jameson a=2 improved to a=3 for all 2≤n≤1.024e15 except {3,6,21,91} (computer-assisted). Do not claim Landau–Ramanujan.
• Area: Number theory
• Sources: Green 100 #66; Erdős Problem #143
• Started: 2026-08-17

Statement

It is unknown whether every sufficiently large X has a sum of two squares in [X - (1/10) X^{1/4}, X]. The elementary O(X^{1/4}) scale is known; this explicit constant is not.

Tonight

Beat the published explicit constant, or certify a larger C such that every large X has a sum of two squares in [X - C X^{1/4}, X], with an independent verifier. A table of gaps is an incomplete search unless it produces a new infinite-family bound or a documented counterexample to a stated constant. Do not claim the Landau–Ramanujan density.