A published paper or arXiv preprint answers the main question of Erdős Problem #689 (theorem erdos_689 in formal-conjectures) in full (proof, disproof, or the exact value asked for) with no hypotheses beyond the problem's own.

The problem, as stated in formal-conjectures (Apache 2.0):

Let `n` be sufficiently large. Is there some choice of congruence class `a_p` for all primes `2 ≤ p ≤ n` such that every integer in `[1,n]` satisfies at least two of the congruences `≡ a_p (mod p)`? Yes: this follows from the greedy-matching reduction of Chojecki [Ch26] combined with an unconditional Fourier-analytic proof of the required three-prime counting estimate.

References formal-conjectures lists for this problem (a starting point, not evidence):

• Ben Green's Open Problem 45
• [Ch26] Chojecki, P., A greedy matching proof of Erdős's two-fold residue-class problem. ulam.ai/research/erdos689.pdf (April 2026).
• PALOMAR-2026-09-20-000002