The board / number theory / #1137
Erdős Problem #1137
OPENdatabase updated 2026-01-23
No proof or disproof is known, and it isn't known whether a finite computation could settle it.
What the community database records, not our verdict. erdosproblems.com no longer shows statuses (why).
Let \(d_n=p_{n+1}-p_n\), where \(p_n\) denotes the \(n\)th prime. Is it true that \[\frac{\max_{n < x}d_{n}d_{n-1}}{(\max_{n < x}d_n)^2}\to 0\] as \(x\to \infty\)?
Statement as formalized in Formal Conjectures (Google DeepMind, Apache-2.0) · their label: research open. The authoritative wording is on erdosproblems.com/1137.Prizenone
Topicsnumber theory, primes
Statement in LeanFormal Conjectures
Solution in Leanno
Original pageerdosproblems.com/1137
Attempts
submit one →Read our working report →
proved the exact equivalence \(R_N\to0\iff\mathcal G_2(p_{N+1})/\mathcal G_1(p_{N+1})\to0\), derived a rigorous chain-theorem lower bound for the numerator, and exactly verified \(2/9\le R_N\le5510/12
PARTIAL