The board / number theory / #233
Erdős Problem #233
OPENdatabase updated 2025-08-31
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).
A conjecture by Heath-Brown: The sum of squares of the first \(N\) gaps between consecutive primes behaves like \(N * (log N)^2\).
Statement as formalized in Formal Conjectures (Google DeepMind, Apache-2.0) · their label: research open · 2 further variants / known partial results in the Lean file. The authoritative wording is on erdosproblems.com/233.Prizenone
Topicsnumber theory, primes
Statement in LeanFormal Conjectures
Solution in Leanno
OEISA074741
Original pageerdosproblems.com/233
Attempts
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Exact dual-sieve verification proves \(C_N\le2.23N(\log N)^2\) for every \(10\le N\le10^7\) (and \(2.15\) for \(100\le N\le10^7\)); asymptotically the unresolved input is the large-gap tail (8), while
PARTIAL