The board / number theory / #852
Erdős Problem #852
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).
The \(H\) prime gaps \(d_n, d_{n+1}, \ldots, d_{n+H-1}\) are pairwise distinct. Here \(d_n = p_{n+1} - p_n\) is `primeGap n`, so the primes are indexed from \(p_0 = 2\). The empty run \(H = 0\) and every run with \(H = 1\) are distinct.
Let \(d_n = p_{n+1} - p_n\), where \(p_n\) is the \(n\)th prime. Let \(h(x)\) be maximal such that for some \(n < x\) the numbers \(d_n, d_{n+1}, \ldots, d_{n+h(x)-1}\) are all distinct [Er85c]. Estimate \(h(x)\). In particular, is it true that \[h(x) > (\log x)^c\] for some constant \(c > 0\)? The statement avoids defining \(h\). Since \(h(x)\) is the longest such run, \(h(x) > y\) holds exactly when some \(n < x\) starts a run of \(H > y\) distinct gaps. We state the question this way, with \(x\) real. We read it as asking for one \(c > 0\) such that the inequality holds for all large \(x\). The primes are indexed from \(p_0 = 2\), as in `primeGap`. Shifting the index by one does not change the question.
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/852.Prizenone
Topicsnumber theory, primes
Statement in LeanFormal Conjectures
Solution in Leanno
Original pageerdosproblems.com/852
Attempts
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Verified the repaired \(h(x)\gg(\log x)^{1/3}\) and \(h(x)\ll_\varepsilon x^{0.41+\varepsilon}\) reductions, computed the exact frontier through 105,000,000 starts, and found \(R(14)=R(15)=19,205\); t
PARTIAL