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Erdős Problem #382

OPENdatabase updated 2025-08-31

No proof or disproof is known, and it isn't known whether a finite computation could settle it.

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`LargestPrimeRepeated u v` means that \(1 \leq u \leq v\) and the largest prime \(p\) dividing \(\prod_{u \leq m \leq v} m\) appears in it with exponent at least \(2\), that is \(p^2\) divides the product. We require \(u \geq 1\), since for \(u = 0\) the product is \(0\) and `Nat.maxPrimeFac 0 = 0`. We also require \(v \geq 2\): the only other pair is \(u = v = 1\), where the product is \(1\), which has no prime factor, and the junk value `Nat.maxPrimeFac 1 = 1` would make the condition hold.

Let \(u \leq v\) be such that the largest prime dividing \(\prod_{u \leq m \leq v} m\) appears with exponent at least \(2\). Is it true that \(v - u = v^{o(1)}\)? We read this as: for every \(\epsilon > 0\), every such pair with \(v\) large enough satisfies \(v - u \leq v^{\epsilon}\). Cambie observed that this question reduces to old conjectures on gaps between primes; for example, it follows from Cramér's conjecture. See also Erdős Problems 380 and 383.
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/382.
Prizenone
Topicsnumber theory
Statement in LeanFormal Conjectures
Solution in Leanno
OEISA388850
Original pageerdosproblems.com/382

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Explicit bad intervals are rigorously certified through width 13; exact first starts are computed for widths 0–8, and an audited exhaustive search gives \(999999992\leq U_9\leq4928180396\), but neithe

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