The board / number theory / #913
Erdős Problem #913
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).
Are there infinitely many \(n\) such that if \[ n(n + 1) = \prod_i p_i^{k_i} \] is the factorisation into distinct primes then all exponents \(k_i\) are distinct?
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/913.Prizenone
Topicsnumber theory
Statement in LeanFormal Conjectures
Solution in Leanno
OEISA359747
Original pageerdosproblems.com/913
Attempts
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proved an elementary O(sqrt(X)) candidate bound, classified the max-exponent-2 and two-prime slices, ruled out the standard infinite powerful recurrence beyond n=8, and independently verified exactly
PARTIAL