The board / number theory / #399
Erdős Problem #399
DISPROVED LEANdatabase updated 2026-02-04
The statement as posed is false. A solution has also been checked by machine, in Lean.
What the community database records, not our verdict. erdosproblems.com no longer shows statuses (why).
Is it true that there are no solutions to `n! = x^k ± y^k` with `x,y,n ∈ ℕ`, `x*y > 1`, and `k > 2`? The answer is no: Jonas Barfield found the counterexample `10! = 48^4 - 36^4` (equivalently, `10! + 36^4 = 48^4`). This is discussed in problem D2 of Guy's collection [Gu04]. This was formalized in Lean by Lu using Codex.
Statement as formalized in Formal Conjectures (Google DeepMind, Apache-2.0) · their label: research solved · 4 further variants / known partial results in the Lean file. The authoritative wording is on erdosproblems.com/399.Prizenone
Topicsnumber theory, factorials
Statement in LeanFormal Conjectures
Solution in Leanyes
Original pageerdosproblems.com/399
Attempts
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