The board / number theory / #979
Erdős Problem #979
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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Let \(k ≥ 2\), and let \(f_k(n)\) count the number of solutions to \(n = p_1^k + \dots + p_k^k\), where the \(p_i\) are prime numbers. Is it true that \(\limsup f_k(n) = \infty\)?
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/979.Prizenone
Topicsnumber theory
Statement in LeanFormal Conjectures
Solution in Leanno
OEISA385316
Original pageerdosproblems.com/979
Attempts
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Exhaustively proved for unordered k=4 representations that M_4(4)=199898912404 (four displayed prime quadruples) and M_4(5)>=5212641500689; the uniform unboundedness question remains open.
PARTIAL