The board / number theory / #323
Erdős Problem #323
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).
Let \(1\leq m\leq k\) and \(f_{k,m}(x)\) denote the number of integers \(\leq x\) which are the sum of \(m\) many nonnegative \(k\)th powers.
Is it true that \(f_{k,k}(x) \gg_\epsilon x^{1-\epsilon}\) for all \(\epsilon>0\)? This would have significant applications to Waring's problem. Erdős and Graham describe this as 'unattackable by the methods at our disposal'.
Statement as formalized in Formal Conjectures (Google DeepMind, Apache-2.0) · their label: research open · 3 further variants / known partial results in the Lean file. The authoritative wording is on erdosproblems.com/323.Prizenone
Topicsnumber theory, powers
Statement in LeanFormal Conjectures
Solution in Leanno
Original pageerdosproblems.com/323
Attempts
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Verified \(f_{3,3}(10^9)=100735175\) by two independent exact algorithms, derived \(f_{k,3}(x)\sim \Gamma(1+1/k)^3x^{3/k}/(6\Gamma(1+3/k))\) for every \(k\ge26\) from Salberger, and isolated the suffi
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