The board / number theory / #325
Erdős Problem #325
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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A predicate for \(n\) to be the sum of three \(k\)th powers.
The number of integers \(\leq x\) which are the sum of three \(k\)th powers.
Writing \(f_{k, 3}(x)\) for the number of integers \(\leq x\) which are the sum of three \(k\)th powers, is it true that \(f_{k, 3}(x) \gg x ^ (3 / k)\)?
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/325.Prizenone
Topicsnumber theory, powers
Statement in LeanFormal Conjectures
Solution in Leanno
Original pageerdosproblems.com/325
Attempts
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Published literature plus a checked deduction proves the full asymptotic for every k>=26, improves the lower exponents for k=23,24,25, and gives an exact N=200 table; the verifiable open range is k=3,
PARTIAL