The board / number theory / #322
Erdős Problem #322
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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For `k ≥ 3`, the number of ordered representations of `n` as a sum of `k` many `k`th powers of positive integers. The bases can be restricted to the interval from `1` to `n`, since `x ≤ x ^ k` for positive `x`.
Let \(k\geq 3\) and \(A\subset \mathbb{N}\) be the set of \(k\)th powers. What is the order of growth of \(1_A^{(k)}(n)\), i.e. the number of representations of \(n\) as the sum of \(k\) many \(k\)th powers? Does there exist some \(c>0\) and infinitely many \(n\) such that \[1_A^{(k)}(n) >n^c?\]
Statement as formalized in Formal Conjectures (Google DeepMind, Apache-2.0) · their label: research open. The authoritative wording is on erdosproblems.com/322.Prizenone
Topicsnumber theory, powers
Statement in LeanFormal Conjectures
Solution in Leanno
Original pageerdosproblems.com/322
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