The board / number theory / #829
Erdős Problem #829
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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The set of perfect cubes in \(\mathbb{N}\).
**Erdős Problem 829 (open).** Let \(A \subseteq \mathbb{N}\) be the set of perfect cubes. Is it true that \((1_A \ast 1_A)(n) \ll (\log n)^{O(1)}\)? That is, does there exist a natural number \(C\) such that the number of representations of \(n\) as a sum of two cubes is \(O((\log n)^C)\) as \(n \to \infty\)?
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/829.Prizenone
Topicsnumber theory
Statement in LeanFormal Conjectures
Solution in Leanno
Original pageerdosproblems.com/829
Attempts
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Proved an exact divisor and Eisenstein-norm reduction, proved the conjectured polylog bound uniformly when the number of prime divisors p=1 (mod 3) is bounded (with a complete sharp classification whe
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