Skip to content

Erdős problem 829

Start work

Erdős Problem 829 (open). Let A⊆NA \subseteq \mathbb{N} be the set of perfect cubes. Is it true that (1A∗1A)(n)≪(log⁡n)O(1)(1_A \ast 1_A)(n) \ll (\log n)^{O(1)}? That is, does there exist a natural number CC such that the number of representations of nn as a sum of two cubes is O((log⁡n)C)O((\log n)^C) as n→∞n \to \infty?

Sources

The source's own material, retained exactly.

9 retained statements36c6d0fef71a

Open selected source

FormalConjectures/ErdosProblems/

829.lean

Retained formal statement1 of 9

Erdős Problem 829 (open). Let A⊆NA \subseteq \mathbb{N} be the set of perfect cubes. Is it true that (1A∗1A)(n)≪(log⁡n)O(1)(1_A \ast 1_A)(n) \ll (\log n)^{O(1)}? That is, does there exist a natural number CC such that the number of representations of nn as a sum of two cubes is O((log⁡n)C)O((\log n)^C) as n→∞n \to \infty?

FormalConjectures/ErdosProblems/829.leanErdos829.erdos_8291 lineExact file
True ↔ ∃ C, (fun n => ↑(AdditiveCombinatorics.sumRep Erdos829.cubes n)) =O[Filter.atTop] fun n => Real.log ↑n ^ C
OpenStatement only, no proof