Erdős problem 829 · problems.science

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 $n \to \infty$ ?

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FormalConjectures/ErdosProblems/

829.lean

Retained formal statement8 of 9

Mordell proved $\limsup_{n \to \infty} (1_A \ast 1_A)(n) = \infty$ , where $A$ is the set of perfect cubes. Equivalently, the number of representations of $n$ as a sum of two cubes is unbounded.

FormalConjectures/ErdosProblems/829.leanErdos829.variants.mordell1 lineExact file

1Filter.limsup (fun n => ↑(AdditiveCombinatorics.sumRep Erdos829.cubes n)) Filter.atTop = ⊤

SolvedStatement only, no proof

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