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 statement9 of 9

Stewart improved Mahler's lower bound to $(1_A \ast 1_A)(n) \gg (\log n)^{11/13}$ for infinitely many $n$ , where $A$ is the set of perfect cubes.

[St08] Stewart, C. L., _Cubic Thue equations with many solutions_. Int. Math. Res. Not. IMRN (2008), Art. ID rnn040, 11.

FormalConjectures/ErdosProblems/829.leanErdos829.variants.stewart1 lineExact file

1∃ C > 0, ∃ᶠ (n : ℕ) in Filter.atTop, C * Real.log ↑n ^ (11 / 13) ≤ ↑(AdditiveCombinatorics.sumRep Erdos829.cubes n)

SolvedStatement only, no proof

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