Erdős problem 126 · problems.science

Let $f(n)$ be maximal such that if $A\subseteq\mathbb{N}$ has $|A| = n$ then $\prod_{a\neq b\in A}(a + b)$ has at least $f(n)$ distinct prime factors. Is it true that $\frac{f(n)}{\log n} \to\infty$ ?

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FormalConjectures/ErdosProblems/

126.lean

Retained formal statement2 of 3

Erdős and Turán proved [ErTu34] in their first joint paper that $\log n \ll f(n) \ll \frac{n}{\log n}$

[ErTu34] Erdős, Paul and Turan, Paul, _On a Problem in the Elementary Theory of Numbers_. Amer. Math. Monthly (1934), 608-611.

FormalConjectures/ErdosProblems/126.leanErdos126.erdos_126.variants.IsBigO4 linesExact file

1∀ (f : ℕ → ℕ),2  Erdos126.IsMaximalAddFactorsCard f →3    ((fun n => Real.log ↑n) =O[Filter.atTop] fun n => ↑(f n)) ∧4      (fun n => ↑(f n)) =O[Filter.atTop] fun n => ↑n / Real.log ↑n

SolvedStatement only, no proof

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