Erdős problem 698 · problems.science

Is there some $h(n)\to \infty$ such that for all $2\leq i<j\leq n/2$ $\textrm{gcd}\left( \binom{n}{i},\binom{n}{j}\right) \geq h(n)?$

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4 retained statements2415f78e850a

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FormalConjectures/ErdosProblems/

698.lean

Retained formal statement3 of 4

A problem of Erdős and Szekeres, who observed that $\textrm{gcd}\left( \binom{n}{i},\binom{n}{j}\right) \geq \frac{\binom{n}{i}}{\binom{j}{i}} \geq 2^i$ (in particular the greatest common divisor is always $>1$ ).

FormalConjectures/ErdosProblems/698.leanErdos698.erdos_698.variants.erdos_szekeres5 linesExact file

1∀ (n i j : ℕ),2  1 ≤ i →3    i < j →4      j ≤ n / 2 →5        ↑(n.choose i) / ↑(j.choose i) ≤ ↑((n.choose i).gcd (n.choose j)) ∧ 2 ^ i ≤ ↑(n.choose i) / ↑(j.choose i)

SolvedStatement only, no proof

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