FormalConjectures.ErdosProblems.«175»

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import FormalConjecturesUtil

Erdős Problem 175

References:

erdosproblems.com/175

[Sa85] A. Sárközy, On divisors of binomial coefficients, I, J. Number Theory 20 (1985), 70–80.

[GrRa96] A. Granville and O. Ramaré, Explicit bounds on exponential sums and the scarcity of squarefree binomial coefficients, Mathematika 43 (1996), 73–107.

[Ve95] G. Velammal, Is the binomial coefficient $\binom{2n}{n}$ square free?, Hardy-Ramanujan Journal 18 (1995), 23–45.

Sárközy proved the assertion for all sufficiently large n; Granville--Ramaré and Velammal independently proved the full range n ≥ 5.

namespace Erdos175

Show that, for any $n\geq 5$, the binomial coefficient $\binom{2n}{n}$ is not squarefree.

@[category research solved, AMS 11]
theorem erdos_175 (n : ℕ) (hn : 5 ≤ n) :
    ¬ Squarefree ((2 * n).choose n) := byn:ℕhn:5 ≤ n⊢ ¬Squarefree ((2 * n).choose n)
  sorryAll goals completed! 🐙

-- TODO: Formalise the related questions and results listed in the additional material.

end Erdos175