Let $C\_n=\frac1{n+1}\binom{2n}n$ be the well-known Catalan numbers. Here is a curiosity.



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> **QUESTION.** Are there infinitely many $C\_n$ that are "radical", i.e. that are square-free?
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"Radical" Catalan numbers?
If $s(n)$ is the largest integer such that $s(n)^2 \mid \binom{2n}{n}$, then the main result of Sárközy, A. (1985). On divisors of binomial coefficients, I. Journal of Number Theory, 20(1), 70-80 is



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> If $\varepsilon > 0$, $n > n\_0(\varepsilon)$ then we have $$e^{(c-\varepsilon) \sqrt{n}} < s(n)^2 < e^{(c+\varepsilon) \sqrt{n}}$$ where $$c = \sqrt{2} \sum\_{k=1}^\infty \frac{1}{\sqrt{2k-1}} - \frac{1}{\sqrt{2k}}$$
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Numerically, $c \approx 0.855$.


Define $s'(n) = \frac{s(n)}{\textrm{gcd}(s(n), n+1)}$. Then $s'(n)^2 \mid C\_n$. Noting that $e^{0.81 \sqrt{x}} > x + 1$ for all $x > 0$, $C\_n$ cannot be square-free for any $n > n\_0(c - 0.81)$, whence the number of square-free $C\_n$ is finite.


