---

Statement

Let $A(n)$ be the least positive integer that does not divide $\binom{2n}{n}$. The paper notes that $A(n)$ is always a prime power (p. 91).

Inequality (8) (p. 91). The paper states ("It is not hard to show") that, except for a set of $n$ of density $0$,

$$\exp\bigl((\log n)^{1/2-\epsilon}\bigr)<A(n)<\exp\bigl((\log n)^{1/2+\epsilon}\bigr).$$

The print does not quantify $\epsilon$; the corpus reads (8) as holding for every fixed $\epsilon>0$, the exceptional set depending on $\epsilon$. No proof is given.

The paper adds that sharper results than (8) would not be difficult, "but an asymptotic formula seems hard" (p. 91), and tabulates $A(n)$ for $1\le n\le100$ (Table I, p. 91).

Source. P. Erdős, R. L. Graham, I. Z. Ruzsa and E. G. Straus, On the prime factors of $\binom{2n}{n}$, Math. Comp. 29 (1975), no. 129, 83--92; (8), the remarks and Table I on p. 91. The edition is identified on the source card.

Read depth. Claims checked: the definition, (8) and the remarks were read clause by clause on the page image. The paper gives no proof; the table was not recomputed here.

Proof pointer

None in the paper.

Dependencies

None stated.

Bears on

• Problem 731: the problem asks for a reasonable $f$ with $A(n)\sim f(n)$ for almost all $n$, the asymptotic formula the paper says "seems hard". (8) locates $A(n)$ only up to the exponent $1/2\pm\epsilon$ and gives no such $f$.