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Statement

Averaging remark (p. 89), called "well-known" in the print: if $\epsilon>0$ and $r\ge r(\epsilon)$, then for any prime $p$ with $x^{1/r}<p<x^{1/(r-1)}$, with the exception of at most $x/c_\epsilon^r$ integers $n\le x$, the exact power $p^\alpha\parallel\binom{2n}{n}$ satisfies $n^{1/2-\epsilon}<p^\alpha<n^{1/2+\epsilon}$, where $c_\epsilon>1$.

Theorem 5 (p. 89). Let $\epsilon>0$ and let $m\le x$ be such that every prime power $p^\alpha$ dividing $m$ satisfies $p^\alpha<x^\epsilon$. Then

$$\Bigl|\Bigl\{n\le x:\ m\nmid\binom{2n}{n}\Bigr\}\Bigr|<\frac{x}{c_7^{1/\epsilon}},$$

where $c_7>1$. The print states no dependence of $c_7$ and no lower bound on $x$.

Unnumbered theorem (p. 89), which the paper says can be proved by the preceding methods: for fixed $p$, the $n\le x$ with $p^\alpha\parallel\binom{2n}{n}$ and $p^\alpha\notin(n^{1/2-\epsilon},n^{1/2+\epsilon})$ number $o(x)$, and the paper adds that this holds for $p=o(x^\eta)$.

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; the averaging remark, Theorem 5 with its proof and the unnumbered theorem on p. 89. The edition is identified on the source card.

Read depth. Claims checked: the three statements were read clause by clause on the page image. The proof of Theorem 5 was read for its structure only and not re-derived; the averaging remark and the unnumbered theorem are stated without proof.

Proof pointer

Page 89. For a prime power $p^\alpha\parallel m$ with $x^{1/(r+1)}<p^\alpha<x^{1/r}$, the averaging remark bounds the $n\le x$ for which $p$ divides $\binom{2n}{n}$ to a power below $n^{1/2-\epsilon}$ by $x/c_\epsilon^r$, and at most $r$ distinct prime powers dividing $m$ lie in that range because their product is at most $x$. When the power of $p$ in $\binom{2n}{n}$ is at least $n^{1/2-\epsilon}$, the prime causes no failure once $n^{1/2-\epsilon}\ge x^\epsilon$, which excludes only $n<x^{2\epsilon/(1-2\epsilon)}$. Summing over $r>1/\epsilon$ gives the bound.

Dependencies

The averaging remark, used without proof.

Bears on

No problem in the corpus asks for this bound. The paper does not apply it to the least non-divisor of Problem 731, which it treats in (8).