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Statement

Theorem 1 (p. 84). Let $p,q>1$ be integers and let $A,B$ be positive integers with

$$\frac{A}{p-1}+\frac{B}{q-1}\ge1 .$$

Then infinitely many integers have every digit of their base-$p$ expansion at most $A$ and every digit of their base-$q$ expansion at most $B$.

The digit criterion (1) (p. 84), stated as a "Fact" for a prime $p$: $\binom{2n}{n}\not\equiv0\pmod p$ if and only if every digit $a_k$ of the base-$p$ expansion $n=\sum_{k\ge0}a_kp^k$, $0\le a_k<p$, satisfies $a_k<p/2$.

Consequence (p. 84). The paper states that the result that "for any two primes $p$, $q$" there are infinitely many $n$ with $\bigl(\binom{2n}{n},pq\bigr)=1$ is a special case of Theorem 1. For odd primes it is the case $A=(p-1)/2$, $B=(q-1)/2$, where the hypothesis holds with equality and, by (1), digits at most $(p-1)/2$ are exactly the digits below $p/2$. The prime $2$ is excluded: the introduction (p. 83) notes that $2$ always divides $\binom{2n}{n}$, and $A=(p-1)/2$ is then not a positive integer.

Open as printed (p. 86). The authors cannot decide whether the hypotheses of Theorem 1 can be weakened, or whether similar results hold for three or more bases instead of two, and suggest "perhaps a new idea will be needed".

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 criterion (1) and Theorem 1 on p. 84, the proof on pp. 84--86, the open questions on p. 86. The edition is identified on the source card.

Read depth. Claims checked: the statement, the criterion (1), the consequence and the remark on p. 86 were read clause by clause on the page images. The proof was read for its structure only and not re-derived.

Proof pointer

Pages 84--86. If $\log p/\log q$ is rational, $p$ and $q$ are powers of a common integer $r$, and suitable sums of distinct powers of $r$ have only digits $0$ and $1$ in both bases. Otherwise the proof calls a number $(p,A)$-good or $(q,B)$-good when its base-$p$ or base-$q$ digits are at most $A$ or $B$, and proves a Lemma (p. 84): a $(p,A)$-good number that is not $(q,B)$-good can be replaced by a $(p,A)$-good number whose base-$q$ expansion is better in a well-ordered sense, so that finitely many steps reach a number good in both bases. The Lemma rests on a second Fact (p. 85): every half-open interval $[x,(p-1)x/A)$ with $x$ a positive integer contains a $(p,A)$-good integer. The starting points are powers $p^\alpha$ whose base-$q$ expansion is controlled by the approximation (2) of p. 84, which has infinitely many solutions $\alpha,\beta$ because $\log p/\log q$ is irrational.

Dependencies

None outside the paper; the criterion (1) is Kummer's theorem on carries, which the paper calls an elementary fact.

Bears on

• Problem 376: the problem asks for infinitely many $n$ with $\binom{2n}{n}$ coprime to $105=3\cdot5\cdot7$, three primes. Theorem 1 gives the two-prime case, so infinitely many $n$ with $\binom{2n}{n}$ coprime to $15$, to $21$ or to $35$; the extension to three bases, which the problem needs, is the question the authors could not decide on p. 86.