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Statement

Notation (p. 24). The paper writes $\binom{2n}{n}=(s(n))^2q(n)$ with $q(n)$ squarefree, $\{x\}=x-[x]$ for the fractional part, and $e(x)=e^{2\pi ix}$.

Theorem (p. 24, quoted). "For $n\geq 2^{8000}$, $\binom{2n}{n}$ is never square free."

The paper prints it as the unnumbered THEOREM; the base-$P$ digit criterion on p. 43 is numbered Theorem 2.

Proof pointer

Pp. 24--43. By (1) and (2) on p. 24, the exponent of a prime $p$ in $\binom{2n}{n}$ is at least 2 when both $\{n/p\}$ and $\{n/p^2\}$ are at least $1/2$. For $\sqrt n<p\le\sqrt{2n}$ the second condition holds automatically, so $\log s(n)$ is at least the sum of $\log p$ over primes in $(\sqrt n,\sqrt{2n}\,]$ with $\{n/p\}\ge1/2$, (4) on p. 25. The paper detects that condition with a smoothed indicator from a lemma of Vinogradov (p. 25), splits the resulting exponential sums over primes with Vaughan's identity into three sums $S_1,S_2,S_3$, and bounds them with exponent pairs whose implied constants it computes explicitly (Lemmas 1 and 2): the pair $(1/2,1/2)$ for $S_1$ and the pair $(1/14,11/14)$, obtained by applying Rule A twice, for $S_2$ and $S_3$, with $Z=n^{7/40}(\log 2n)^{7/2}$ and $\epsilon=1/8$ (p. 42). With the Rosser--Schoenfeld bounds for $\theta(m)$ this gives $\log s(n)>0$, so $s(n)>1$, for all $n\ge2^{8000}$ (p. 43).

Read depth

Claims checked: the notation and the Theorem were read on the page images of the print, and the outline of the proof was followed. The explicit constants of Lemmas 1 and 2 and the numerical bounds on p. 42 were not rechecked. On p. 42 the print sets "$n\leq 2^{8000}$" while computing the bounds whose conclusion on p. 43 is stated for $n\geq2^{8000}$. Nothing here is independently reviewed.

Dependencies

None in the corpus. External inputs named by the paper: Vinogradov's lemma on smoothed periodic indicators, Vaughan's identity, the exponent-pair processes of Ivić's The Riemann Zeta-Function (chapter 2), and the Rosser--Schoenfeld bounds for $\theta(x)$.

Source. G. Velammal, Is the binomial coefficient $\binom{2n}{n}$ squarefree?, Hardy-Ramanujan J. 18 (1995), 23--45, DOI 10.46298/hrj.1995.132; the edition read is named on the source card.

Bears on

• Problem 175: proves the problem's statement for every $n\ge2^{8000}$. The paper extends it to every $n>4$ with Theorem 2 and a computation on p. 43, as the main theorem page records.