---

Statement

Main theorem (abstract, p. 23). For every integer $n>4$ the binomial coefficient $\binom{2n}{n}$ is not squarefree. The abstract states that the paper proves "the Erdös conjecture that the binomial coefficient $\binom{2n}{n}$ is never squarefree, for all $n > 4$"; the paper gives the result no number and ends the argument on p. 43 with "This proves the conjecture."

Proof pointer

The range $n\ge2^{8000}$ is the Theorem on p. 24. For $4<n<2^{8000}$ the paper uses Theorem 2 (p. 43): its $P=2$ step leaves only $n=2^j$ with $2<j\le8000$, and a computer check finds for each such $j$ a prime $P<100$ meeting the hypothesis of Theorem 2, except $j=4$, where $3^2$ divides $\binom{32}{16}$.

A postscript (p. 45) records that J. W. Sander, J. Number Theory 46 (1994), 372--384, points out that G. Velammal, A. Granville and O. Ramaré proved the conjecture independently of each other.

Read depth

Claims checked: the abstract, the closing argument on p. 43 and the postscript on p. 45 were read on the page images of the print. The analytic constants and the computer check were not rechecked; the two pages above record what was read of each. Nothing here is independently reviewed.

Dependencies

• Theorem (p. 24).
• Theorem 2 (p. 43) and the computation reported after it.

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: the theorem is the problem's statement, that $\binom{2n}{n}$ is not squarefree for any $n\ge5$, proved for every such $n$.