*(English)*Zbl 0868.11009

The interest in prime factors of binomial coefficients during the last ten years or so has been motivated to a great extent by a conjecture of Erdös asserting that $\left(\genfrac{}{}{0pt}{}{2n}{n}\right)$ is not squarefree for any $n>4$. This was recently proved by *G. Velammal* [Hardy-Ramanujan J. 18, 23-45 (1995; Zbl 0817.11011)], and another proof is given in the present paper, among many other interesting results. As a sharpening of the Erdös conjecture, it is shown that the coefficient in question is divisible even by the square of a prime $\ge \sqrt{n/5}$ for all $n\ge 2082$. On the other hand, $\left(\genfrac{}{}{0pt}{}{1572}{786}\right)$ is not divisible by the square of any odd prime (it is divisible by ${2}^{4}$), and it is the largest coefficient of this kind.

In addition to the middle of the Pascal triangle, the authors consider it as a whole, in particular its edges. Squarefree values (other than 1) do occur near the edges, and only there. For instance, there are infinitely many integers $n$ such that $\left(\genfrac{}{}{0pt}{}{n}{k}\right)$ is squarefree for all $k\le (1/5)logn$. On the other hand, it is shown that if $\left(\genfrac{}{}{0pt}{}{n}{k}\right)$ is squarefree, then $n$ or $n-k$ is $\ll exp\left(c{(logn)}^{2/k}{(loglogn)}^{1/3}\right)$ for some constant $c$, and it is conjectured that this bound can be reduced to $\ll {(lognloglogn)}^{2}$, which would be close to being best possible. A curious statistical result indicating the scarcity of the squarefree binomial coefficients is that the average number of these in a row of the Pascal triangle is about $10\xb766$.

An important tool in previous work related to the Erdös conjecture has been the exponential sums of the type ${\sum}_{n}{\Lambda}\left(n\right)e(x/n)$, where $n$ thus runs essentially over primes, and the same is the case also in the present paper, where explicit estimates for such sums are given and applied as a key ingredient of the argument.