ON PRIME DIVISORS OF BINOMIAL COEFFICIENTS

(2n\ In 1985, Sarkozy proved a conjecture of Erdos by showing that) is never square-free for sufficiently large n. By applying a new estimate on exponential sums, we prove that this also holds for I I, if d is no t ' t oo big'. \ n I Let 0 < e < 1, pQe N. For m ^ m0 and 1 ̂ k ^ m satisfying \m — 2k \ < m1"8, there is a prime p> p0 such that / r r (2w\I is never square-free for sufficiently largenJ n, thus proving a conjecture of Erdos. In [5], the author obtained a result, which shows that for d 4 n^y (y> 0 small), binomial coefficients I ~ I are also not square-free. THEOREM 1. There is a computable constant y> 0 such that, given p0, there is an m0 with the following property. For all m ^ mQ and 1 ̂ k ^ m with \m-2k\<m&, there is a prime p> p0 satisfying p2 kf Recently, the author [6] established an exponential sum estimate, which implies an asymptotic formula for DJLo) = card|/>eP:/> < P,0 ^ | ^ | < oY where 0 < a ^ 1, and {z} denotes the fractional part of the real number z. THEOREM 2. Let j ^ 1 and P ^ xlli. Then there is a positive constant c1 such that Dt{&) = <jn(P) + O((P1-ciQoePnoex)i + Pu+*)/2x-*)(\ogxy). This gives the following improvement of Theorem 1, which answers another question of Erdos and Graham