ON PRIME DIVISORS OF BINOMIAL COEFFICIENTS

In 1985, Sarkozy proved a conjecture of Erdos by showing that (2n(n)) is never square-free for sufficiently large n. By applying a new estimate on exponential sums, we prove that this also holds for (2n +/- d(n)), if d is not 'too big'. Let 0 < epsilon < 1, p0 is-an-element-of N. For m greater-than-or-equal-to m0 and 1 less-than-or-equal-to k less-than-or-equal-to m satisfying \m - 2k\ < m1-epsilon, there is a prime p > p0 such that p2\(m(k)).