On the Number of Times an Integer Occurs as a Binomial Coefficient

Abstract

Let N(t) denote the number of times the integer t > 1 occurs as a binomial co-efficient ; that is, N(t) is the number of solutions of t = (n in integers n and r. r We have N(2) = 1, N(3) = N(4) = N(5) = 2, N(6) = 3, etc . In a recent note in the research problems section of the MONTHLY, D . Singmaster [1] proved that N(t) = O(log t) . He conjectured that N(t) = 0(1) but pointed out that this conjecture, if it is in fact true, is perhaps very deep . In [1] and [5], Singmaster points out that N(t) = 6 for the following values of t _ 6 inflaitely often . Sing-master has verified that the only value of t _ 8 is t = 3003, for which N(t) = 8 . In this note we obtain some additional information about the behavior of N(t) . In Theorem 1 we prove that the average and normal order of N(t) is 2 ; in fact, we prove somewhat more than this, namely, the number of integers t, 1 < t 5 x, for which N(t) > 2 is 0(\/x). (See [4] p . 263 and p . 356, for the definitions of average and normal order .) In Theorem 2 we give an upper bound for N(t) in terms of the number of distinct prime factors of t . Our main result is Theorem 3, in which we show that (1) can be improved to N(t) = O(log t/log log t) . Finally, in Theorem 4, we consider the related problem of determining the number of representations of an integer as a product of consecutive integers . THEOREM 1 . The average and normal order of N(t) = 2 . Proof. For integral x, let n be defined by(nn 12) < x < (kk) 5 _ s 2r5m 256 so that n = 0(log x) .