On products of consecutive integers

Squares from Products of Consecutive Integers

Sums of Consecutive Integers

Wai Yan Pong ([email protected]) received his B.Sc. from the Chinese University of Hong Kong and his M.Sc. and Ph.D. from the University of Illinois at Chicago. He was a Doob Research Assistant Professor at the University of Illinois at Urbana-Champaign for three years. He then moved to California and is now teaching at California State University, Dominguez Hills. His research interests are in m...

On the Product of Consecutive Integers

of k consecutive integers is never an l-th power if k > 1, 1 > 1 2 ) . RIGGE 3 ) and a few months later I 1 ) proved that Ak(n) is never a square, and later RIDGE and 14) proved using the Thue-Siegel theorem that for every l > 2 there exists a k0(l) so that for every k > k0(l) A k(n) is not an l-th power. In 1940 SIEGEL and I proved that there is a constant c so that for k > c, l > 1 A k(n) is ...

On the factorization of consecutive integers

A classical result of Sylvester [21] (see also [16], [17]), generalizing Bertrand’s Postulate, states that the greatest prime divisor of a product of k consecutive integers greater than k exceeds k. More recent work in this vein, well surveyed in [18], has focussed on sharpening Sylvester’s theorem, or upon providing lower bounds for the number of prime divisors of such a product. As noted in [...

On products of integers. II

1. Throughout this paper, c 1 , c2 , . . . denote absolute constants ; ko (a, fl, . . .), kr (a, f3, . . .), . . ., xo (a, /3, . . .), . . . denote constants depending only on the parameters a, /l, . . . ; v(n) denotes the number of the prime factors of the positive integer n, counted according to their multiplicity . The number of the elements of a finite set S is denoted by I S I . Let k, n b...