Journal of the Australian Mathematical Society - Series A Vol. 64 Part 2 (1998) On strings of consecutive integers with no large prime factors Antal Balog Mathematical Institute Budapest 1364 Hungary e-mail: balog@math-inst.hu and Trevor D. Wooley Department of Mathematics University of Michigan East Hall 525 East University Avenue Ann Arbor, Michigan 48109-1109 USA e-mail: wooley@math.lsa.umich.edu Abstract: We investigate conditions which ensure that systems of binomial polynomials with integer coefficients are simultaneously free of large prime factors. In particular, for each positive number $\varepsilon$, we show that there are infinitely many strings of consecutive integers of size about n, free of prime factors exceeding $n^\varepsilon$, with the length of the strings tending to infinity with speed $\log \log \log \log n$. PDF file size: 59K © Copyright 1998, Australian Mathematical Society