A Representation of Large Integers from Combinatorial Sieves Original Research Article

For any positive integers k and m, and any l, 0 ≤ l < m, we show that there is a number β = β(k, m) > 0 such that any sufficiently large integer x can be represented as x = ƒ1 · · · ƒk + r · m + l where ƒ1, ..., ƒk and r are nonnegative integers and r · m + l ≤ xβ and ƒi ≥ xβ for each I = l,..., k. This says one can find numbers with certain factorizations in "short arithmetic sequences". The representation is proven by way of the number sieve of Brun and its generalization to multiplicative functions by Alladi; by studying the distribution of the arithmetic function ν(n), the number of distinct prime divisors of n, on sieved short arithmetic sequences. This has applications in Combinatorial Design Theory and Coding Theory.