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A Representation of Large Integers from Combinatorial Sieves

Blanchard, John L. (1995) A Representation of Large Integers from Combinatorial Sieves. Journal of Number Theory, 54 (2). pp. 287-296. ISSN 0022-314X. doi:10.1006/jnth.1995.1119.

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For any positive integers k and m, and any /, 0 ≤ / < 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 = ƒ_1··· ƒ_k + r · m + / where ƒ_1,..., ƒ_k and r are nonnegative integers and r·m + / ≤ x^β and ƒ_i≥ x^β 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.

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Additional Information:© 1995 Academic Press. Received July 23, 1993: revised February 21, 1994.
Issue or Number:2
Record Number:CaltechAUTHORS:20170906-091239879
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Official Citation:J.L. Blanchard, A Representation of Large Integers from Combinatorial Sieves, Journal of Number Theory, Volume 54, Issue 2, 1995, Pages 287-296, ISSN 0022-314X, (
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:81180
Deposited By: Ruth Sustaita
Deposited On:06 Sep 2017 17:20
Last Modified:15 Nov 2021 19:41

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