Shifted products that are coprime pure powers

Rainer Dietmann; Christian Elsholtz; Katalin Gyarmati; Miklós Simonovits

doi:10.1016/j.jcta.2004.11.006

Shifted products that are coprime pure powers

Rainer Dietmann^*, Christian Elsholtz, Katalin Gyarmati, Miklós Simonovits

^*Corresponding author for this work

Research output: Contribution to journal › Article › peer-review

Abstract

A set A of positive integers is called a coprime Diophantine powerset if the shifted product ab + 1 of two different elements a and b of A is always a pure power, and the occurring pure powers are all coprime. We prove that each coprime Diophantine powerset A ⊂ {1,..., N} has A ≤ 8000 log N/log log N for sufficiently large N. The proof combines results from extremal graph theory with number theory. Assuming the famous abc-conjecture, we are able to both drop the coprimality condition and reduce the upper bound to c log log N for a fixed constant c.

Original language	English
Pages (from-to)	24-36
Number of pages	13
Journal	Journal of Combinatorial Theory, Series A
Volume	111
Issue number	1
DOIs	https://doi.org/10.1016/j.jcta.2004.11.006
Publication status	Published - Jul 2005

Keywords

abc-conjecture
Applications of extremal graph theory to number theory
Diophantine m-tupules

ASJC Scopus subject areas

Discrete Mathematics and Combinatorics
Theoretical Computer Science

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10.1016/j.jcta.2004.11.006

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Shifted products that are coprime pure powers

Abstract

Keywords

ASJC Scopus subject areas

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