Shifted products that are coprime pure powers

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

*Corresponding author for this work

    Research output: Contribution to journalArticle

    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 languageEnglish
    Pages (from-to)24-36
    Number of pages13
    JournalJournal of Combinatorial Theory / A
    Volume111
    Issue number1
    DOIs
    Publication statusPublished - 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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