Shifted products that are coprime pure powers

Rainer Dietmann, Christian Elsholtz, Katalin Gyarmati, M. Simonovits

Research output: Contribution to journalArticle

5 Citations (Scopus)

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, Series A
Volume111
Issue number1
DOIs
Publication statusPublished - Jul 2005

Fingerprint

Number theory
Coprime
Graph theory
ABC Conjecture
Extremal Graph Theory
Upper bound
Integer

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

Cite this

Shifted products that are coprime pure powers. / Dietmann, Rainer; Elsholtz, Christian; Gyarmati, Katalin; Simonovits, M.

In: Journal of Combinatorial Theory, Series A, Vol. 111, No. 1, 07.2005, p. 24-36.

Research output: Contribution to journalArticle

Dietmann, Rainer ; Elsholtz, Christian ; Gyarmati, Katalin ; Simonovits, M. / Shifted products that are coprime pure powers. In: Journal of Combinatorial Theory, Series A. 2005 ; Vol. 111, No. 1. pp. 24-36.
@article{4aca88673ff243a6b61813c7d4448c81,
title = "Shifted products that are coprime pure powers",
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.",
keywords = "abc-conjecture, Applications of extremal graph theory to number theory, Diophantine m-tupules",
author = "Rainer Dietmann and Christian Elsholtz and Katalin Gyarmati and M. Simonovits",
year = "2005",
month = "7",
doi = "10.1016/j.jcta.2004.11.006",
language = "English",
volume = "111",
pages = "24--36",
journal = "Journal of Combinatorial Theory - Series A",
issn = "0097-3165",
publisher = "Academic Press Inc.",
number = "1",

}

TY - JOUR

T1 - Shifted products that are coprime pure powers

AU - Dietmann, Rainer

AU - Elsholtz, Christian

AU - Gyarmati, Katalin

AU - Simonovits, M.

PY - 2005/7

Y1 - 2005/7

N2 - 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.

AB - 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.

KW - abc-conjecture

KW - Applications of extremal graph theory to number theory

KW - Diophantine m-tupules

UR - http://www.scopus.com/inward/record.url?scp=20344398646&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=20344398646&partnerID=8YFLogxK

U2 - 10.1016/j.jcta.2004.11.006

DO - 10.1016/j.jcta.2004.11.006

M3 - Article

AN - SCOPUS:20344398646

VL - 111

SP - 24

EP - 36

JO - Journal of Combinatorial Theory - Series A

JF - Journal of Combinatorial Theory - Series A

SN - 0097-3165

IS - 1

ER -