### 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 | |

Publication status | Published - Jul 2005 |

### Fingerprint

### 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

*Journal of Combinatorial Theory, Series A*,

*111*(1), 24-36. https://doi.org/10.1016/j.jcta.2004.11.006

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

Research output: Contribution to journal › Article

*Journal of Combinatorial Theory, Series A*, vol. 111, no. 1, pp. 24-36. https://doi.org/10.1016/j.jcta.2004.11.006

}

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 -