### 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 |
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Pages (from-to) | 24-36 |

Number of pages | 13 |

Journal | Journal of Combinatorial Theory / A |

Volume | 111 |

Issue number | 1 |

DOIs | |

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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## Cite this

Dietmann, R., Elsholtz, C., Gyarmati, K., & Simonovits, M. (2005). Shifted products that are coprime pure powers.

*Journal of Combinatorial Theory / A*,*111*(1), 24-36. https://doi.org/10.1016/j.jcta.2004.11.006