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.
Originalsprache | englisch |
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Seiten (von - bis) | 24-36 |
Seitenumfang | 13 |
Fachzeitschrift | Journal of Combinatorial Theory, Series A |
Jahrgang | 111 |
Ausgabenummer | 1 |
DOIs | |
Publikationsstatus | Veröffentlicht - Juli 2005 |
ASJC Scopus subject areas
- Diskrete Mathematik und Kombinatorik
- Theoretische Informatik