Shifted products that are coprime pure powers

  • Authors:

  • Rainer Dietmann;Christian Elsholtz;Katalin Gyarmati;Miklós Simonovits

  • Affiliations:

  • Institut für Algebra und Zahlentheorie, Pfaffenwaldring 57, D-70550 Stuttgart, Germany;Department of Mathematics, Royal Holloway, University of London, Egham, Surrey TW20 0EX, UK;Department of Algebra and Number Theory, Eötvös Loránd University, H-1117 Budapest, Pazmany Peter Setany 1/C, Hungary;Alfréd Rényi Institute of Mathematics, Hungarian Academy of Sciences, H-1053 Budapest, Reáltanoda u. 13-15, Hungary

  • Venue:
  • Journal of Combinatorial Theory Series A
  • Year:
  • 2005

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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.