On the size of sets whose elements have perfect power $n$-shifted products

Abstract

Abstract. We show that the size of sets Ahaving the property that with somenon-zero integer n, a 1 a 2 + n is a perfect power for any distinct a 1 ;a 2 2A, cannot bebounded by an absolute constant. We give a much more precise statement as well,showing that such a set Acan be relatively large. We further prove that under the abc-conjecture a bound for the size of Adepending on n can already be given. Extending aresult of Bugeaud and Dujella, we also derive an explicit upper bound for the size of Awhen the shifted products a 1 a 2 + n are k-th powers with some xed k 2. The latterresult plays an important role in some of our proofs, too. 1. IntroductionA set A= fa 1 ;:::;a m gof positive integers is called a Diophantine m-tuple,if for any 1 i < j mwe have a i a j + 1 = x 2ij for an integer x ij . The Mathematics Subject Classi cation: 11B75, 11D99.Key words and phrases: shifted product, perfect power, abc-conjecture, Diophantine m-tuple.The rst three authors are supported by the Hungarian-Croatian bilateral project Numbertheory and cryptography. The rst and third authors are supported in part by the OTKA grantsK67580 and K75566, and by the TAMOP 4.2.1./B-09/1/KONV-2010-0007 project. The projectis implemented through the New Hungary Development Plan, co nanced by the European SocialFund and the European Regional Development Fund. The second author is supported by theMinistry of Science, Education and Sports, Republic of Croatia, grant 037-0372781-2821. Thefourth author was supported in part by Grants SEP-CONACyT 79685 and PAPIIT 100508.