Publ. Math. Debrecen 79/3-4 (2011), 325–339 DOI: 10.5486/PMD.2011.5155 On the size of sets whose elements have perfect power n-shifted products By ATTILA B ´ ERCZES (Debrecen), ANDREJ DUJELLA (Zagreb), LAJOS HAJDU (Debrecen) and FLORIAN LUCA (Morelia) Dedicated to Professors K. Gy˝ory and A. S´ark¨ozy on their 70th birthdays and Professors A. Peth˝o and J. Pintz on their 60th birthdays Abstract. We show that the size of sets A having the property that with some non-zero integer n, a1a2 + n is a perfect power for any distinct a1,a2 ∈A, cannot be bounded by an absolute constant. We give a much more precise statement as well, showing that such a set A can be relatively large. We further prove that under the abc- conjecture a bound for the size of A depending on n can already be given. Extending a result of Bugeaud and Dujella, we also derive an explicit upper bound for the size of A when the shifted products a 1 a 2 + n are k-th powers with some fixed k ≥ 2. The latter result plays an important role in some of our proofs, too. Mathematics Subject Classification: 11B75, 11D99. Key words and phrases: shifted product, perfect power, abc-conjecture, Diophantine m-tuple. The first three authors are supported by the Hungarian-Croatian bilateral project Number theory and cryptography. The first and third authors are supported in part by the OTKA grants K67580 and K75566, and by the T ´ AMOP 4.2.1./B-09/1/KONV-2010-0007 project. The project is implemented through the New Hungary Development Plan, cofinanced by the European Social Fund and the European Regional Development Fund. The second author is supported by the Ministry of Science, Education and Sports, Republic of Croatia, grant 037-0372781-2821. The fourth author was supported in part by Grants SEP-CONACyT 79685 and PAPIIT 100508.