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

Autori
Berczes, Attila ; Dujella, Andrej ; Hajdu, Lajos ; Luca, Florian

Izvornik
Publicationes mathematicae (0033-3883) 79 (2011), 3-4; 325-339

Vrsta, podvrsta i kategorija rada
Radovi u časopisima, članak, znanstveni

Ključne riječi
shifted product ; perfect power ; abc-conjecture ; Diophantine m-tuple

Sažetak
We show that the size of sets A having the property that with some non-zero integer n, a_1a_2 + n is a perfect power for any distinct a_1, a_2 in 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_1a_2 + n are k-th powers with some fixed k >= 2. The latter result plays an important role in some of our proofs, too.

Izvorni jezik
Engleski

Znanstvena područja
Matematika

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