Glasnik Matematicki, Vol. 54, No. 2 (2019), 279-319.

ON THE EXISTENCE OF S-DIOPHANTINE QUADRUPLES

Volker Ziegler

Institute of Mathematics, University of Salzburg, Hellbrunnerstrasse 34/I, A-5020 Salzburg, Austria
e-mail: volker.ziegler@sbg.ac.at

Abstract.   Let S be a set of primes. We call an m-tuple (a1,… ,am) of distinct, positive integers S-Diophantine, if for all i≠ j the integers si,j:=aiaj+1 have only prime divisors coming from the set S, i.e. if all si,j are S-units. In this paper, we show that no S-Diophantine quadruple (i.e. m=4) exists if S={3,q}. Furthermore we show that for all pairs of primes (p,q) with p < q and p ≡ 3 mod 4 no {p,q}-Diophantine quadruples exist, provided that (p,q) is not a Wieferich prime pair.

2010 Mathematics Subject Classification.   11D61, 11D45

Key words and phrases.   Diophantine equations, S-unit equations, Diophantine tuples, S-Diophantine quadruples

Full text (PDF) (free access)

https://doi.org/10.3336/gm.54.2.03

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