Geodesic
A note on the number of $S$-Diophantine quadruples
Communications in Mathematics, Tome 22 (2014) no. 1, pp. 49-55.

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Let $(a_1,\dots , a_m)$ be an $m$-tuple of positive, pairwise distinct integers. If for all $1\leq i j \leq m$ the prime divisors of $a_ia_j+1$ come from the same fixed set $S$, then we call the $m$-tuple $S$-Diophantine. In this note we estimate the number of $S$-Diophantine quadruples in terms of $|S|=r$.
Classification : 11D45, 11N32
Keywords: Diophantine equations; $S$-unit equations; $S$-Diophantine tuples 
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Luca, Florian; Ziegler, Volker. A note on the number of $S$-Diophantine quadruples. Communications in Mathematics, Tome 22 (2014) no. 1, pp. 49-55. http://geodesic.mathdoc.fr/item/COMIM_2014__22_1_a3/