On the size of Diophantine \(m\)-tuples (Q2777972)

A set of distinct positive integers \(\{a_1,a_2,\dots,a_m\}\) is said to have the property \(D(n)\) if \(a_ia_j+n\) is a perfect square for all pairs \((i,j)\) with \(1 \leq i<j\leq m\). Such a set is known as a Diophantine \(m\)-tuple (with the property \(D(n))\). So has the Diophantine quadruple \(\{1, 33,68,105\}\) the property \(D (256)\) (known to Diophantus), and the Diophantine quadruple \(\{1,3,8, 120\}\) has the property \(D(1)\) (attributed to Fermat). A well-known conjecture is that there is no Diophantine quintuple with the property \(D(1)\).NEWLINENEWLINEBut Diophantine quintuples, such as \(\{1,33,105,320,18240\}\) with the property \(D(256)\) (due to the author), and sextuples, such as \(\{99,315,9920, 32768,44460,19534284\}\) with the property \(D(2985984)\) (due to P. Gibbs) are known to exist. Recently, the author has shown that Diophantine 9-tuples with the property \(D(1)\) cannot exist [J. Number Theory 89, 126--150 (2001; Zbl 1010.11019)].NEWLINENEWLINE The purpose of the present paper is to find upper bounds for the number NEWLINE\[NEWLINEM_n=\sup \bigl\{ | S|:S\text{ has the property }D(n)\bigr\},NEWLINE\]NEWLINE where \(| S|\) is the cardinality of \(S\). The author proves that \(M_n\leq 32\) for \(| n|\leq 400\), and \(M_n<267.81\log| n|(\log \log | n|)^2\) for \(| n| >400\). For the proof he makes use of theorems of \textit{M. A. Bennett} [J. Reine Angew. Math. 498, 173--199 (1998; Zbl 1044.11011)] on simultaneous approximations of algebraic numbers, and of \textit{P. X. Gallagher} [Acta Arith. 18, 77--81 (1971; Zbl 0231.10028)] on a large sieve method.