A Diophantine \(m\)-tuple is a set of \(m\) positive integers such that the product of any two increased by \(1\) is a perfect square. A Diophantine quadruple \(\{a,b,c,d\}\) is called regular if for \(d>\max\{a,b,c\}\) we have \(d=a+b+c+2abc+2\sqrt{(ab+1)(ac+1)(bc+1)}\). It is widely believed that all Diophantine quadruples are regular. The authors show that every Diophantine quadruple of the form \(\{k-1,k+1,16k^3-4k,d\}\) is regular. As the authors point out this combined with a recent result due to Y. Fujita [J. Number Theory 128, No. 2, 322–353 (2008; Zbl 1151.11015)] proves that all Diophantine quadruple of the form \(\{k-1,k+1,c,d\}\) are regular.
The proof is based on the resolution of simultaneous Pell equations. In particular they use an improved congruence method an a new bound for linear forms in three logarithms [Y. Bugeaud, M. Mignotte and S. Siksek, Compos. Math. 142, No. 1, 31–62 (2006; Zbl 1128.11013)].