Extensions of a Diophantine triple by adjoining smaller elements II

Abstract

Let \(\{a_1,b,c\}\) and \(\{a_2,b,c\}\) be Diophantine triples with \(a_1<b<a_2<c\) and \(a_2\ne b+c-2\sqrt{bc+1}\). Put \(d_2=a_2+b+c+2a_2bc-2r_2st\), where \(r_2=\sqrt{a_2b+1}\), \(s=\sqrt{ac+1}\) and \(t=\sqrt{bc+1}\). In this paper, we prove that if \(c \le 16\mu ^2 b^3\), where \(\mu =\min \{a_1,d_2\}\), then \(\{a_1,a_2,b,c\}\) is a Diophantine quadruple. Combining this result with one of our previous results implies that if \(\{a_i,b,c,d\}\) \((i\in \{1,2,3\})\) are Diophantine quadruples with \(a_1<a_2<b<a_3<c<d\), then \(a_3=b+c-2\sqrt{bc+1}\). It immediately follows that there does not exist a septuple \(\{a_1,a_2,a_3,a_4,b,c,d\}\) with \(a_1<a_2<b<a_3<a_4<c<d\) such that \(\{a_i,b,c,d\}\) \((i \in \{1,2,3,4\})\) are Diophantine quadruples. Moreover, it is shown that there are only finitely many sextuples \(\{a_1,a_2,a_3,b,c,d\}\) with \(a_1<b<a_2<a_3<c<d\) such that \(\{a_i,b,c,d\}\) \((i \in \{1,2,3\})\) are Diophantine quadruples.