Given a positive integer \(m\), a Diophantine \(m\)-tuple is a set of integers \(\{a_1, a_2, \ldots, a_m\}\) with the property that \(a_ia_j+ 1\) is a perfect square for all \(1\leq i<j\leq m.\) In [A. Dujella, J. Reine Angew. Math. 566, 183–214 (2004; Zbl 1037.11019)] it is proved by the second author that the definition above is satisfied for no \(m\)-tuple with \(m\leq 6\). Then, in [B. He et al., Trans. Am. Math. Soc. 371, No. 9, 6665–6709 (2019; Zbl 1430.11044)], it is shown that it necessarily holds \(m\leq 4.\) To see a complete bibliography on Diophantine sets, one can see [A. Dujella, Diophantine \(m\)-tuples. https://web.math.pmf.unizg.hr/~duje/ref.html]. The most of the works in the literature are devoted to extensions of Diophantine triples by adjoining a fourth element greater than the three already known. The following conjecture is the currently open question of greatest interest in this are, which was posed implicitly in [J. Arkin et al., Fibonacci Q. 17, 333–339 (1979; Zbl 0418.10021); D. E. Gibbs, Proc. Am. Math. Soc. 70, 103–108 (1978; Zbl 0404.18011); A. Dujella, J. Number Theory 89, No. 1, 126–150 (2001; Zbl 1010.11019)].
Conjecture 1.1: Any Diophantine triple \(\{a, b, c\}\) has unique extension to a Diophantine quadruple \(\{a, b, c, d\}\) by an element \(d > \max\{a, b, c\}.\)
In their previous work [Period. Math. Hung. 82, No. 1, 56–68 (2021; Zbl 1474.11078)], by studding on the extendibility of Diophantine triples by an integer smaller than all elements of the initial triple, the present authors stated that:
Conjecture 1.2: Suppose that \(\{a_1,b,c,d\}\) is a Diophantine quadruple with \(a_1 <b<c<d.\) Then, \(\{a_2,b,c,d\} \) is not a Diophantine quadruple for any integer \(a_2\) with \(a_1\neq a_2<b.\)
In the paper under review, continuing their studied, the authors proved:
Main Theorem: Assume that \(\{a_1,b,c,d\}\) and \(\{a_2,b,c,d\}\) are Diophantine quadruples with \(a_1<a_2< b < c <d\). Then, the following holds:

(1)

\(a_2> \max \{36a_1^3,300a_1^2\}.\)

(2)

\(b< a_2^{3/2} \) for \(a_1 \geq 1\), and \(b< a_2^{4/3}\) for \(a_1 \geq 2\) or \(a_1=1\) and \(a_2 < 4 \cdot 10^5\).

(3)

\(a_2 > 24^3 = 13824.\)

(4)

\( 16 a_1^2 b^3 < c < 16 a_2 b^3.\)

They, also showed that there are only finitely many quintuples \(\{a_1,a_2,b,c,d\}\) as above, and have obtained some interesting consequences of their theorem, including the Corollary 1.6, which states that Conjecture 1.2 implies Conjecture 1.1.