There are only finitely many Diophantine quintuples

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The extensibility of Diophantine pairs {k−1,k+1} (2008, background): “(Note that the symbols “d−” and “d+” in [7] respectively coincide with cν−1 and cν+1 in our notation, and that d = cν−1, cν+1 are the solutions of the equation (2k − c − d)2 = 4k2(cd + 1); see [7, Section 1].”
There is no Diophantine quintuple (2016, background): “In 2004, Dujella [16] proved that there does not exist a Diophantine sextuple.”
Complete Solution of a Problem of Diophantus and Euler (2005, background): “Recently, the first author proved that there does not exist a D(1)-sextuple and there are only finitely many D(1)-quintuples (see [11]).”
Effective solution of the D(-1)-quadruple conjecture (2007, background): “Recently, the first author proved that there does not exist a D(1)-sextuple and there are only finitely many D(1)-quintuples (see [15], this refined previous results [22, 12]).”
There are infinitely many rational Diophantine sextuples (2015, background): “{k − 1, k + 1, 4k, 16k3 − 4k} for k ≥ 2), while it was proved in [8] that an integer Diophantine sextuple does not exist and that there are only finitely many such quintuples (for additional information and references see e.”