This is a very old problem, dating to Diophantus of Alexandria in the 3rd century, who first observed that the quadruple of rational numbers {1/16, 33/16, 17/4, 105/16} has the property that the product of any two of them plus one is the square of a rational number.

In the 17th century Pierre de Fermat found the integer solution {1,3,8,120}.

Leonhard Euler extended Fermat's solution using the rational number 777480/8288641.  Euler also found an infinite family of such sets:  

{a, b, a + b + 2r, 4r(r + a)(r + b) }, where ab + 1 = r^2. 

In 1969 Baker and Davenport used 'Baker's theory on linear forms in logarithms of algebraic numbers and a reduction method based on continued fractions' to prove if d is a positive integer such that {1, 3, 8, d} forms a Diophantine quadruple, then d = 120.  By implication, Fermat's set cannot be extended to a Diophantine quintuple.

In 1979 a constructive proof was found to extend any Diophantine triple to a Diophantine quadruple.  

In 2004 Dujella proved no Diophantine 6-tuple exists.  That capped the upward search since any 6 terms of a 7-tuple would be a 6-tuple.

No Diophantine quintuple has been found nor has one been proved impossible.

https://plus.google.com/101584889282878921052/posts/FNDF7e8pW9X

https://web.math.pmf.unizg.hr/~duje/intro.html