Some polynomial formulas for Diophantine quadruples

The Greek mathematician Diophantus of Alexandria studied the following problem: Find four (positive rational) numbers such that the product of any two of them increased by 1 is a perfect square. He obtained the following solution: 1 16 , 33 16 , 17 4 , 105 16 (see [4]). Fermat obtained four positive integers satisfying the condition of the problem above: 1, 3, 8, 120. For example, 3 · 120+1 = 19. Later, Davenport and Baker [3] showed that if d is a positive integer such that the set {1, 3, 8, d} has the property of Diophantus, then d has to be 120. There are two direct generalizations of the set {1, 3, 8, 120}: the sets {n, n + 2, 4n + 4, 4(n + 1)(2n + 1)(2n + 3)}, (1)