Triples (x,y,z) with 1 < x < y < z and z = y + 1 such that x! y! z! is a square.

3, 5, 6, 4, 5, 6, 2, 7, 8, 2, 17, 18, 6, 19, 20, 3, 23, 24, 4, 23, 24, 5, 29, 30, 10, 27, 28, 2, 31, 32, 7, 34, 35, 6, 44, 45, 2, 49, 50, 3, 53, 54, 4, 53, 54, 10, 62, 63, 2, 71, 72, 8, 69, 70, 9, 69, 70, 11, 76, 77, 6, 79, 80, 3, 95, 96, 4, 95, 96, 2, 97, 98

1

It appears that this sequence is infinite.

T. D. Noe, Plot of 1000 triples

T. D. Noe, Table of 1000 triples

(Mma) (using the code in S001000) Select[t, #[[3]] - #[[2]] == 1 &]

Cf. S001000.

nonn,tabl

T. D. Noe, Apr 19 2017