A sequence starts 1,3,21 and then every subsequent term is 5 times the sum of the previous two terms, less the term before that, so that the fourth term is 5*(21+3)-1 = 119, and the fifth term is 5*(119+21)-3 = 697, etc. The sequence continues:

1,3,21,119,697,4059,23661,137903,803761,4684659…

None of those numbers after 3 is prime.

In fact I know for certain that 3 is the only prime number in this sequence and that each of the infinity of subsequent terms is composite.

How can I possibly know that?