Show that there exist an infinite number of infinite sequences of distinct positive integers a, b, c, d, ... for which a+1, ab+1, abc+1, abcd+1, ... are all squares.

Start with n^2-1; obviously, if you add 1 to it, you get a square.Multiply n^2-1 by n^2+1; you get n^4-1, and adding 1 to it gets a square. Multiply n^4-1 by n^4+1; you get n^8-1... and so on.

The sequences are n^2-1, n^2+1, n^4+1, n^8+1, n^16+1... and it's easy to see that there are no repeated numbers for n>0.

Edited on December 1, 2004, 5:20 pm

Posted by e.g. on 2004-12-01 17:19:53