   SEARCH HOME Math Central Quandaries & Queries  Question from Cillian, a student: In a certain sequence, to get from one term to the other you multiply by 2 and add 1, i.e. This is a difference equation of form: U(n+1) = 2Un + 1. prove that there is a maximum of 2 perfect squares in this sequence Cillian,

As a warm-up, you can do the following:

(a) Prove that there is a maximum of one even number in the sequence (the very first).

This should be easy enough, and it prepares for the follow-up:

(b) Prove that there is a maximum of one odd square in the sequence.

To do this, you suppose that $U(n+1) = 2Un + 1$ is a perfect square, that is

$X^2 = 2Un + 1.$

This can be rewritten as

$X^2 - 1 = 2Un.$

You can then factor the left hand side, and check the parity of the terms involved to draw a conclusion. This should evenhelp you to show that there is only one such sequence with two perfect squares.

Claude     Math Central is supported by the University of Regina and The Pacific Institute for the Mathematical Sciences.