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 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

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