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        riddles >> medium >> Counting Squares
        
(Message started by: Michael Dagg on Apr 25^th, 2009, 8:59am)

Title: Counting Squares
Post by Michael Dagg on Apr 25^th, 2009, 8:59am

For n>=1 , how many squares are in the sequence

18, 69, 154, ... , 17n^2 + 1 ?

Title: Re: Counting Squares
Post by towr on Apr 25^th, 2009, 2:14pm

Aren't there infinitely many if there is at least one?
[hide]It's a Pell equation
one solution to k² - 17n² = 1 is k₁=33, n₁=8

more solutions i=2..inf:
k_i = k_i-1 k₁ + 17 n_i-1 n₁
n_i = k_i-1 n₁ + n_i-1 k₁
[/hide]

[edit]Oh wait, you probably mean up to a given n.
[hide]There should be solutions regularly spaced among the convergents of the continued fraction of sqrt(17)[/hide][/edit]