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2*P + 2 = Perfect Square? (Posted on 2010-05-01) Difficulty: 4 of 5
Each of P and Q is a nonnegative integer satisfying the equation: P2 = 28*Q2 + 1

Is 2*P + 2 always a perfect square?

If so, prove it. Otherwise, provide a counterexample.

No Solution Yet Submitted by K Sengupta    
Rating: 4.0000 (1 votes)

Comments: ( Back to comment list | You must be logged in to post comments.)
re(2): a computerized start but no proof and too small a sample size | Comment 4 of 5 |
(In reply to re: a computerized start but no proof and too small a sample size by Jer)

P^2=28Q^2+1 is a pell equation when put in the form
P^2-28Q^2=1
and there are established methods for finding all solutions to these types of equations (just google pell equation).  In fact the method of solution is to find a recurrence relation like you did.  So it is fairly easy to establish that your recurrence denotes all of the solutions.  The hard part is the inductive proof which I as well have been attempting with no success
  Posted by Daniel on 2010-05-03 10:31:08

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