perplexus.info :: Just Math : 2*P + 2 = Perfect Square?

2*P + 2 = Perfect Square? (Posted on 2010-05-01)

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.

See The Solution Submitted by K Sengupta

Rating: 4.5000 (2 votes)

Comments: ( Back to comment list | You must be logged in to post comments.)

a computerized start but no proof and too small a sample size

| Comment 1 of 5

list
   10   for Q=0 to 10000000
   20   P2=28*Q*Q+1
   30   P=int(sqrt(P2)+0.5)
   40   if P*P=P2 then
   50     :Perfsq=2*P+2
   60     :Sr=sqrt(Perfsq)
   70     :print Q;P,Perfsq;Sr
   80   next
OK

 Q        P         2P+2      Sqrt(2P+2)
 0        1         4         2.0
 24       127       256       16.0
 6096     32257     64516     254.0
 1548360  8193151   16386304  4048.0

Every entry in column 3 is a perfect square, giving some credibility to the assertion, but not even a statistical proof.

Posted by Charlie on 2010-05-01 16:12:05

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