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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)

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Solution Finishing what I started Comment 5 of 5 |

Ah for the heady days when I still had not heard of the Pell equation!

Using trusty Dario Alpern supplies the recurrence relation

Xn+1 = P Xn + Q Yn
Yn+1 = R Xn + S Yn

P = 127
Q = 672
R = 24
S = 127

The first few terms for P are {1,127,32257,...} etc. WolframAlpha resolves this to P = 1/2 (-(127-48 sqrt(7))^n-(127+48 sqrt(7))^n), and we need to take the negative value, multiply by 2 and add 2:

(127-48 sqrt(7))^n+(127+48 sqrt(7))^n+2 (I)

Next we look at the values of the square roots of this: {2,16,254,4048,64514 ...}, A090727 in Sloane with formula (8+sqrt(63))^n + (8-sqrt(63))^n.

Squaring this gives: (8-3 sqrt(7))^(2n)+(8+3 sqrt(7))^(2n)+2((8-3 sqrt(7))(8+3 sqrt(7)))^n.

More conveniently:

(8-3 sqrt(7))^(2n)+(8+3 sqrt(7))^(2n)+2 (II)

Now, (8-3 sqrt(7))^(2) = 127-48 sqrt(7); while (8+3 sqrt(7))^(2) =127+48 sqrt(7)   (III)

So at last we have the equality:

(127-48 sqrt(7))^n+(127+48 sqrt(7))^n+2  = (127-48 sqrt(7))^n + (127+48 sqrt(7))^n+2. (I),(III)

This number is always square since it is the square of (8+sqrt(63))^n + (8-sqrt(63))^n, see (II).

And the mystery is solved at last.

Edited on September 30, 2013, 2:14 am
  Posted by broll on 2013-09-30 02:09:06

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