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 PX + 144 = Perfect Square (Posted on 2010-01-12)
X is a positive integer > 1 and, P is a prime number. Determine all possible pairs (X, P) such that PX + 144 is a perfect square.

 No Solution Yet Submitted by K Sengupta No Rating

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 Solution Comment 9 of 9 |
`P^X + 144 = k^2P^X = (k+12)(k-12)Let k+12 = P^A -- (1)k-12 = P^B -- (2)SubtractP^A - P^B = 24P^B(P^(A-B)-1) = 24P^B(P^(A-B)-1) = 2^3 * 3 --- (3)This implies1) P is a divisor of 24 => P = 2 or 3 (B is not 0)2) B = 0Case 1)i) P = 2From (3) B = 1, 2 or 3From (2) k = P^B + 12=> k = 14, 16 or 20=> k+12 = 26, 28 or 32Only 32 is a power of 2 => X = A+B = 5+3= 8 (8,2)ii) P = 3From (3) B = 1From (2) k = P^B + 12 = 3+12 = 15=> k+12 = 27 = 3^3=> X = A+B = 3+1 = 4 (4,3)Case 2)B = 0 From (2) k = 1+12 = 13k+12 = 25 = 5^2X = A+B = 2+0 = 2 (2,5)Only Solutions are (8,2), (4,3) and (2,5)`

 Posted by Praneeth on 2010-01-13 03:28:30

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