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PX + 144 = Perfect Square (Posted on 2010-01-12) Difficulty: 3 of 5
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    
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Some Thoughts an observation, partial proof (spoiler) | Comment 8 of 9 |
If X were given as 2, we would have the equation P2 + 122 = n2, where P is the prime number and n is the root of the resulting perfect square. The equation is very much like that for a Pythagorean triple (a,b,c) where a2 + b2 = c2.

There are four Pythagorean triples that fits Euclid's formula
[a = m2 - n2 : b = 2mn : c = m2 + n2] with 12 as a side:

(5,12,13), which corresponds with 52 = P2 :: (2,5);
(9,12,20), which corresponds with 92 = 34 = P2 :: (4,3);
(12,16,20), which corresponds with 162 = 2= P2 :: (8,2); and
(12,35,37)... but, as 35 is neither prime nor a perfect square, it is not a solution and does not lead to a solution.

Thus, no other solutions exist for X as 2 or as a multiple of 2.
  Posted by Dej Mar on 2010-01-12 14:41:51
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