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.
If X were given as 2, we would have the equation P
2 + 12
2 = n
2, 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 a
2 + b
2 = c
2.
There are four Pythagorean triples that fits Euclid's formula
[a = m
2 - n
2 : b = 2mn : c = m
2 + n
2] with 12 as a side:
(5,12,13), which corresponds with 5
2 = P
2 :: (2,5);
(9,12,20), which corresponds with 9
2 = 3
4 = P
2 :: (4,3);
(12,16,20), which corresponds with 16
2 = 2
8 = P
2 :: (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.
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Posted by Dej Mar
on 2010-01-12 14:41:51 |
an observation, partial proof (spoiler)
