X is a positive integer > 1 and, P is a prime number.
Determine all possible pairs (X, P) such that
P^{X} + 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.

Posted by Dej Mar
on 20100112 14:41:51 