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x^6 = y^2 + 127

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by sanju09 » Thu Dec 10, 2009 10:08 pm
Find the number of pairs of positive integers (x, y) such that x^6 = y^2 + 127.

(A) 0
(B) 1
(C) 2
(D) 3
(E) 4
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by thephoenix » Thu Dec 10, 2009 10:27 pm
IMO B

x=4 and y=63
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by loi.fu.yogi » Fri Dec 11, 2009 5:25 am
At first glance, i thought, how could one possibly count all valid combination and say the answer with confidence in just 2 mins ?? So went defining an approach...... Please let me know if there is any other quick approach for these type of questions...

x^6 -y^2 =127

Assume z=x^3

z^2 - y^2 =127
(z+y)(z-y) =127

since 127 is a prime number the only possible combination is 1, 127.
since y and x are positive, z+y =127 and z-y =1 solving which we get x=4, y=63...
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