Consider the equation x^2+y^5=z^3 where x, y, and z, are positive integers.

(A) Can you give at least three solutions to it?

(B) Determine whether or not there is an infinite number of solutions.

(In reply to

No Subject by Ferdinand)

2^(k-1) + 2^(k-1)

= (2^k)/2 + (2^k)/2

= (1/2)*2*(2^k)

= 2^k

This proves that 2^(k-1) + 2^(k-1) = 2^k as has been asserted in the solution text.

NOTE:

The first line of the solution text is

" Substituting: x = 2a, y = 5b, and z = 3c, in the original equation, one obtains" instead of "Substituting: x = 2a, y = 2b, and z = 2c....".

The error is regretted.

*Edited on ***December 31, 2006, 12:46 pm**