# Integer solutions to equation p^3=q^2

• January 25th 2010, 01:54 AM
borophyll
Integer solutions to equation p^3=q^2
Hi, I am wondering if there are solutions to the equation p^2 = q^3, where p and q are integers (ignoring zero and one). If so, can someone give me a hint on how to find them. If not, can someone show me a proof that this is the case?
• January 25th 2010, 01:59 AM
tonio
Quote:

Originally Posted by borophyll
Hi, I am wondering if there are solutions to the equation p^2 = q^3, where p and q are integers (ignoring zero and one). If so, can someone give me a hint on how to find them. If not, can someone show me a proof that this is the case?

$8^2=4^3\,,\,27^2=9^3\,,\,125^2=25^3$ ...can you see a pattern here?

Tonio
• January 25th 2010, 02:07 AM
borophyll
Quote:

Originally Posted by tonio
$8^2=4^3\,,\,27^2=9^3\,,\,125^2=25^3$ ...can you see a pattern here?

Tonio

ah of course, $(n^2)^3 = (n^3)^2$, for any n

How could I be so dumb?