Determine all possible integer solutions, whether positive or negative, to this equation:

5y⁴ + 2560y² = x⁵ - 65536

x = 16, y = 16    or    x = 16, y = -16  (x has to be positive, but y could go either way, of course)

I wish I had some formal mathematical proof for this, but I don't, I'll leave that to the smart people.  I sort of happened upon it by accident, noticing that 2560 = 10(16^2) and 65536 = 16^4.  I figured if x or y was 16 it would be easy to simplify...turns out it was. 

One of those I just "saw", can't really explain the inspiration.  Just figured I'd check out the "16" theme, normally this is the type of puzzle I won't even try - I'll wait for someone else to show me how it's done. :)

 

Posted by tomarken on 2006-03-15 15:15:43