The integers x,y,z are such that 29 divides the sum x⁴+y⁴+z⁴. Determine if 29⁴ also divides x⁴+y⁴+z⁴.

Obvious solutions
x=y=z=0

If x=y=z=29 then the sum is 3*29^4 which is divisible by 29^4.

You can generalize this result to:

(29a)^4 + (29b)^4 + (29c)^4 = 29^4(a^4+b^4+c^4) for any integers a,b,c.

So the answer to the problem is either "sometimes" or "yes, always"

Are there solutions where x,y,z are not all multiples of 29?

Posted by Jer on 2013-05-15 12:25:00