True or False?

Number Theory Level 3

True or False?

There exists integer solutions $(x, y, z)$ to $(x-y)^3+3(y-z)^2+5|z-x|=2017$ .

False True

1 solution

Linkin Duck
Mar 16, 2017

Since $(x-y)+(y-z)+(z-x)=0$ is an even number, the set $[(x-y),(y-z),(z-x)]$ contains 3 even numbers or 1 even number and 2 odd numbers, which always leads to the fact that $M=(x-y)^3+3(y-z)^2+5|z-x|$ must be even, hence $M$ cannot be $2017$ .

True or False?

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