Fermat's Last Theorem

Number Theory Level 5

Andrew Wiles, pictured above, was just awarded the 2016 Abel Prize for his 1994 proof of Fermat's Last Theorem.

Suppose $$n=2016$$ and $$|x|,|y|,|z|$$ are all less than 10. How many solutions in integers are there to the equation on the chalkboard?

Note: A solution here refers to an ordered triple $$(x,y,z).$$

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