# Many many tuples

Number Theory Level 5

Find the last three digits of the number of 599-tuples $$(x_1,x_2, \ldots, x_{599})$$ of integers where $$0\leq x_i \leq 598$$, for $$i=1$$ to 599, such that $x_1^2-x_2^2+x_3^2-x_4^2+\cdots-x_{598}^2+x_{599}^2 \equiv 1 \pmod{599}.$

×