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The number of solution of $x_1^2 + \cdots + x_k^2 \equiv \lambda \bmod q$
I'm playing with exponential sums...
If $q$ is an odd prime and $a$ an integer such that $q \nmid a$, then the following formula for the Gaussian sum is known
$$\sum_{x=0}^{q-1} e_q(ax^2) = \left(\...
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