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2011c.00466 H\'ezard, David Sums of squares in a finite field. (Sommes de carr\'es dans un corps fini.) Quadrature 76, 44-48 (2010). 2010 Quadrature, Revigny-sur-Ornain FR F60 sum of squares finite field nonhomogeneous linear recurrence

doi:10.1051/quadrature/2010001

The author computes, for each element $x$ of a finite field $\mathbb F_q$ and each positive integer $k$, the number $N_k(x)$ of $k$-tuples formed by elements from $\mathbb F_q$ whose squares add up to $x$. This is easy in case of characteristic $2$, where the answer is given by simple formulas. In odd characteristic case, one first studies $N_1(x)$ and $N_2(x)$, then one finds a nonhomogeneous linear recurrence of order two for the sequence $\left(N_k(x)\right)_{k\ge 1}$. Characters of the additive group $\mathbb F_q$ make appearance in the proof. Mihai Cipu (Bucure\c{s}ti)