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Friedlander and Iwaniec investigated integral solutions $(x_1,x_2,x_3)$ ofthe equation $x_1^2+x_2^2-x_3^2=D$, where $D$ is square-free and satisfies thecongruence condition $D\equiv 5\bmod{8}$. They obtained an asymptotic formulafor solutions with $x_3\asymp M$, where $M$ is much smaller than $\sqrt{D}$. Tobe precise, their condition is $M\ge D^{1/2-1/1332}$. Their analysis led themto averages of certain Weyl sums. The condition of $D$ being square-free isessential in their work. We investigate the "opposite" case when $D=n^2$ is asquare of an odd integer $n$. This case is different in nature and leads tosums of Kloosterman sums. We obtain an asymptotic formula for solutions with$x_3\asymp M$, where $M\ge D^{1/2-1/16+\varepsilon}$ for any fixed$\varepsilon>0$.

Solutions of $x_1^2+x_2^2-x_3^2=n^2$ with small $x_3$