Geodesic
The distribution of Hardy--Littlewood numbers in arithmetic progressions
Izvestiya. Mathematics , Tome 34 (1990) no. 1, pp. 213-228.

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An asymptotic formula is obtained for the number of solutions of the congruence $$ p+n^2\equiv l\ (\operatorname{mod}D),\qquad p\leqslant x,\quad n\leqslant\sqrt x,\quad(l,D)=1, $$ where $D$ is a sufficiently large prime. Bibliography: 7 titles. 
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Z. Kh. Rakhmonov. The distribution of Hardy--Littlewood numbers in arithmetic progressions. Izvestiya. Mathematics , Tome 34 (1990) no. 1, pp. 213-228. http://geodesic.mathdoc.fr/item/IM2_1990_34_1_a9/

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