Geodesic
The distribution of pairs of quadratic residues and nonresidues of a~special form
Izvestiya. Mathematics , Tome 31 (1988) no. 2, pp. 307-323

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Nontrivial estimates are obtained for sums of Legendre symbols of a quadratic polynomial over primes in an arithmetic progression. These estimates are used to prove a theorem concerning the number of pairs of the form $(p+a,p+b)$, $p\equiv l(\operatorname{mod}k)$, $p\leqslant N$, for which $p+a$ is a quadratic residue (nonresidue), $p+b$ is a quadratic residue (nonresidue) modulo the prime $q$, and $N>k^3q^{0.75+\varepsilon}$. Bibliography: 27 titles. 
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     author = {A. A. Karatsuba},
     title = {The distribution of pairs of quadratic residues and nonresidues of a~special form},
     journal = {Izvestiya. Mathematics },
     pages = {307--323},
     publisher = {mathdoc},
     volume = {31},
     number = {2},
     year = {1988},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/IM2_1988_31_2_a3/}
}
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A. A. Karatsuba. The distribution of pairs of quadratic residues and nonresidues of a~special form. Izvestiya. Mathematics , Tome 31 (1988) no. 2, pp. 307-323. http://geodesic.mathdoc.fr/item/IM2_1988_31_2_a3/