The distribution of pairs of quadratic residues and nonresidues of a special form

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 $N>k^3q^{0.75+\varepsilon}$.
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