Geodesic
Sums of characters with prime numbers in an arithmetic progression
Izvestiya. Mathematics , Tome 5 (1971) no. 3, pp. 485-501.

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One of the corollaries of the fundamental theorem in this paper is a theorem about power residues and nonresidues $\operatorname{mod}q$ in sequences of the form $p+k$, where the prime numbers $p$ belong to the beginning of an arithmetic progression. 
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A. A. Karatsuba. Sums of characters with prime numbers in an arithmetic progression. Izvestiya. Mathematics , Tome 5 (1971) no. 3, pp. 485-501. http://geodesic.mathdoc.fr/item/IM2_1971_5_3_a1/

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[2] Karatsuba A. A., “Summy kharakterov s prostymi chislami”, Izv. AN SSSR. Ser. matem., 34 (1970), 229–321

[3] Linnik Yu. V., “On the least prime in an arithmetic progression. I. The bacic theorem”, Matem. sb., 15(57) (1944), 139–178 | MR | Zbl

[4] Fogels E. K., “O prostykh chislakh v nachale arifmeticheskoi progressii”, Dokl. AN SSSR, 102:3 (1955), 455–456 | MR | Zbl