On representations of a given number as the sum of two primes and the square of a third prime in an arithmetic progression

This paper considers the problem of representing a given natural number as combination of two prime numbers and the square of a third prime number taken from an arithmetic progression. It was the rst to establish the solvability of the equation under consideration in prime numbers from the arithmetic progression, and prove a lower estimate for the number of solutions to this equation. The results obtained are important in the study of additive problems with prime numbers. The proof of the obtained results uses the Hardy-Littlewood circular method and the Vinogradov method of trigonometric sums.

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