On Diophantine Systems with Sum of Squares and Linear Form Satisfying a Congruential Condition of a Special Form

Abstract

An asymptotic formula with a residual term is obtained for the number of solutions of a Diophantine system containing the sum of squares of variables, as well as a linear form, and each solution of the system satisfies a congruential condition of a special kind. In addition, in the case of an even number of variables an upper estimate is given for the special series of the Diophantine system under study, relying on the formula for the number of solutions of congruences of the second degree modulo the power of a prime number.