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We solve a nonlinear additive problem in prime and natural numbers whose binary expansion is of a special form.

Let N0 be the set of natural numbers whose binary expansions have an even number of 1’s, and let N1 = N\N0. In this paper, we obtain asymptotic formulas for the number of primes p not exceeding X and such that p ∈ N i , p + 1 ∈ N j , where i and j take values 0 and 1 independently of each other.

We obtain an asymptotic formula for the mean value of the function τk(n), which is the number of solutions of the equation x1 … xk = n in natural numbers x1, …, xk, in some special sequences of natural numbers.

We obtain estimates of the trigonometric sum $$S_N (\alpha ) = \sum\nolimits_{n = 1}^N {\varepsilon (n){\text{exp(}}} 2\pi i\alpha n)$$ .