We prove a Bombieri-Vinogradov type theorem for the number of representations of an integer $N$ in the form $N=p_1^g+p_2^g+...+p_s^g$ with $p_1,p_2,...,p_s$ prime numbers such that $p_1\equiv l\left(\mathrm{mod}k\right)$, under suitable hypothesis on $s=s\left(g\right)$ for every integer $g\ge 2$.

In this paper we consider the asymptotic formula for the number of the solutions of the equation $p_1+p_2+p_3=N$ where $N$ is an odd integer and the unknowns $p_i$ are prime numbers of the form $p_i=\left[n^{1/{\gamma }_i}\right]$. We use the two-dimensional van der Corput’s method to prove it under less restrictive conditions than before. In the most interesting case ${\gamma }_1={\gamma }_2={\gamma }_3=\gamma$ our theorem implies that every sufficiently large odd integer $N$ may be written as the sum of three Piatetski-Shapiro primes of type $\gamma$ for $50/53$ < $\gamma$ < $1$.

Romanoff (1934) showed that integers that are the sum of a prime and a power of 2 have positive lower asymptotic density in the positive integers. We adapt his method by showing more generally the existence of a positive lower asymptotic density for integers that are the sum of a prime and a term of a given nonconstant nondegenerate integral linear recurrence with separable characteristic polynomial.

For $\epsilon \>0$ and any sufficiently large odd $n$ we show that for almost all $k\le R:=n^{1/5-\epsilon }$ there exists a representation $n=p_1+p_2+p_3$ with primes $p_i\equiv b_i$ mod $k$ for almost all admissible triplets $b_1,b_2,b_3$ of reduced residues mod $k$.

Let $N$ be a sufficiently large integer. We prove that almost all sufficiently large even integers $n\in [N-6U,N+6U]$ can be represented as $$n=p_1^2+p_2^3+p_3^3+p_4^3+p_5^3+p_6^3,\left|p_1^2-{\displaystyle \frac{N}{6}}\right|\le U,\left|p_i^3-{\displaystyle \frac{N}{6}}\right|\le U,i=2,3,4,5,6,$$ where $U=N^{1-\delta +\epsilon }$ with $\delta \le 8/225$.