Abstract:
The article deals with two problems of approximating a given positive number $N$ by the sum of two primes, and by the sum of a prime and two squares of primes.
In 2001, R. Baker, G. Harman, and J. Pintz proved for the number of solutions of the inequality $|p-N|\leqslant H$ in primes $p$ a lower bound for $H\geqslant N^{21/40+\varepsilon}$, where $\varepsilon$ is an arbitrarily small positive number. Using this result and the density technique, in this paper we prove a lower bound for the number of solutions of the inequality $|p_1+p_2-N| \leqslant H$ in prime numbers $p_1$, $p_2$ for $H\geqslant N^{7/80+\varepsilon}$.
Also based on the density technique, we prove a lower bound for the number of solutions of the inequality $\left|p_1^2+p_2^2+p_3-N\right| \leqslant H$ in prime numbers $p_1$, $p_2$ and $p_3$ for $H\geqslant N^{7/72+\varepsilon}$.
Keywords:diophantine inequalities, prime numbers, density theorems.
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