Diophantine approximation by special primes

We show that whenever $\delta>0$, $\eta$ is real and constants $\lambda_i$ satisfy some necessary conditions, there are infinitely many prime triples $p_1,\, p_2,\, p_3$ satisfying the inequality $|\lambda_1p_1 + \lambda_2p_2 + \lambda_3p_3+\eta|<(\max p_j)^{-1/12+\delta}$ and such that, for each $i\in\{1,2,3\}$, $p_i+2$ has at most $28$ prime factors

Square-free values of n2+n+1\mathbf{n^2+n+1}

In this paper we show that there exist infinitely many square-free numbers of the form $n^2+n+1$. We achieve this by deriving an asymptotic formula by improving the reminder term from previous results.Comment: arXiv admin note: text overlap with arXiv:2004.0997

The quaternary Piatetski-Shapiro inequality with one prime of the form p=x2+y2+1\mathbf{p=x^2+y^2+1}

In this paper we show that, for any fixed $1<c<967/805$, every sufficiently large positive number $N$ and a small constant $\varepsilon>0$, the diophantine inequality \begin{equation*} |p_1^c+p_2^c+p_3^c+p_4^c-N|<\varepsilon \end{equation*} has a solution in prime numbers $p_1,\,p_2,\,p_3,\,p_4$, such that $p_1=x^2 + y^2 +1$.Comment: arXiv admin note: substantial text overlap with arXiv:2011.0396

Pairs of square-free values of the type n2+1\mathbf{n^2+1}, n2+2\mathbf{n^2+2}

In the present paper we show that there exist infinitely many consecutive square-free numbers of the form $n^2+1$, $n^2+2$. We also establish an asymptotic formula for the number of such square-free pairs when $n$ does not exceed given sufficiently large positive integer