arXiv subjects
Publications and source records attributed to S. I. Dimitrov.
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.
In this paper we study the distribution of consecutive square-free numbers of the forms $x^2+y^2+z+1$, $x^2+y^2+z+2$ and $x^2+y^2+z^2+z+1$, $x^2+y^2+z^2+z+2$, respectively. We establish asymptotic formulas for each of these two cases.
Let $γ<1<c$ and $19(c-1)+171(1-γ)<9$. In this paper, we establish an asymptotic formula for exponential sums over Piatetski-Shapiro primes $p=[n^{1/γ}]$ in arithmetic progressions.
Let $[\, \cdot\,]$ be the floor function. In this paper we show that when $1<c<\frac{3849}{3334}$, then there exist infinitely many prime numbers of the form $[n^c]$, where $n$ is square-free.
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.
In this paper, we establish a theorem of Bombieri -- Vinogradov type for exponential sums over Piatetski-Shapiro primes $p= [n^{1/γ}]$ with $\frac{865}{886}<γ< 1$.
In this paper we introduce a new diophantine equation with prime numbers. Let $[\, \cdot\,]$ be the floor function. We prove that when $1 1$ is a fixed, then every sufficiently large positive integer $N$ can be represented in the form \begin{equation*} N=\big[p^c_1\tan^θ(\log p_1)\big]+ \big[p^c_2\tan^θ(\log p_2)\big]+ \big[p^c_3\tan^θ(\log p_3)\big]\,, \end{equation*} where $p_1,\, p_2,\, p_3$ are prime numbers. We also establish an asymptotic formula for the number of such representations.
Let $[\, \cdot\,]$ be the floor function. In the present paper we prove that when $1 1$ is a fixed, then there exist infinitely many prime numbers of the form $[n^c \tan^θ(\log n)]$.
In this paper we introduce a new diophantine inequality with prime numbers. Let $1 1$, every sufficiently large positive number $N$ and a small constant $\varepsilon>0$, the tangent inequality \begin{equation*} \big|p^c_1\tan^θ(\log p_1)+ p^c_2\tan^θ(\log p_2)+ p^c_3\tan^θ(\log p_3) -N\big|<\varepsilon \end{equation*} has a solution in prime numbers $p_1,\,p_2,\,p_3$.
Let $\varepsilon>0$ be a small constant. In the present paper we prove that whenever $η$ is real and constants $λ_i$ satisfy some necessary conditions, then there exist infinitely many prime triples $p_1,\, p_2,\, p_3$ satisfying the inequality \begin{equation*} |λ_1p_1 + λ_2p_2 + λ_3p_3+η|<\varepsilon \end{equation*} and such that $p_3=x^2 + y^2 +1$.
Let $[\,\cdot\,]$ be the floor function. In this paper we show that whenever $η$ is real, the constants $λ_i$ satisfy some necessary conditions, then for any fixed $1<c<38/37$ there exist infinitely many prime triples $p_1,\, p_2,\, p_3$ satisfying the inequality \begin{equation*} |λ_1p_1 + λ_2p_2 + λ_3p_3+η|<(\max p_j)^{{\frac{37c-38}{26c}}}(\log\max p_j)^{10} \end{equation*} and such that $p_i=[n_i^c]$, $i=1,\,2,\,3$.
Let $[\, \cdot\,]$ be the floor function. In this paper we show that every sufficiently large positive integer $N$ can be represented in the form \begin{equation*} N=[p_1\log p_1]+[p_2\log p_2]+[p_3\log p_3], \end{equation*} where $p_1,\, p_2,\, p_3$ are prime numbers. We also establish an asymptotic formula for the number of such representations, when $p_1,\, p_2,\, p_3$ do not exceed given sufficiently large positive number.
Assume that $N$ is a sufficiently large positive number. In this paper we show that for a small constant $\varepsilon>0$, the logarithmic inequality \begin{equation*} \big|p_1\log p_1+p_2\log p_2+p_3\log p_3-N\big|<\varepsilon \end{equation*} has a solution in prime numbers $p_1,\,p_2,\,p_3$.
Let $[\, \cdot\,]$ be the floor function. In this paper, we show that when $1<c<37/36$, then every sufficiently large positive integer $N$ can be represented in the form \begin{equation*} N=[p^c_1]+[p^c_2]+[p^c_3]\,, \end{equation*} where $p_1,p_2,p_3$ are primes close to squares.
Let $c$ be fixed with $1 0$, the diophantine inequality \begin{equation*} |p_1^c+p_2^c+p_3^c-N|<\varepsilon \end{equation*} is solvable in primes $p_1,\,p_2,\,p_3$ near to squares.
In the present paper we show that there exist infinitely many consecutive square-free numbers of the form $x^2+y^2+1$, $x^2+y^2+2$. We also give an asymptotic formula for the number of pairs of positive integers $x, y \leq H$ such that $x^2+y^2+1$, $x^2+y^2+2$ are square-free.
In this short paper we shall prove that there exist infinitely many consecutive square-free numbers of the form $[αp]$, $[αp]+1$, where $p$ is prime and $α>0$ is irrational algebraic number. We also establish an asymptotic formula for the number of such square-free pairs when $p$ does not exceed given sufficiently large positive integer.
In the present paper we show that there exist infinitely many consecutive square-free numbers of the form $[αn]$, $[αn]+1$, where $α>1$ is irrational number with bounded partial quotient or irrational algebraic number.