Related papers: Prescribing the binary digits of primes

We consider Upper Domination, the problem of finding the minimal dominating set of maximum cardinality. Very few exact algorithms have been described for solving Upper Domination. In particular, no binary programming formulations for Upper…

Based on new explicit estimates for the prime counting function, we improve the currently known estimates for the particular sequence $C_n = np_n - \sum_{k \leq n}p_k$, $n \geq 1$, involving the prime numbers.

We investigate the distribution of the digits of quotients of randomly chosen positive integers taken from the interval $[1,T]$, improving the previously known error term for the counting function as $T\to+\infty$. We also resolve some…

We put a new conjecture on primes from the point of view of its binary expansions and make a step towards justification.

In the recent preprint [3], Goldston, Pintz, and Y{\i}ld{\i}r{\i}m established, among other things, $$ \liminf_{n\to\infty}{p_{n+1}-p_n\over\log p_n}=0,\leqno(0) $$ with $p_n$ the $n$th prime. In the present article, which is essentially…

By establishing an improved level of distribution we study almost primes of the form $f(p,n)$ where $f$ is an irreducible binary form over $\mathbb Z$.

The boundedness from $L^p \times L^q$ to $L^r$, $1<p,q \le \infty$, $0<1/p+1/q=1/r \le 1$, of bilinear pseudo-differential operators with symbols in the bilinear H\"ormander class $BS^m_{\rho,\rho}$, $0 \le \rho <1$, is proved for the…

Let $X$ be a large parameter. We will first give a new estimate for the integral moments of primes in short intervals of the type $(p,p+h]$, where $p\leq X$ is a prime number and $h=\odi{X}$. Then we will apply this to prove that for every…

We provide approximations to the prime counting function by various discretized versions of the logarithmic integral function, expressed solely in terms of the harmonic numbers. We demonstrate with explicit error bounds that these…

Currently there is no known efficient formula for primes. Besides that, prime numbers have great importance in e.g., information technology such as public-key cryptography, and their position and possible or impossible functional generation…

For two relatively prime square-free positive integers $a$ and $b$, we study integers of the form $a p+b P_{2}$ and give a new lower bound for the number of such representations, where $a p$ and $b P_{2}$ are both square-free, $p$ denote a…

Let $p_n$ be $n$th prime, and let $(S_n)_{n=1}^\infty:=(S_n)$ be the sequence of the sums of the first $2n$ consecutive primes, that is, $S_n=\sum_{k=1}^{2n}p_k$ with $n=1,2,\ldots$. Heuristic arguments supported by the corresponding…

We prove that there is a small but fixed positive integer e such that for every prime larger than a fixed integer, every subset S of the integers modulo p which satisfies |2S|<(2+e)|S| and 2(|2S|)-2|S|+2 < p is contained in an arithmetic…

BBP-type formulas are usually discovered experimentally, one at a time and in specific bases, through computer searches. In this paper, however, we derive directly, without doing any searches, explicit digit extraction BBP-type formulas in…

Here, we give upper and lower bounds on the count of positive integers $n\le x$ dividing the $n$th term of a nondegenerate linearly recurrent sequence with simple roots.

We study the equidistribution of multiplicatively defined sets, such as the squarefree integers, quadratic non-residues or primitive roots, in sets which are described in an additive way, such as sumsets or Hilbert cubes. In particular, we…

We introduce two new classes of covering codes in graphs for every positive integer $r$. These new codes are called local $r$-identifying and local $r$-locating-dominating codes and they are derived from $r$-identifying and…

In 1992, Erd$\H{o}$s and Hegyv$\'{a}$ri showed that for any prime p, there exist infinitely many length 3 weakly prime-additive numbers divisible by p. In 2018, Fang and Chen showed that for any positive integer m, there exists infinitely…

Building on previous results of Xing, we give new lower bounds on the rate of intersecting codes over large alphabets. The proof is constructive, and uses algebraic geometry, although nothing beyond the basic theory of linear systems on…

Positive bases, which play a key role in understanding derivative free optimization methods that use a direct search framework, are positive spanning sets that are positively linearly independent. The cardinality of a positive basis in…