Related papers: Almost-prime $k$-tuples

Let $P$ be a subset of the primes of lower density strictly larger than $\frac12$. Then, every sufficiently large even integer is a sum of four primes from the set $P$. We establish similar results for $k$-summands, with $k\geq 4$, and for…

Let $m \in \mathbb{N}$ be large. We show that there exist infinitely many primes $q_{1}< \cdot\cdot\cdot < q_{m+1}$ such that \[ q_{m+1}-q_{1}=O(e^{7.63m}) \] and $q_{j}+2$ has at most \[ \frac{7.36m}{\log 2} + \frac{4\log m}{\log 2} + 21…

This work proposes elementary proofs of several related primes counting problems, based on an elementary weighted sieve. The subsets of primes considered here are the followings: the subset of twin primes PT = {p and p + 2 are primes}, the…

We prove that analogues of the Hardy-Littlewood generalised twin prime conjecture for almost primes hold on average. Our main theorem establishes an asymptotic formula for the number of integers $n=p_1p_2 \leq X$ such that $n+h$ is a…

The results of the study provide guidelines for the development and applications of algorithms. When the number of steps for calculating an assumption tends to infinity, probability theory can be applied to predict whether the assumption…

Suppose P is a set of primes, such that for every p in P, every prime factor of p-1 is also in P. If P does not contain all primes, we apply a new sieve method to show that the counting function of P is O(x^{1-c}) for some c>0, where c…

For two coprime positive integers $a,b$, let $T(a,b)=\{ ax+by : x,y\in \mathbb{Z}_{\ge 0} \} $ and let $s(a,b)=ab-a-b$. It is well known that all integers which are greater than $s(a,b)$ are in $T(a,b)$. Let $\pi (a, b)$ be the number of…

Let \beta be a real number. Then for almost all irrational \alpha>0 (in the sense of Lebesgue measure) \limsup_{x\to\infty}\pi_{\alpha,\beta}^*(x)(\log x)^2/x>=1, where \pi_{\alpha,\beta}^*(x)={p<=x: both p and [\alpha p+\beta] are primes}.

It was asked by E. Szemer\'edi if, for a finite set $A\subset\mathbb{Z}$, one can improve estimates for $\max\{|A+A|,|A\cdot A|\}$, under the constraint that all integers involved have a bounded number of prime factors -- that is, each…

Let $f(n,k)$ be the largest number of positive integers not exceeding $n$ from which one cannot select $k+1$ pairwise coprime integers, and let $E(n,k)$ be the set of positive integers which do not exceed $n$ and can be divided by at least…

We show that for every $r \geq 1$, and all $r$ distinct (sufficiently large) primes $p_1,..., p_r > p_0(r)$, there exist infinitely many integers $n$ such that ${2n \choose n}$ is divisible by these primes to only low multiplicity. From a…

New bounds on the number of similar or directly similar copies of a pattern within a finite subset of the line or the plane are proved. The number of equilateral triangles whose vertices all lie within an $n$-point subset of the plane is…

We say that $(a_1,...,a_k)$ is pairwise non-coprime if $\gcd(a_i,a_j) \ne 1$ for all $1 \le i <j \le k$. Let $a_1,a_2,a_3$ be positive integers less than $H$. We obtain an asymptotic formula for the number of $(a_1,a_2,a_3)$ that are…

The well-known Hardy--Ramanujan inequality states that if $\omega(n)$ denotes the number of distinct prime factors of a positive integer $n$, then there is an absolute constant $C>0$ such that uniformly for $x\ge2$ and $k\in\mathbb{N}$,…

We prove an asymptotic formula for the number of primes of the shape $a^2 +p^4$, thereby refining the well known work of Friedlander and Iwaniec. Along the way, we prove a result on equidistribution of primes up to $x$, in which the moduli…

If a sequence $(a_n)$ of non-negative real numbers has ``best possible'' distribution in arithmetic progressions, Bombieri showed that one can deduce an asymptotic formula for the sum $\sum_{n\le x} a_n \Lambda_k(n)$ for $k\ge 2$. By…

We derive asymptotic estimates for the coefficient of $z^{k}$ in $\left( f\left( z\right) \right) ^{n}$ when $n\rightarrow \infty $ and $k$ is of order $n^{\delta }$, where $0<\delta <1,$ and $f\left( z\right) $ is a power series satisfying…

We present the first fixed-length elementary closed-form expressions for the prime-counting function, $\pi(n)$, and the $n$-th prime number, $p(n)$. These expressions are arithmetic terms, requiring only a finite and fixed number of…

Let $f(n)$ be the number of distinct exponents in the prime factorization of the natural number $n$. We prove some results about the distribution of $f(n)$. In particular, for any positive integer $k$, we obtain that $$ \#\{n \leq x : f(n)…

Let $n$ be a positive integer, and let $k$ be a field (of arbitrary characteristic) accessible to symbolic computation. We describe an algorithmic test for determining whether or not a finitely presented $k$-algebra $R$ has infinitely many…