Related papers: Pairwise non-coprimality of triples

We obtain an upper bound for the number of pairs $ (a,b) \in {A\times B} $ such that $ a+b $ is a prime number, where $ A, B \subseteq \{1,...,N \}$ with $|A||B| \, \gg \frac{N^2}{(\log {N})^2}$, $\, N \geq 1$ an integer. This improves on a…

Let k>2 be a fixed integer exponent and let \theta > 9/10. We show that a positive integer N can be represented as a non-trivial sum or difference of 3 k-th powers, using integers of size at most B, in O(B^{\theta}N^{1/10}) ways, providing…

We prove that suitable asymptotic formulae in short intervals hold for the problems of representing an integer as a sum of a prime and a square, or a prime square. Such results are obtained both assuming the Riemann Hypothesis and in the…

For any positive integer r, let pi_{2r}(x) denote the number of prime pairs (p, p+2r) with p not exceeding (large) x. According to the prime-pair conjecture of Hardy and Littlewood, pi_{2r}(x) should be asymptotic to 2C_{2r}li_2(x) with an…

A nonempty finite set of positive integers A is relatively prime if gcd(A) = 1 and it is relatively prime to n if gcd(A [ fng) = 1. The number of nonempty subsets of A which are relatively prime to n is \Phi(A, n) and the number of such…

We consider the problem of counting $k$-tuples of positive integers satisfying any arbitrary set of gcd conditions, where every integer is not larger than $x$. We first establish the conditions to guarantee the existence of such tuples, and…

For an integer $k\ge2$, a tuple of $k$ positive integers $(M_i)_{i=1}^{k}$ is called an amicable $k$-tuple if the equation \[ \sigma(M_1)=\cdots=\sigma(M_k)=M_1+\cdots+M_k \] holds. This is a generalization of amicable pairs. An amicable…

The Euler's totient function $ \varphi(n) $ counts the positive integers up to a given integer $ n$ that are relatively prime to $ n $. We solve a problem due to Lehmer that there is no composite number $ n $ such that $ \varphi(n)\mid n-1…

We obtain an asymptotic formula with a power-saving error term for counting the integer points $(a,b,c,d)$ in an expanding box $[-X,X]^4$ that satisfy the determinant equation $x_1x_2-x_3x_4=r$ for $r \neq 0$ with two of entries to be…

A hypergraph is simple if it has no loops and no repeated edges, and a hypergraph is linear if it is simple and each pair of edges intersects in at most one vertex. For $n\geq 3$, let $r= r(n)\geq 3$ be an integer and let $\boldsymbol{k} =…

Every natural number greater than $2$ can be written as the sum of a prime and a square-free number, and recent work has imposed additional divisibility conditions on the square-free number. We overcome limitations in these works to prove…

Given a power $q$ of a prime number $p$ and "nice" polynomials $f_1,...,f_r\in\bbF_q[T,X]$ with $r=1$ if $p=2$, we establish an asymptotic formula for the number of pairs $(a_1,a_2)\in\bbF_q^2$ such that…

Let $1 < k < 33 / 29$. We prove that if $\lambda_1$, $\lambda_2$ and $\lambda_3$ are non-zero real numbers, not all of the same sign and that $\lambda_1 / \lambda_2$ is irrational and $\varpi$ is any real number, then for any $\eps > 0$ the…

Fix $\left\{a_1, \dots, a_n \right\} \subset \mathbb{N}$, and let $x$ be a uniformly distributed random variable on $[0,2\pi]$. The probability $\mathbb{P}(a_1,\ldots,a_n)$ that $\cos(a_1 x), \dots, \cos(a_n x)$ are either all positive or…

Let $E_0,\ldots,E_n$ be a partition of the set of prime numbers, and define $E_j(x) := \sum_{p \in E_j \atop p \leq x} \frac{1}{p}$. Define $\pi(x;\mathbf{E},\mathbf{k})$ to be the number of integers $n \leq x$ with $k_j$ prime factors in…

The paper solves the problems of determining the asymptotics of the number of primes and the sums of functions of primes in a subset of the natural series that satisfies the conditions that the asymptotic density of the number of primes in…

A recent conjecture by C. Carlet on the sum-freedom of the binary multiplicative inverse function can be stated as follows: For each pair of positive integers $(n,k)$ with $3\le k\le n-3$, there is a $k$-dimensional $\Bbb F_2$-subspace $E$…

Let $a_{1},...,a_{n}, b_{1},...,b_{n}$ be random variables in some (non-commutative) probability space, such that $\{a_{1}, ..., a_{n} \}$ is free from $\{b_{1}, ..., b_{n} \}$. We show how the joint distribution of the $n$-tuple $(a_{1}…

Every integer greater than two can be expressed as the sum of a prime and a square-free number. Expanding on recent work, we provide explicit and asymptotic results when divisibility conditions are imposed on the square-free number. For…

Fix an integer $l$ such that $|l|$ is a prime $3$ modulo $4$. Let $d > 0$ be a squarefree integer and let $N_d(x, y)$ be the principal binary quadratic form of $\mathbb{Q}(\sqrt{d})$. Building on a breakthrough of Alexander Smith, we give…