Related papers: Very small intervals containing at least three pri…

We prove the existence of primitive sets (sets of integers in which no element divides another) in which the gap between any two consecutive terms is substantially smaller than the best known upper bound for the gaps in the sequence of…

We study the properties of certain graphs involving the sums of primes. Their structure largely turns out to relate to the distribution of prime gaps and can be roughly seen in Cram\'er's model as well. We also discuss generalizations to…

We prove that (under the assumption of the generalized Riemann hypothesis) a totally real multiquadratic number field $K$ has a positive density of primes $p \in \mathbb{Z}$ for which the image of the unit group $(\mathcal{O}_K)^{\times})$…

We construct families of prime ideals in polynomial rings for which the number of associated primes of the second power (or higher powers) is exponential in the number of variables in the ring. We give a lower bound on the Ananyan-Hochster…

In this paper, we show a new upper bound of prime gaps, that is the gap between a prime number and its consecutive prime number. We show that the gap between a prime number $p_n$ and its consecutive prime number is not larger than…

The set of primes where a hypergeomeric series with rational parameters is $p$-adically bounded is known by [10] to have a Dirichlet density. We establish a formula for this Dirichlet density and conjecture that it is rare for the density…

The sequence $(x_n)_{n\in\mathbb N} = (2,5,15,51,187,\dots)$ given by the rule $x_n=(2^n+1)(2^{n-1}+1)/3$ appears in several seemingly unrelated areas of mathematics. For example, $x_n$ is the density of a language of words of length $n$…

Let $X$ be large and let $\mathcal{P}$ denote the set of primes. Fix positive real parameters $r_1,\dots,r_s$ and a parameter $\lambda\geqslant 1$ determined by a balancing relation, and let $\mathcal{A}_{\lambda}(X)\subset[1,2X]$ be the…

Let $\mathcal{A}$ denote a finite set of arithmetic progressions of positive integers and let $s \geq 2$ be an integer. If the cardinality of $\mathcal{A}$ is at least 2 and $U$ is the union formed by taking certain arithmetic progressions…

The appearance of primes in a family of linear recurrence sequences labelled by a positive integer $n$ is considered. The terms of each sequence correspond to a particular class of Lehmer numbers, or (viewing them as polynomials in $n$)…

Let $f(n)=\min_{p} |n-p|$, where $p$ is a prime. We show that there is a positive constant $\delta$ such that for any large integer $N$ there exist two positive integers $n_1$ and $n_2$ such that $N=n_1 + n_2$ and $f(n_i)\gg \ln N (\ln\ln…

A prime p is called Sophie Germain prime if 2p+1 is also prime. A formula for the density of such primes is given in a more general setting using a new approach. This method uses the Ramanujan-Fourier series for a modified von Mangoldt…

Let us call a simple graph on $n\geq 2$ vertices a prime gap graph if its vertex degrees are $1$ and the first $n-1$ prime gaps. We show that such a graph exists for every large $n$, and in fact for every $n\geq 2$ if we assume the Riemann…

Let $p_n$ denote the $n$-th prime number, $\{q_n\}$ be a sequence of positive numbers and $x\in\mathbb{R}$. In this note we prove that the inequality $$q_n p_{n+1}^{x}-q_{n+1}p_{n}^{x}<p_{n}^{x}p_{n+1}^{x-1}, $$ holds for infinitely many…

Let $d_n = p_{n+1} - p_n$, where $p_n$ denotes the $n$th smallest prime, and let $R(T) = \log T \log_2 T\log_4 T/(\log_3 T)^2$ (the "Erd{\H o}s--Rankin" function). We consider the sequence $(d_n/R(p_n))$ of normalized prime gaps, and show…

Let \psi(x) be a polynomial with rational coefficients. Suppose that \psi has the positive leading coefficient and zero constant term. Let A be a set of positive integers with the positive upper density. Then there exist x,y\in A and a…

By extending the notion of spin of prime ideals, we show that a short character sum conjecture implies that the set of primes raising the level of a certain even Galois representation has density 2/3, as conjectured by Ramakrishna in 1998.

In this paper we review the properties of families of numbers of the form $6n\pm1$, with $n$ integer (in which there are all prime numbers greater than 3 and other compound numbers with particular properties) to later use them in a new…

For integers $k\geq 3,d\geq 2,$ we consider the abundance property of pinned $k$-point patterns occurring in $E\subseteq \mathbb R^d$ with positive upper density $\delta(E)$. We show that for any fixed $k$-point pattern $V$, there is a set…

We develop a unified density-based framework for primality, coprimality, and prime pairs, and introduce an intrinsic normalized model for prime gaps constrained by the Prime Number Theorem. Within this setting, a structural tension between…