相关论文: On sums of primes from Beatty sequences

For every integer $n\ge 1$ let $a_n$ be the smallest positive integer such that $n+a_n$ is prime. We investigate the behavior of the sequence $(a_n)_{n\ge 1}$, and prove asymptotic results for the sums $\sum_{n\le x} a_n$, $\sum_{n\le x}…

The paper gives some criteria for partial sums of rational number sequences to be not rational functions and to be not algebraic functions. As an application, we study partial sums of some famous rational number sequences in mathematical…

We extend a result by Ikeda and Suriajaya (2025) to find the asymptotic behaviour of the average number of representations of an integer $n$, over multiples of a fixed $q\ge 2$, as a sum of two prime $k$-th powers, for $k\ge 2$.

Let $\alpha = (1+\sqrt{5})/2$ and define the lower and upper Wythoff sequences by $a_i = \lfloor i \alpha \rfloor$, $b_i = \lfloor i \alpha^2 \rfloor$ for $i \geq 1$. In a recent interesting paper, Kawsumarng et al. proved a number of…

We have found several summation formulas that extend Ramanujan's psi sum. First contains a parameter $\alpha=1/N$, $N$ is a positive integer, and transforms to $q$-beta integral in the limit $N\to\infty$. The other is a $q$-analogue of…

We refine a recent result of Parsell on the values of the form $\lambda_1p_1 + \lambda_2p_2 + \mu_1 2^{m_1} + ...m + \mu_s 2^{m_s}, $ where $p_1,p_2$ are prime numbers, $m_1,...c, m_s$ are positive integers, $\lambda_1 / \lambda_2$ is…

Let $m$ be a positive integer larger than $1$, let $w$ be a finite word over $\left\{0,1,...,m-1\right\}$ and let $a_{m;w}(n)$ be the number of occurrences of the word $w$ in the $m$-expansion of $n$ mod $p$ for any non-negative integer…

We study the averages of ratios of characteristic polynomials over circular $\beta$-ensembles, where $\beta$ is a positive real number. Using Jack polynomial theory, we obtain three expressions for ratio averages. Two of them are given as…

We speculate on the distribution of primes in exponentially growing, linear recurrence sequences $(u_n)_{n\geq 0}$ in the integers. By tweaking a heuristic which is successfully used to predict the number of prime values of polynomials, we…

Denote by $\sigma_k(n)$ the sum of the $k$-th powers of the divisors of $n$, and let $S_k=\sum_{n\geq 1}\frac{\sigma_k(n)}{n!}$. We prove that Schinzel's conjecture H implies that $S_k$ is irrational, and give an unconditional proof for the…

Using a method of H. Davenport and W. M. Schmidt, we show that, for each positive integer n, the ratio 2/n is the optimal exponent of simultaneous approximation to real irrational numbers 1) by all conjugates of algebraic numbers of degree…

By using Beta Dirichlet series and then Eisenstein series we ca represent primes with first a good approximation and an exact expression. This can be done with arbitrary prime (up to 10^101).

Let $p_n$ denote the $n$th smallest prime number, and let $\boldsymbol{L}$ denote the set of limit points of the sequence $\{(p_{n+1} - p_n)/\log p_n\}_{n = 1}^{\infty}$ of normalized differences between consecutive primes. We show that for…

A folklore proof of Euclid's theorem on the infinitude of primes uses the Euler product and the irrationality of $\zeta(2) = \pi^2/6$. A quantified form of Euclid's Theorem is Bertrand's postulate $p_{n+1} < 2p_n$. By quantifying the…

Let $A, B$ be polynomial rings over a field $k$, and $I\subseteq A, J\subseteq B$ proper homogeneous ideals. We analyze the associated primes of powers of $I+J\subseteq A\otimes_k B$ given the data on the summands. The associated primes of…

Let $\alpha=0.a_1a_2a_3\ldots$ be an irrational number in base $b>1$, where $0\leq a_i<b$. The number $\alpha \in (0,1)$ is a \textit{normal number} if every block $(a_{n+1}a_{n+2}\ldots a_{n+k})$ of $k$ digits occurs with probability…

For $x\ge0$ let $\pi(x)$ be the number of primes not exceeding $x$. The asymptotic behaviors of the prime-counting function $\pi(x)$ and the $n$-th prime $p_n$ have been studied intensively in analytic number theory. Surprisingly, we find…

For $b>1$ and $\alpha\beta$ a string of two digits in base $b$, let $K_1$ be the subsum of the harmonic series with only those integers having exactly one occurrence of $\alpha\beta$. We obtain a theoretical representation of such $K_1$…

In this paper we show that certain sets are dense in $\mathbb{R}$. We give some applications. For example, we show an analytical proof that $q^{\frac{1}{n}}$, $q$ is a prime number and $e$; are irrational numbers. As another application we…

Prime numbers play a key role in number theory and have applications beyond Mathematics. In particular, in the Theory of Codes and also in Cryptography, the properties of prime numbers are relevant, because, from them, it is possible to…