Related papers: Mertens' prime product formula, dissected

Let $K$ be a quadratic field which is not an imaginary quadratic field of class number one. We describe an algorithm to compute the primes $p$ for which there exists an elliptic curve over $K$ admitting a $K$-rational $p$-isogeny. This…

We introduce a new class of pseudoprimes-so called "overpseudoprimes to base $b$", which is a subclass of strong pseudoprimes to base $b$. Denoting via $|b|_n$ the multiplicative order of $b$ modulo $n$, we show that a composite $n$ is…

Consider a strongly $b$-multiplicative sequence and a prime $p$. Studying its $p$-rarefaction consists in characterizing the asymptotic behaviour of the sums of the first terms indexed by the multiples of $p$. The integer values of the…

We give a simple proof of a major index determinant formula in the symmetric group discovered by Krattenthaler and first proved by Thibon using noncommutative symmetric functions. We do so by proving a factorization of an element in the…

In this paper we use the framework of automatic sequences to study combinatorial sequences modulo prime powers. Given a sequence whose generating function is the diagonal of a rational power series, we provide a method, based on work of…

Given a finitely generated multiplicative subgroup of rational numbers $\Gamma$, assuming the Generalized Riemann Hypothesis, we determine an asymptotic formula for average over prime numbers, powers of the order of the reduction group…

In this paper, we are going to prove a famous problem concerning prime numbers. Bertrand postulate states that there is always a prime p with n < p < 2n, if n > 1. Bertrand postulate is not a newer one to be proven, in fact, after his…

Let $p>3$ be a prime. Gauss first introduced the polynomial $S_p(x)=\prod_{c}(x-\zeta_p^c),$ where $0<c<p$ and $c$ varies over all quadratic residues modulo $p$ and $\zeta_p=e^{2\pi i/p}$. Later Dirichlet investigated this polynomial and…

It is known that there are infinitely-many prime numbers which take the form of a polynomial of degree one with integer coefficients, this is Dirichlet's theorem. We use an elementary sieving argument together with bounds on the prime…

Let $p>3$ be a prime, and let $d\in\mathbb Z$ with $p\nmid d$. For the determinants $$S_m(d,p)=\det\left[(i^2+dj^2)^{m}\right]_{1\leqslant i,j \leqslant (p-1)/2}\ \ \left(\frac{p-1}2\leqslant m\leqslant p-1\right),$$ Sun recently determined…

Let g be a non-zero rational number. Let N_{g,t}(x) denote the number of primes p<=x for which the subgroup of the multiplicative group of the finite field having p elements that is generated by g mod p is of residual index t. In Part I,…

Let $K$ be a number field and let $G$ be a finitely generated subgroup of $K^\times$. For all but finitely many primes $\mathfrak p$ of $K$, the reduction $(G \bmod \mathfrak p)$ generates a well-defined subgroup of the multiplicative group…

For a positive integer $n$, let $p(n)$ be the number of ways to express $n$ as a sum of positive integers. In this note, we revisit the derivation of the Rademacher's convergent series for $p(n)$ in a pedagogical way, with all the details…

For a random permutation sampled from the stationary distribution of the TASEP on a ring, we show that, conditioned on the event that the first entries are strictly larger than the last entries, the order of the first entries is independent…

Assuming a $q$-variant of the prime $k$-tuple conjecture uniformly, we compute mixed moments of the number of primes in disjoint short intervals and progressions, respectively. This involves estimating the mean of singular series along…

A famous conjecture of Artin asserts that any integer $a$ that is neither $-1$ nor a square should be a primitive root (mod $p$) for a positive proportion of primes $p$. Moreover, using a heuristic argument, Artin guessed an explicit…

Let $p$ be a sufficiently large prime number, $r$ be any given positive integer. Suppose that $a_1,\,\dots,\,a_r$ are pairwise distinct and not zero modulo $p$. Let $N(a_1,\,\dots,\,a_r;\,p)$ denote the number of…

Let $p_n$ be the $n$th prime, and consider the sequence $s_n = (2\cdot3\cdots p_n)^{1/n} = (p_n\#)^{1/n}$, the geometric mean of the first $n$ primes. We give a short proof that $p_n/s_n \to e$, a result conjectured by Vrba (2010) and…

Under the assumption of Heath-Brown's conjecture on the first prime in an arithmetic progression, we prove that there are infinitely many Carmichael numbers $n$ such that the number of prime factors of $n$ is prime.

We establish a lower bound of 2/p(p-1) for the asymptotic density of the Motzkin numbers divisible by a general prime number p > 3. We provide a criteria for when this asymptotic density is actually 1. We also provide a partial…