Related papers: ABC implies primitive prime divisors in arithmetic…

Let $\theta$ be an algebraic integer and $f(x)=x^{n}+ax^{n-1}+bx+c$ be the minimal polynomial of $\theta$ over the rationals. Let $K=\mathbb{Q}(\theta)$ be a number field and $\mathcal{O}_{K}$ be the ring of integers of $K.$ In this…

We show that integer partitions, the fundamental building blocks in additive number theory, detect prime numbers in an unexpected way. Answering a question of Schneider, we show that the primes are the solutions to special equations in…

Let p be a rational prime and let F be a number field. Then, for each i>0, there is a short exact localization sequence for K_{2i}(F). If p is odd or F is nonexceptional, we find necessary and sufficient conditions for this exact sequence…

Given a prime power $q$ and an integer $n\geq2$, we establish a sufficient condition for the existence of a primitive pair $(\alpha,f(\alpha))$ where $\alpha \in \mathbb{F}_q$ and $f(x) \in \mathbb{F}_q(x)$ is a rational function of degree…

We prove the following: there is a primitive recursive function f_-^*(-,-), in the three variables, such that: for every natural numbers t,n>0, and c, for any natural number k>=f^*_t(n,c) the following holds. Assume L is an alphabet with…

Let $m$ be any positive integer and let $\delta_1,\delta_2\in\{1,-1\}$. We show that for some constanst $C_m>0$ there are infinitely many integers $n>1$ with $p_{n+m}-p_n\le C_m$ such that $$\left(\frac{p_{n+i}}{p_{n+j}}\right)=\delta_1\…

Let $G$ be a finite group. Let $K/k$ be a Galois extension of number fields with Galois group isomorphic to $G$, and let $C \subseteq \mathrm{Gal}(K/k) \simeq G$ be a conjugacy invariant subset. It is well known that there exists an…

Let $k$ be a given positive odd integer and $p$ an odd prime. In this paper, we shall give a sufficient condition when a prime $p$ divides the order of the groups $K_{2k}(\mathbb{Z}[\zeta_m+\zeta_m^{-1}])$ and $K_{2k}(\mathbb{Z}[\zeta_m])$,…

We prove that the first-order theory of any function field K of characteristic p>2 is undecidable in the language of rings without parameters. When K is a function field in one variable whose constant field is algebraic over a finite field,…

Schinzel and W\'ojcik have shown that if $\alpha, \beta$ are rational numbers not $0$ or $\pm 1$, then $\mathrm{ord}_p(\alpha)=\mathrm{ord}_p(\beta)$ for infinitely many primes $p$, where $\mathrm{ord}_p(\cdot)$ denotes the order in…

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 k be an algebraically closed field of characteristic zero. An element F from k(x_1,...,x_n) is called a closed rational function if the subfield k(F) is algebraically closed in the field k(x_1,...,x_n). We prove that a rational function…

In this article, we derive lower bounds for the number of distinct prime divisors of families of non-zero Fourier coefficients of non-CM primitive cusp forms and more generally of non-CM primitive Hilbert cusp forms. In particular, for the…

We study the primitive divisors of the terms of $(\Delta_n)_{n \geq 1}$, where $\Delta_n=N_{K/ \mathbb{Q}}(u^n-1)$ for $K$ a real quadratic field, and $u>1$ a unit element of its ring of integers. The methods used allow us to find the terms…

Let $b \ge 2$ be an integer. Not much is known on the representation in base $b$ of prime numbers or of numbers whose prime factors belong to a given, finite set. Among other results, we establish that any sufficiently large integer which…

Let $D>1$ be an integer, and let $b=b(D)>1$ be its smallest divisor. We show that there are infinitely many number fields of degree $D$ whose primitive elements all have relatively large height in terms of $b$, $D$ and the discriminant of…

The $abc$ conjecture predicts a highly non trivial upper bound for the height of an algebraic point in terms of its discriminant and its intersection with a fixed divisor of the projective line counted without multiplicity. We describe the…

Let b be an odd integer such that b=+/-1 (mod 8) and let q be a prime with primitive root 2 such that q does not divide b. We show that if (p(k)) is a sequence of odd primes, with 0<=k<=q-2 such that p(k)=2p(k-1)+b for all 1<=k<=q-2, then…

Let $f(x)\in \mathbb{Z}[x]$ be an irreducible polynomial of degree $d\ge 1$. Let $k\ge2$ be an integer. The number of integers $n$ such that $f(n)$ is $k$-free is widely studied in the literature. In principle, one expects that $f(n)$ is…

Let $K$ be a number field, and let $G$ be a finitely generated subgroup of $K^\times$. Without relying on the Generalized Riemann Hypothesis we prove an asymptotic formula for the number of primes $\mathfrak p$ of $K$ such that the order of…