Related papers: Completely $p$-primitive binary quadratic forms

Let $K$ be a totally real field, and $r\geq 5$ a fixed rational prime. In this paper, we use the modular method as presented in the recent work of Freitas and Siksek to study non-trivial, primitive solutions $(x,y,z) \in \mathcal{O}_K^3$ of…

We consider integer recurrences of the form a_n = f(a_{n-1}), where f is a quadratic polynomial with integer coefficients. We show, for four infinite families of f, that the set of primes dividing at least one term of such a sequence must…

Suppose $ m,n\geq 2 $ are co prime integers. We prove certain new symmetries of the base $ n $ representation of $ 1/m $, and in particular characterize the subgroup generated by $ n $ inside $ (\mathbb{Z}/m\mathbb{Z})^\times $. As an…

In this paper we charatcterize primes of the form $x^2+dy^2$ with $x\equiv 0\pmod{N}$ or $y\equiv 0\pmod{N}$ for positive integer $N$ and $d$ with $d$ being square free.

Using elementary number theory we study Diophantine equations over the rational integers of the following form, $y^2=(x+a)(x+a+k)(x+b)(x+b+k)$, $y^2=c^2x^4+ax^2+b$ and $y^2=(x^2-1)(x^2-\alpha^2)(x^2-(\alpha+1)^2).$ We express their integer…

In this paper we study the Diophantine equation $x^{4}-q^{4}=py^{5},$ with the following conditions: $p$ and $q$ are different prime natural numbers, $y$ is not divisible with $p$, $p\equiv3$ (mod20), $q\equiv4$ (mod5), $\overline{p}$ is a…

Given any number field, we prove that there exist arbitrarily shaped constellations consisting of pairwise non-associate prime elements of the ring of integers. This result extends the celebrated Green-Tao theorem on arithmetic progressions…

Let $Q(x,y)$ be a quadratic form with discriminant $D\neq 0$. We obtain non trivial upper bound estimates for the number of solutions of the congruence $Q(x,y)\equiv\lambda \pmod{p}$, where $p$ is a prime and $x,y$ lie in certain intervals…

A quadratic form over a non-archimedian local field of characteristic zero $F$ is called universal if it is integral and it represents all non-zero integers of $F$. Xu Fei and Zhang Yang determined all universal quadratic forms in the case…

let $R$ be a regular local ring of a mixed characteristic $(0,p)$ where $p\neq 2$ is a prime number. Suppose that the quotient ring $R/pR$ is also regular. Fix a non-degenerate Pfister form $Q(T_{1},\ldots,T_{2^{m}})$ over $R$ and an…

Let $\mathbb{F}_{q^n}$ be a finite field with $q^n$ elements, and let $m_1$ and $m_2$ be positive integers. Given polynomials $f_1(x), f_2(x) \in \mathbb{F}_q[x]$ with $\textrm{deg}(f_i(x)) \leq m_i$, for $i = 1, 2$, and such that the…

For finite-dimensional Hopf algebras, their classification in characteristic $0$ (e.g. over $\mathbb{C}$) has been investigated for decades with many fruitful results, but their structures in positive characteristic have remained elusive.…

The study of finiteness or infiniteness of integer solutions of a Diophantine equation has been considered as a standard problem in the literature. In this paper, for f(x) in Z[x] monic and q1 ,...., qm in Z, we study the conditions for…

Let F(z) be a rational function in Q(z) of degree at least 2 with F(0) = 0 and such that F does not vanish to order d at 0. Let b be a rational number having infinite orbit under iteration of F, and write F^n(b) = A_n/B_n as a fraction in…

We improve an $L^2\times L^2\to L^2$ estimate for a certain bilinear operator in the finite field of size $p$, where $p$ is a prime sufficiently large. Our method carefully picks the variables to apply the Cauchy-Schwarz inequality. As a…

We study involutive non-degenerate set-theoretic solutions (X,r) of the Yang-Baxter equation on a finite set X. The emphasis is on the case where (X,r) is indecomposable, so the associated permutation group acts transitively on X. One of…

Erd\"os and Obl\'ath proved that the equation $n!\pm m!=x^p$ has only finitely many integer solutions. More general, under the ABC-conjecture, Luca showed that $P(x)=An!+Bm!$ has finitely many integer solutions for polynomials of degree…

Let $Q$ be a positive-definite quaternary quadratic form with prime discriminant. We give an explicit lower bound on the number of representations of a positive integer $n$ by $Q$. This problem is connected with deriving an upper bound on…

In this paper, we prove that a binary definite quadratic form over F_q[t], where q is odd, is completely determined up to equivalence by the polynomials it represents up to degree 3m-2, where m is the degree of its discriminant. We also…

Let $p$ be an odd prime and let $f(x)=\sum_{i=1}^ka_ix^{p^{\alpha_i}+1}\in\Bbb F_{p^n}[x]$, where $0\le \alpha_1<...<\alpha_k$. We consider the exponential sum $S(f,n)=\sum_{x\in\Bbb F_{p^n}}e_n(f(x))$, where $e_n(y)=e^{2\pi…