Related papers: Remarks on a Problem of Eisenstein

Let $\LL^+=\mathbb{Q}(\sqrt{2}, \sqrt{pq}, \sqrt{ps})$ and $\LL=\mathbb{Q}(\sqrt{2}, \sqrt{pq}, \sqrt{ps}, \sqrt{-\ell})$ be two fields, where $q$, $p$ and $s$ three different prime integers and $\ell\geq1$ be a positive odd square-free…

Let $d\geq 2$ be a squarefree integer, let $\omega\in\{\sqrt{d},\frac{1+\sqrt{d}}{2}\}$ be such that $\mathbb{Z}[\omega]$ is the ring of algebraic integers of the real quadratic number field $\mathbb{Q}(\sqrt{d})$, let $\varepsilon>1$ be…

It is shown, subject to the abc-conjecture, that \[\sum_{n\le N}\exp(2\pi i\alpha n^3)\ll_{\epsilon,\alpha}N^{5/7+\epsilon}\] for any $\epsilon>0$ and any quadratic irrational $\alpha$.

Let $x$ be a positive real number, and $\mathcal{P} \subset [2,\lambda(x)]$ be a set of primes, where $\lambda(x) \in \Omega(x^\varepsilon)$ is a monotone increasing function with $\varepsilon \in (0,1)$. We examine $Q_{\mathcal{P}}(x)$,…

In this paper, we sharpen and simplify our earlier results based on Thue's Fundamentaltheorem and use it to obtain effective irrationality measures for certain roots of polynomials of the form $(x-\sqrt{t})^{n}+(x+\sqrt{t})^{n}$, where $n…

We discuss two independent constructions to introduce an N-extended Supersymmetric Quantum Mechanics. The first one makes use of the Fierz identities while the second one (divided into two subcases) makes use of the Schur lemma. The N…

The representation dimension of a finite group $G$ is the minimal dimension of a faithful complex linear representation of $G$. We prove that the representation dimension of any finite group $G$ is at most $\sqrt{|G|}$ except if $G$ is a…

Let $K$ be a number field of degree $n$ over ${\mathbb Q}$. Then the 4-rank of the strict class group of $K$ is at least ${\text{rank}_2 \, } ({ E_{K}^{+} } / E_K^2) - \lfloor n /2 \rfloor$ where $E_K$ and ${ E_{K}^{+} }$ denote the units…

Let $K$ be a number field with ring of integers $\mathcal{O}_K$. Let $\mathcal{N}_K$ be the set of positive integers $n$ such that there exist units $\varepsilon, \delta \in \mathcal{O}_K^\times$ satisfying $\varepsilon + \delta = n$. We…

Suppose $F=\mathbb{Q}(\sqrt{-p_1\dotsm p_t})$ is an imaginary quadratic number field with distinct primes $p_1,\dots,p_{t}$, where $p_i\equiv 1\pmod{4}$ ($i=1,\dots,t-1$) and $p_t\equiv 3\pmod{4}$. We express the possible values of the…

A corner is a set of three points in $\mathbf{Z}^2$ of the form $(x, y), (x + d, y), (x, y + d)$ with $d \neq 0$. We show that for infinitely many $N$ there is a set $A \subset [N]^2$ of size $2^{-(c + o(1)) \sqrt{\log_2 N}} N^2$ not…

In this article, we study necessary conditions for certain square-free integers to be congruent numbers. Our method uses divisibility properties of class numbers of related imaginary quadratic fields. We first consider positive square-free…

Let $\mathfrak{B}$ denote the collection of odd primitive Gaussian integers and $n\mapsto b(n)$ denote the characteristic function of elements of $\mathfrak{B}$. We prove that the exponential sum $ S(\alpha; N)=\sum_{n\le…

Unit-generated orders of a quadratic field are orders of the form $\mathcal{O} = \mathbb{Z}[\varepsilon]$, where $\varepsilon$ is a unit in the quadratic field. If the order $\mathcal{O}$ is a maximal order of a real quadratic field, then…

Let $q$ be a large prime number, $a$ be any integer, $\epsilon$ be a fixed small positive quantity. Friedlander and Shparlinksi \cite{FSh} have shown that there exists a positive integer $n\ll q^{5/2+\epsilon}$ such that $\phi(n)$ falls…

For a positive integer $N$ and $\mathbb{A}$ a subset of $\mathbb{Q}$, let $\mathbb{A}$-$\mathcal{KS}(N)$ denote the set of $\alpha=\dfrac{\alpha_{1}}{\alpha_{2}}\in \mathbb{A}\setminus \{0,N\}$ verifying $\alpha_{2}p-\alpha_{1}$ divides…

In this paper we prove that inf_{|z_k| => 1} max_{v=1,...,n^2} |sum_{k=1}^n z_k^v| = sqrt n+O(n^{0.2625+epsilon}). This improves on the bound O(sqrt (n log n)) of Erdos and Renyi. In the special case of $n+1$ being a prime we have…

The $q$-calculus for generic $q$ is developed and related to the deformed oscillator of parameter $q^{1/2}$. By passing with care to the limit in which $q$ is a root of unity, one uncovers the full algebraic structure of ${{\cal…

Let $M(x)$ denote the median largest prime factor of the integers in the interval $[1,x]$. We prove that $$M(x)=x^{\frac{1}{\sqrt{e}}\exp(-\text{li}_{f}(x)/x)}+O_{\epsilon}(x^{\frac{1}{\sqrt{e}}}e^{-c(\log x)^{3/5-\epsilon}})$$ where…

In this paper, we analyze the sum of squares hierarchy (SOS) on the ordering principle on $n$ elements. We prove that degree $O(\sqrt{n}log(n))$ SOS can prove the ordering principle. We then show that this upper bound is essentially tight…