Related papers: On the negative Pell equation

In 2013, Zhi-Wei Sun proposed a Romanov-type conjecture stating that every integer $n > 1$ can be written as $n = k + m$ with $k, m \ge 1$ such that $2^k + m$ is a prime. In this paper, we unconditionally prove that the natural numbers…

Based on previous results on the classification of finite-dimensional Nichols algebras over dihedral groups and the characterization of simple modules of Drinfeld doubles, we compute the irreducible characters of the Drinfeld doubles of…

We propose a method to determine the solvability of the diophantine equation $x^2-Dy^2=n$ for the following two cases: $(1)$ $D=pq$, where $p,q\equiv 1 \mod 4$ are distinct primes with $(\frac{q}{p})=1$ and…

Given a finite family F of linear forms with integer coefficients, and a compact abelian group G, an F-free set in G is a measurable set which does not contain solutions to any equation L(x)=0 for L in F. We denote by d_F(G) the supremum of…

We consider the prescribed scalar curvature problem on $ {\mathbb{S}}^N $ $$ \Delta_{{\mathbb S}^N} v-\frac{N(N-2)}{2} v+\tilde{K}(y) v^{\frac{N+2}{N-2}}=0 \quad \mbox{on} \ {\mathbb S}^N, \qquad v >0 \quad \mbox{on} \ {\mathbb S}^N, $$…

We deeply investigate the Diophantine equation $cx^2+d^{2m+1}=2y^n$ in integers $x, y\geq 1, m\geq 0$ and $n\geq 3$, where $c$ and $d$ are given coprime positive integers such that $cd\not\equiv 3 \pmod 4$. We first solve this equation for…

In this paper, we introduce a new and direct approach to study the solvability of systems of equations generated by bilinear forms. More precisely, let $B (\cdot, \cdot)$ be a non-degenerate bilinear form and $E$ be a set in…

The determinantal complexity of a polynomial $P \in \mathbb{F}[x_1, \ldots, x_n]$ over a field $\mathbb{F}$ is the dimension of the smallest matrix $M$ whose entries are affine functions in $\mathbb{F}[x_1, \ldots, x_n]$ such that $P =…

We show that for every fixed $\ell\in\mathbb{N}$, the set of $n$ with $n^\ell|\binom{2n}{n}$ has a positive asymptotic density $c_\ell$, and we give an asymptotic formula for $c_\ell$ as $\ell\to \infty$. We also show that $\# \{n\le x,…

In this paper, we prove some new thickness theorems with partial derivatives. We give some applications. First, we give a simple criterion that can judge whether two scaled Cantor sets have non-empty intersection. Second, we prove under…

Let $n$ be a non negative integer, and define $D_n$ to be the family of all finite groups having precisely $n$ conjugacy classes of nontrivial subgroups that are not self-normalizing. We are interested in studying the behavior of $D_n$ and…

Finite groups are said to be isospectral if they have the same sets of element orders. A finite nonabelian simple group $L$ is said to be almost recognizable by spectrum if every finite group isospectral to $L$ is an almost simple group…

Let $G$ be a transitive permutation group on a finite set with solvable point stabiliser and assume that the solvable radical of $G$ is trivial. In 2010, Vdovin conjectured that the base size of $G$ is at most 5. Burness proved this…

We consider a special subsequence of the Fourier coefficients of powers of the Dedekind $\eta$-function, analogous to the sequence $\delta_\ell := 24^{-1} \pmod{\ell}$ on which exceptional congruences of the partition function are…

A classical result in number theory is Dirichlet's theorem on the density of primes in an arithmetic progression. We prove a similar result for numbers with exactly k prime factors for k>1. Building upon a proof by E.M. Wright in 1954, we…

For fixed $q\in\{3,7,11,19, 43,67,163\}$, we consider the density of primes $p$ congruent to $1$ modulo $4$ such that the class group of the number field $\mathbb{Q}(\sqrt{-qp})$ has order divisible by $16$. We show that this density is…

In the conformal class of the standard metric on the $3$-sphere, we prove a quantitative refinement of the Andrews-De Lellis-Topping inequality in terms of a two-term distance to the set of minimizing conformal factors. This inequality is…

Under the Riemann Hypothesis for Dirichlet L-functions, we improve on the error term in a smoothed version of an estimate for the density of elliptic curves with square-free $\Delta=D/16$, where D is the discriminant, by T.D. Browning and…

For any positive integer $n$, let $\sigma (n)$ be the sum of all positive divisors of $n.$ In this paper, it is proved that the set of positive integers $ n $ for which $ \sigma(30n+1)\geq \sigma(30n) $ has a density less than $ 0.0371813,…

We consider the problem of finding a $d$-dimensional spectral density through a moment problem which is characterized by an integer parameter $\nu$. Previous results showed that there exists an approximate solution under the regularity…