Related papers: Binary quadratic forms that represent almost the s…

For a congruence subgroup $\Gamma$, we define the notion of $\Gamma$-equivalence on binary quadratic forms which is the same as proper equivalence if $\Gamma = \mathrm{SL}_2(\mathbb Z)$. We develop a theory on $\Gamma$-equivalence such as…

We develop dualities for complete perfect distributive quasi relation algebras and complete perfect distributive involutive FL-algebras. The duals are partially ordered frames with additional structure. These frames are analogous to the…

We obtain an asymptotic formula for the number of $\operatorname{GL}_2(\mathbb{Z})$-equivalence classes of irreducible binary quartic forms with integer coefficients with vanishing $J$-invariant and whose Hessians are proportional to the…

We study the distribution of palindromic numbers (with respect to a fixed base $g\ge 2$) over certain congruence classes, and we derive a nontrivial upper bound for the number of prime palindromes $n\le x$ as $x\to\infty$. Our results show…

In this paper we study arithmetical and structural features of a finite group that possesses exactly two conjugacy class sizes that are composite numbers.

A positive-definite integral quadratic form is called regular if it represents every positive integer which is locally represented. In this article, we classify all regular diagonal quadratic forms of rank greater than 3.

Let $f$ be a primitive positive definite integral binary quadratic form of discriminant $-D$ and let $\pi_f(x)$ be the number of primes up to $x$ which are represented by $f$. We prove several types of upper bounds for $\pi_f(x)$ within a…

Let $-D$ be a fundamental discriminant. We express the number of representations of an integer by a positive definite binary quadratic form of discriminant $-D$ with an odd class number $h(-D)$ as a rational linear expression involving the…

The N distinct prime numbers that make up a composite number M allow $2^{N-1}$ bi partioning into two relatively prime factors. Each such pair defines a pair of conjugate representations. These pairs of conjugate representations, each of…

We define a canonical quadratic pair on the Clifford algebra of an algebra with quadratic pair over a field. This allows us to extend to the characteristic 2 case the notion of trialitarian triples, from which we derive a characterization…

We call a positive definite Hermitian lattice regular if it represents all integers which can be represented locally by the lattice. We investigate binary regular Hermitian lattices over imaginary quadratic fields $\mathbb{Q}(\sqrt{-m})$…

We examine a bias towards the zero residue class for the integers represented by binary quadratic forms. In many cases, we are able to prove that the bias comes from a secondary term in the associated asymptotic expansion (unlike…

We study totally positive definite quadratic forms over the ring of integers $\mathcal{O}_K$ of a totally real biquadratic field $K=\mathbb{Q}(\sqrt{m}, \sqrt{s})$. We restrict our attention to classical forms (i.e., those with all…

In this paper, we find formulas for the number of representations of certain diagonal octonary quadratic forms with coefficients $1,2,3,4$ and $6$. We obtain these formulas by constructing explicit bases of the space of modular forms of…

We show that the class of every primitive indefinite binary quadratic form is naturally represented by an infinite graph (named \c{c}ark) with a unique cycle embedded on a conformal annulus. This cycle is called the spine of the \c{c}ark.…

We prove that for every irrational number $\alpha$, real number $\beta$, real number $c$ satisfying $1<c<9/8$ and positive real number $\theta$ satisfying $\theta<(9/c-8)/10$, there exist infinitely many primes of the form…

Let $\mathcal{P}_r$ denote an almost-prime with at most $r$ prime factors, counted according to multiplicity. In this paper, it is proved that, for $0.989<\gamma<1$, there exist infinitely many primes $p$ such that…

We study cliques in graphs arising from quadratic forms where the vertices are the elements of the module of the quadratic form and two vertices are adjacent if their difference represents some fixed scalar. We determine structural…

In this paper we study quadratic forms which are universal when restricted to almost prime inputs, establishing finiteness theorems akin to the Conway--Schneeberger 15 theorem.

We consider $48$ parastrophically uncancellable quadratic functional equations with four object variables and two quasigroup operations in two classes: balanced non--Belousov (consists of 16 equations) and non--balanced non--gemini…