Related papers: The class number formula for imaginary quadratic f…

We give a precise description of how the class group of a number field measures the failure of unique factorization in its ring of integers. Specifically, following ideas of Kummer, we determine the structure of all irreducible…

In this paper, we will introduce Bell numbers $D(n)$ of type $D$ as an analogue to the classical Bell numbers related to all the partitions of the set $[n]$. Then based on a signed set partition of type $D$, we will construct the recurrence…

This paper describes a simple method for estimating lower bounds on the number of classes of equivalence for a special kind of integer sequences, called division sequences. The method is based on adding group structure to classes of…

Sarnak obtained the asymptotic formula of the sum of the class numbers of indefinite binary quadratic forms from the prime geodesic theorem for the modular group. In the present paper, we show several asymptotic formulas of partial sums of…

In this paper, we establish the explicit lower bound estimates for the rank of universal quadratic forms in some certain families of real cubic fields under the condition of density one. The more general results that represent all multiples…

Let $n>1$ be an odd integer. We prove that there are infinitely many imaginary quadratic fields of the form $\mathbb{Q}(\sqrt{x^2-2y^n})$ whose ideal class group has an element of order $n$. This family gives a counter example to a…

Departing from a class of infinite series with central binomial coefficients in the numerator and depending on a positive integer parameter, we first extend known identities to all complex parameters. Then we use various methods, including…

For a number field $K$, we extend the notion of the ring class field of an order in $K$ [C. Lv and Y. Deng, SciChina. Math., 2015] to that of an arbitrary number ring in $K$. We give both ideal-theoretic and idele-theoretic description of…

In this paper, we introduce the notion of unit reducibility for number fields, that is, number fields in which all positive unary forms attain their nonzero minimum at a unit. Furthermore, we investigate the link between unit reducibility…

Prime number multiplet classifications and patterns are extended to negative integers. The extension from prime numbers to single prime powers is also studied. Prime number septets at equal distance are given. It is also shown that each…

In this paper we study the distribution of orders of bounded discriminants in number fields. We give an asymptotic formula for the number of orders contained in the ring of integers of a quintic number field.

Diffusive representations of fractional derivatives have proven to be useful tools in the construction of fast and memory efficient numerical methods for solving fractional differential equations. A common challenge in many of the known…

A new kind of numbers called Hyper Space Complex Numbers and its algebras are defined and proved. It is with good properties as the classic Complex Numbers, such as expressed in coordinates, triangular and exponent forms and following the…

In this paper we present an explicit formula for the number of permutations with a given number of alternating descents. Moreover, we study the interlacing property of the real parts of the zeros of the generating polynomials of these…

We prove that the representations numbers of a ternary definite integral quadratic form defined over F_q[t], where F_q is a finite field of odd characteristic, determine its integral equivalence class when q is large enough with respect to…

Let $K$ be an imaginary quadratic field of discriminant $d_K$ with ring of integers $\mathcal{O}_K$. When $K$ is different from $\mathbb{Q}(\sqrt{-1})$ and $\mathbb{Q}(\sqrt{-3})$, we consider a certain specific model for the elliptic curve…

We construct a class of negative spin irreducible representations of the su(2) Lie algebra. These representations are infinite-dimensional and have an indefinite inner product. We analyze the decomposition of arbitrary products of positive…

Although fractional powers of non-negative operators have received much attention in recent years, there is still little known about their behavior if real-valued exponents are greater than one. In this article, we define and study the…

Counting integral binary quadratic forms with certain restrictions is a classical problem. In this paper, we count binary quadratic forms of fixed discriminant given restrictions on the size of their coefficients. We accomplish this by…

Generalizing the notion of the degree of a finite-to-one factor code from a shift of finite type, the class degree of a possibly infinite-to-one factor extends many important properties of degree. In this paper, introducing class degree, we…