首页/论文/2301.07375

arXiv 2301.073752023-01math.NTmath.RA

通过配立方求解三次方程及通过配幂求解高次方程

Solving cubic equations by completing the cube and higher degree equations by completing powers

Hua-Lin Huang, Shengyuan Ruan, Xiaodan Xu, Yu Ye

备注：: 11 pages. Minor revision. A new version with more details will be published in the American Mathematical Monthly

阅读 PDF ↗

arXiv 页面 ↗HTML 原文 ↗

摘要英文摘要相关论文

AI总结：

该研究基于Harrison高次型中心理论提出配幂思想，用配立方推导三次方程Cardano公式，给出部分高次代数方程的根式解及简单判别准则，计算仅需求解线性和二次方程。

摘要：

我们通过配立方推导了三次方程的Cardano公式，并通过配幂给出了部分高次代数方程的根式解。配幂的核心思想源自Harrison的高次型中心理论。本文提出了这类代数方程的一个极为简单的判别准则，且其计算可归结为求解线性方程与二次方程。

英文摘要：

We derive the Cardano formula of cubic equations by completing the cube, and provide radical solutions to some algebraic equations of higher degree by completing powers. The main idea of completing powers arises from Harrison's center theory of higher degree forms. A very simple criterion for such algebraic equations is presented, and the computation amounts to solving linear equations and quadratic equations.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

阅读原文

阅读 PDF ↗

arXiv 页面 ↗HTML 原文 ↗

本文内容

摘要英文摘要相关论文

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

继续探索

相关论文

二元三次型与有理数立方和问题

Binary Cubic Forms and Rational Cube Sum Problem

本文利用整二元三次型研究有理数立方和问题，无条件证明了在特定剩余类中存在无穷多个素数可表示为两个有理数的立方和。

2024-05-31math.NT

Infinite series of quaternionic 1-vertex cube complexes, the doubling construction, and explicit cubical Ramanujan complexes

We construct vertex transitive lattices on products of trees of arbitrary dimension $d \geq 1$ based on quaternion algebras over global fields with exactly two ramified places. Starting from arithmetic examples, we find non-residually finite groups generalizing earlier results of Wise, Burger and Mozes to higher dimension. We make effective use of the combinatorial language of cubical sets and the doubling construction generalized to arbitrary dimension. Congruence subgroups of these quaternion lattices yield explicit cubical Ramanujan complexes, a higher dimensional cubical version of Ramanuj…

2018-08-09math.GRmath.NT

Arithmetic properties of cubic and biquadratic theta series

A cubic (resp. biquadratic) theta series is a power series whose n-th coefficient is equal to 1 if n is a perfect cube (resp. fourth power) and zero otherwise. We improve on a result of Bradshaw by showing that such series is not a cubic (resp. biquadratic) algebraic number when evaluated at reciprocals of integers. The proof relies on a "nested gaps technique" for linear independence and on recent results by the author on Waring's problem for cubes and biquadrates.

2019-10-11math.NT

On the sums of two cubes

We solve the equation $f(x,y)^3 + g(x,y)^3 = x^3 + y^3$ for homogeneous $f, g \in \mathbb C(x,y)$, completing an investigation begun by Viète in 1591. The usual addition law for elliptic curves and composition give rise to two binary operations on the set of solutions. We show that a particular subset of the set of solutions is ring-isomorphic to $\mathbb Z[e^{2 πi / 3}]$.

2011-02-14math.AGmath.NT

On the Diophantine equation in the form that a sum of cubes equals a sum of quintics

We show how sums of some $5th$ powers can be written as sums of some cubics

2017-03-30math.NT

Primitive divisors on twists of the Fermat cubic

We show that for an elliptic divisibility sequence on a twist of the Fermat cubic, u^3+v^3=m, with m cube-free, all the terms beyond the first have a primitive divisor.

2007-03-19math.NT

更多相关论文收起更多论文

Cyclic Cubic Extensions of Q

In this article we explicitly describe irreducible trinomials X^3-aX+b which gives all the cyclic cubic extensions of Q. In doing so, we construct all integral points (x,y,z) with GCD(y,z)=1, of the curves X^2+3Y^2 = 4DZ^3 and X^2+27Y^2=4DZ^3 as D varies over cube-free positive integers. We parametrise these points using well known parametrisation of integral points (x,y,z) of the curve X^2+3Y^2=4Z^3 with GCD(y,z)=1. As an accidental byproduct of our result we show that there are infinitely many primes congruent to 1 or 8 modulo 9, can be expressed as sum of two rational cubes.

2022-11-30math.NT

倒数平衡数立方和的无穷级数

The infinite sum of the cubes of reciprocal balancing numbers

VIT-AP University（维特-AP大学）；KIIT Deemed to be University（KIIT认证大学）

本文研究平衡数倒数立方和的无穷级数，给出其显式界，并据此确定该级数倒数的整数部分。

2026-09-29math.NT

Beyond endoscopy for the Symmetric Cube L-function

This paper is a first attempt at getting information on a symmetric power representation of a $GL_2$ automorphic form via a trace formula that is beyond endoscopic techniques. In particular, we study the symmetric third power representation for all forms over $\Q$ adjoined the cube roots of unity. A key tool needed in the study is an identity relating cubic exponential sums to Kloosterman sums. This very same identity is crucial to the fundamental lemma of a trace formula comparison in work of Mao and Rallis.

2012-08-08math.NT

Ranks of elliptic curves in cubic extensions

For an elliptic curve E over a number field K, we prove that the algebraic rank of E goes up in infinitely many extensions of K obtained by adjoining a cube root of an element of K. As an example, we briefly discuss E=X_1(11) over Q, and how the result relates to Iwasawa theory.

2005-10-25math.NT