存在完美的蜈蚣

与完美长方体相关的反问题

Inverse problems associated with perfect cuboids

完美长方体是具有整数边、整数面对角线和整数空间对角线的矩形平行六面体。这样的长方体还没有被发现，但它们的存在也没有被证明是错误的。完美长方体由具有内在$S_3$对称性的丢番图方程组描述。最近，这些方程被因式分解关于这个$S_3$对称性和因子方程被转换成$E$ -形式。如所示，变换因子方程是显式可解的。在此基础上，本文建立了多项式反问题。

一个与完美长方体相关的四次丢番图方程

A biquadratic Diophantine equation associated with perfect cuboids

一个完美的欧拉长方体是一个长方体，它有整数条边和整数条面对角线，其空间对角线也是整数。这样的长方体还没有被发现，它们的不存在也没有被证明。理想欧拉长方体由具有自然$S_3$对称性的四个丢番图方程组成的方程组描述。最近这些方程因式分解关于这个$S_3$对称性和因子方程推导。本文将因子方程转化为$E$ -形式，然后化为一个单一的双二次方程。

与完美长方体相关的椭圆曲线的二次和三次下降

Two and three descent for elliptic curves associated with perfect cuboids

一个有理完美长方体是一个长方体，它的边和面对角线由有理数给出，并且它的空间对角线等于1。最近，它表明，丢番图方程描述这样一个长方体导致一对夫妇的参数族的椭圆曲线。本文讨论了计算它们的秩的两种和三种下降法。每个参数族中的椭圆曲线被细分为两个子集，分别允许2-下降法和3-下降法。

关于两条与完美长方体相关的椭圆曲线

On two elliptic curves associated with perfect cuboids

一个有理完美长方体是一个长方体，它的边和面对角线由有理数给出，并且它的空间对角线等于1。找到这样一个长方体就等价于找到一个完美的长方体，它的所有边和对角线都是整数，这是一个古老的未解决的问题。最近，基于对称方法，证明了有理完美长方体的边和面对角线是两个系数依赖于两个有理参数的三次方程的根。已发现这些三次方程可约的六种特殊情况，本文又考虑了这些三次方程可约的两种可能情况。它们导致一对椭圆曲线。

关于与完美长方体有关的一对三次方程

On a pair of cubic equations associated with perfect cuboids

完美长方体是具有整数边和整数面对角线的矩形平行六面体，其空间对角线也是整数。这种长方体的存在既没有被证明，也没有被证伪。有理完全长方体是与完全长方体绝对等价的完全长方体的自然伙伴。它的边和面对角线是有理数，而它的空间对角线等于1。最近，基于对称约化，证明了有理完美长方体的边是某个有理系数依赖于两个有理参数的三次方程的根。这个长方体的面对角线是另一个三次方程的根，该方程的系数是取决于相同的两个有理参数的有理数。本文研究这两个三次方程的约化问题。找到了它们在有理数域上约简的六种特殊情形。

关于长方体方程及其因子方程的等价性

On the equivalence of cuboid equations and their factor equations

欧拉长方体是具有整数边和整数面对角线的矩形平行六面体。如果欧拉长方体的空间对角线也是整数，则称其为完美长方体。一些欧拉长方体已经被发现。至于完美长方体，目前还没有人知道，也没有证明它们的不存在。欧拉长方体和完美长方体由某些丢番图方程系统描述。这些方程具有固有的$S_3$对称性。最近，他们因式分解关于这个$S_3$对称性和因子方程推导。在本文中的因子方程是等价的，以原长方体方程的搜索完美的长方体和选择欧拉长方体。

A fast modulo primes algorithm for searching perfect cuboids and its implementation

A perfect cuboid is a rectangular parallelepiped whose all linear extents are given by integer numbers, i. e. its edges, its face diagonals, and its space diagonal are of integer lengths. None of perfect cuboids is known thus far. Their non-existence is also not proved. This is an old unsolved mathematical problem. Three mathematical propositions have been recently associated with the cuboid problem. They are known as three cuboid conjectures. These three conjectures specify three special subcases in the search for perfect cuboids. The case of the second conjecture is associated with solutions…

长方体方程有理解的两种代数参数化

On two algebraic parametrizations for rational solutions of the cuboid equations

一个有理完美长方体是一个长方体，它的边和面对角线由有理数给出，并且它的空间对角线等于1。它的存在性等价于一个具有所有整数边和对角线的完美长方体的存在性。找到这样一个长方体或证明它不存在是一个古老的未解决的问题。最近，基于对称性方法，理想长方体的方程被转化为因子方程。因子方程被证明是可解的，并且被求解，导致了一对逆问题。本文的工作就是解决这些反问题。代数参数化的解决方案，使用代数函数的两个合理的论点。

Reverse asymptotic estimates for roots of the cuboid characteristic equation in the case of the second cuboid conjecture

A perfect cuboid is a rectangular parallelepiped whose edges, whose face diagonals, and whose space diagonal are of integer lengths. The second cuboid conjecture specifies a subclass of perfect cuboids described by one Diophantine equation of tenth degree and claims their non-existence within this subclass. This Diophantine equation has two parameters. Previously asymptotic expansions and estimates for roots of this equation were obtained in the case where the first parameter is fixed and the other tends to infinity. In the present paper reverse asymptotic expansions and estimates are derived …

Asymptotic estimates for roots of the cuboid characteristic equation in the linear region

A perfect cuboid is a rectangular parallelepiped whose edges, whose face diagonals, and whose space diagonal are of integer lengths. The second cuboid conjecture specifies a subclass of perfect cuboids described by one Diophantine equation of tenth degree and claims their non-existence within this subclass. This Diophantine equation is called the cuboid characteristic equation. It has two parameters. The linear region is a domain on the coordinate plane of these two parameters given by certain linear inequalities. In the present paper asymptotic expansions and estimates for roots of the charac…

完美长方体与同余数方程解

Perfect Cuboid and Congruent Number Equation Solutions

完美长方体（PC）是具有有理边$a,b,c$的矩形平行六面体，其面对角线$d_{ab}$、$d_{bc}$、$d_{ac}$和空间（体）对角线$d_s$是有理数。PC的存在与否至少从莱昂哈德·欧拉时代起就是一个已知的问题。利用同余方程$ y^2=x^3-N^2x$的非平凡有理解$(X,Y)$ }和$(Z,W)$ }建立了PC的等价条件，其中乘积$XZ$为平方.利用这对解，得到了近似完美长方体（NPC）（只有一个面对角线是无理的）的五种参数化形式和PC的五个等价条件，每种参数化形式给出了所有可能的NPC。例如，利用其中一种方法，得到了NPC的边和对角线的不变参数化：$a=2XZN$，$b=|YW|$，$c=|X-Z|\sqrt{XZ}\,N$，$d_{bc}=|XZ-N^2|\sqrt{XZ}$，$d_{ac}=|X+Z|\sqrt{XZ}\,N$，$d_s=(XZ+N^2)\sqrt{XZ}$，PC存在的条件是$d_{ab} = \sqrt{Y^2W^2+4N^2X^2Z^2}$的合理性.由于每个参数化都是完备的，因此讨论了反问题。对于给定的NPC，求出了相应的全等数方程（即全等数）及其解.

与长方体因子方程有关的有理曲线和椭圆曲线的注记

A note on rational and elliptic curves associated with the cuboid factor equations

一个有理完美长方体是一个长方体，它的边和面对角线由有理数给出，并且它的空间对角线等于1。它由一个关于六个变量的四个方程组成的系统描述。由这四个方程经对称化处理，导出了长方体因子方程。它们构成了一个由八个多项式方程组成的系统，该系统已被参数化求解。本文用有理曲线与椭圆曲线的交线表示其参数解。

寻找完美立方体的数值搜索策略（第二立方体猜想情况)

A strategy of numeric search for perfect cuboids in the case of the second cuboid conjecture

本文基于第二立方体猜想，研究了一个十次迪ophantine方程，并提出优化的计算机辅助搜索策略以寻找完美立方体。

On the Existence of Odd Perfect Numbers

In this paper, we prove the conjecture that if there is an odd perfect number, then there are infinitely many of them.

No perfect triangle is isosceles

A perfect triangle is a triangle with rational sides, medians, and area. In this article, we use a similar strategy due to Pocklington to show that if $Δ$ is a perfect triangle, then it cannot be an isosceles triangle. It gives a partial answer to a question of Richard Guy, who asked whether any perfect triangles exist. No example has been found to date. It is widely believed that such a triangle does not exist.

关于完满数之间的距离

On the distance between perfect numbers

证明了存在无穷多个三角数（全等于3 mod 12），它们不能是两个完满数之间的距离。

Difference of powers of consecutive primes which are perfect squares

We investigate the consecutive primes $p$ and $q$ ($p > q$) for which there exists a pair of natural numbers $(x,y)$ such that $p^x-q^y$ is a perfect square and make some conjectures.