On the abundance of the irrational numbers

Statistical approach based on 1D Voronoi tessellation as a tool for studying the randomness of fractional digits of some irrational numbers

An "experimental" study on the randomness of the fractional digits of $π$, $e$ and $ϕ$ irrational numbers are presented. This is done by exploiting the $1D$ Poisson-Voronoi tessellation. We employed two approaches and in both cases, within the numerical error, no differences have been detected between the irrational fractional digits and an equivalent random sequence of digits. The number of tested digits is $1.6 \times 10^7$ and $4 \times 10^7$ for the first and second approach, respectively. Although not shown here, we investigated several irrational numbers and all of them have displayed a …

The Product $e π$ Is Irrational

This note shows that the product $e π$ of the natural base $e$ and the circle number $π$ is an irrational number.

A Proof that Euler's Constant Gamma is an Irrational Number

The attributes of Euler's constant Gamma have been a baffling problem to the world's mathematicians in the number theory field. In 1900, when German mathematician D. Hilbert addressed the 2nd International Congress of Mathematicians, he suggested twenty-three previously unsolved problems to the international mathematical field. The 7th of these problems pertained to Euler's constant Gamma. After investigating this problem for many years, the author has proved that Euler's constant Gamma is an irrational number.

三个数 $\zeta(5)$, $\zeta(7)$, $\zeta(9)$ 中至少有一个是无理数

At least one of the three numbers $ζ(5)$, $ζ(7)$, $ζ(9)$ is irrational

本文通过构造线性形式并利用积分量改进估计，证明了黎曼zeta函数在奇数点5、7、9处的值中至少有一个为无理数。

The Zeta Quotient $ζ(3)/ π^3$ is Irrational

This note proves that the first odd zeta value does not have a closed form formula $ζ(3)\ne r π^3$ for any rational number $r \in \mathbb{Q}$. Furthermore, assuming the irrationality of the second odd zeta value $ζ(5)$, it is shown that $ζ(5)\ne rπ^5$ for any rational number $r \in \mathbb{Q}$.

某些无理数的线性独立性

Linear Independence Of Some Irrational Numbers

本文提出一种分析技术，用于证明某些实数子集在有理数上的线性独立性，应用于证明{1,e,π}、{1,e,π⁻¹}和{1,π^r,π^s}等三元组的线性独立性。