Related papers: The Real Number n-Degree Pythagorean Theorem

Given a positive integer $n$, a sufficient condition on a field is given for bounding its Pythagoras number by $2^n+1$. The condition is satisfied for $n=1$ by function fields of curves over iterated formal power series fields over…

A sequence of real numbers $\{x_{n}\}_{n\in \mathbb{N}}$ is said to be $\alpha \beta$-statistically convergent of order $\gamma$ (where $0<\gamma\leq 1$) to a real number $x$ \cite{a} if for every $\delta>0,$ $$\underset{n\rightarrow…

In his seminal 1961 paper, Wirsing studied how well a given transcendental real number $\xi$ can be approximated by algebraic numbers $\alpha$ of degree at most $n$ for a given positive integer $n$, in terms of the so-called naive height…

In this paper we study practical numbers of some special forms. For any integers $b\ge0$ and $c>0$, we show that if $n^2+bn+c$ is practical for some integer $n>1$, then there are infinitely many nonnegative integers $n$ with $n^2+bn+c$…

The notion of $\theta$-congruent numbers generalizes the classical congruent number problem. Recall that a positive integer $n$ is $\theta$-congruent if it is the area of a rational triangle with an angle $\theta$ whose cosine is rational.…

Given any polynomial with real coefficients, the existence of a real quadratic polynomial factor is proven using only basic real analysis. The aim is to provide an approachable proof to anybody who is familiar with the least upper bound…

Under Cram\'er's conjecture concerning the prime numbers, we prove that for any $x>1$, there exists a real $A=A(x)>1$ for which the formula $[A^{n^x}]$ (where $[]$ denotes the integer part) gives a prime number for any positive integer $n$.…

Given n >= 4 positive real numbers, we prove in this note that they are the face areas of a convex polyhedron if and only if the largest number is not more than the sum of the others.

Let $\Gamma$ be an infinite discrete subgroup of Gl$_n(\mathbb{C})$. Then either $(\mathbb{R}, <, +, \cdot, \Gamma)$ is interdefinable with $(\mathbb{R}, <, +, \cdot, \lambda^\mathbb{Z})$ for some $\lambda \in \mathbb{R}$, or $(\mathbb{R},…

Lower and upper bounds $B_a(x)$ on the incomplete gamma function $\Gamma(a,x)$ are given for all real $a$ and all real $x>0$. These bounds $B_a(x)$ are exact in the sense that $B_a(x)\underset{x\downarrow0}\sim\Gamma(a,x)$ and…

A "practical number" is a positive integer $n$ such that every positive integer less than $n$ can be written as a sum of distinct divisors of $n$. We prove that most of the binomial coefficients are practical numbers. Precisely, letting…

Let $\alpha$ be a real algebraic number of degree $d \geq 3$ and let $\beta \in \mathbb Q(\alpha)$ be irrational. Let $\mu$ be a real number such that $(d/2) + 1 < \mu < d$ and let $C_0$ be a positive real number. We prove that there exist…

Some aspects of the development of physics and the mathematics set one think about relation between complex numbers and reality around us. If number to spot as the relation of two quantities, from the fact of existence of complex numbers…

It has been conjectured for some time that, for any integer n\ge 2, any real number \epsilon >0 and any transcendental real number \xi, there would exist infinitely many algebraic integers \alpha of degree at most n with the property that…

If n points B_1,---,B_n$ in the standard simplex \Delta_n are affinely independent, then they can span an (n-1)-simplex denoted by \Lambda=Con(B_1,---,B_n). Here \Lambda corresponds to an n*n matrix [\Lambda] whose columns are B_1,---,B_n.…

A positive integer $n$ is called a $\theta$-congruent number if there is a triangle with sides $a,b$ and $c$ for which the angle between $a$ and $b$ is equal to $\theta$ and its area is $n\sqrt{r^2 - s^2}$, where $0 < \theta < \pi$, $\cos…

In the present paper and as an application of Roth's theorem concerning the rational approximation of algebraic numbers, we give a sufficient condition that will assure us that a series of positive rational terms is a transcendental number.…

We obtain general criteria for giving a lower bound on the degree of numbers of the form $\prod_{n=1}^\infty \left(1+\frac{b_n}{\alpha_n}\right)$ or of the form $\prod_{m=1}^\infty \left(1+ \sum_{n=1}^\infty…

We improve a result of H. L. Montgomery and J. P. Weinberger by establishing the existence of infinitely many fundamental discriminants $d>0$ for which the class number of the real quadratic field $\mathbb{Q}(\sqrt{d})$ exeeds…

Following de Loera and Santos, the P\'olya exponent of a $n$-ary real form (i.e. a homogeneous polynomial in $n$ variables with real coefficients) $f$ is the infimum of the upward closed set of nonnegative integers $m$ such that $(x_1 +…