The Product eπe \pi Is Irrational

This note shows that the product $e \pi$ of the natural base $e$ and the circle number $\pi$ is an irrational number.Comment: Twelve Pages. Improved Version. keywords: Irrational number; Natural base, Circle number; Unbounded partial quotients. arXiv admin note: text overlap with arXiv:1212.408

Geometric Phase Integrals and Irrationality Tests

Let $F(x)$ be an analytical, real valued function defined on a compact domain $\mathcal {B}\subset\mathbb{R}$. We prove that the problem of establishing the irrationality of $F(x)$ evaluated at $x_0\in \mathcal{B}$ can be stated with respect to the convergence of the phase of a suitable integral $I(h)$, defined on an open, bounded domain, for $h$ that goes to infinity. This is derived as a consequence of a similar equivalence, that establishes the existence of isolated solutions of systems equations of analytical functions on compact real domains in $\mathbb{R}^p$, if and only if the phase of a suitable ``geometric'' complex phase integral $I(h)$ converges for $h\rightarrow \infty$. We finally highlight how the method can be easily adapted to be relevant for the study of the existence of rational or integer points on curves in bounded domains, and we sketch some potential theoretical developments of the method

Apery limits and special values of L-functions

We describe a general method to determine the Apery limits of a differential equation that have a modular-function origin. As a by-product of our analysis, we discover a family of identities involving the special values of L-functions associated with modular forms. The proof of these identities is independent of differential equations and Apery limits.Comment: 24 page

Exact enumeration of 1342-avoiding permutations: A close link with labeled trees and planar maps

Solving the first nonmonotonic, longer-than-three instance of a classic enumeration problem, we obtain the generating function $H(x)$ of all 1342-avoiding permutations of length $n$ as well as an {\em exact} formula for their number $S_n(1342)$. While achieving this, we bijectively prove that the number of indecomposable 1342-avoiding permutations of length $n$ equals that of labeled plane trees of a certain type on $n$ vertices recently enumerated by Cori, Jacquard and Schaeffer, which is in turn known to be equal to the number of rooted bicubic maps enumerated by Tutte in 1963. Moreover, $H(x)$ turns out to be algebraic, proving the first nonmonotonic, longer-than-three instance of a conjecture of Zeilberger and Noonan. We also prove that $\sqrt[n]{S_n(1342)}$ converges to 8, so in particular, $lim_{n\rightarrow \infty}(S_n(1342)/S_n(1234))=0$

Euclid's theorem on the infinitude of primes: a historical survey of its proofs (300 B.C.--2017) and another new proof

In this article, we provide a comprehensive historical survey of 183 different proofs of famous Euclid's theorem on the infinitude of prime numbers. The author is trying to collect almost all the known proofs on infinitude of primes, including some proofs that can be easily obtained as consequences of some known problems or divisibility properties. Furthermore, here are listed numerous elementary proofs of the infinitude of primes in different arithmetic progressions. All the references concerning the proofs of Euclid's theorem that use similar methods and ideas are exposed subsequently. Namely, presented proofs are divided into 8 subsections of Section 2 in dependence of the methods that are used in them. {\bf Related new 14 proofs (2012-2017) are given in the last subsection of Section 2.} In the next section, we survey mainly elementary proofs of the infinitude of primes in different arithmetic progressions. Presented proofs are special cases of Dirichlet's theorem. In Section 4, we give a new simple "Euclidean's proof" of the infinitude of primes.Comment: 70 pages. In this extended third version of the article, 14 new proofs of the infnitude of primes are added (2012-2017