"Divergent" Ramanujan-type supercongruences

"Divergent" Ramanujan-type series for $1/\pi$ and $1/\pi^2$ provide us with new nice examples of supercongruences of the same kind as those related to the convergent cases. In this paper we manage to prove three of the supercongruences by means of the Wilf--Zeilberger algorithmic technique.Comment: 14 page

New approach to λ-Stirling numbers

The aim of this paper is to study the $\lambda$-Stirling numbers of both kinds, which are $\lambda$-analogues of Stirling numbers of both kinds. These numbers have nice combinatorial interpretations when $\lambda$ are positive integers. If $\lambda = 1$, then the $\lambda$-Stirling numbers of both kinds reduce to the Stirling numbers of both kinds. We derive new types of generating functions of the $\lambda$-Stirling numbers of both kinds which are related to the reciprocals of the generalized rising factorials. Furthermore, some related identities are also derived from those generating functions. In addition, all the corresponding results to the $\lambda$-Stirling numbers of both kinds are obtained for the $\lambda$-analogues of $r$-Stirling numbers of both kinds, which are generalizations of those numbers

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

Open Conjectures on Congruences

We collect here various conjectures on congruences made by the author in a series of papers, some of which involve binary quadratic forms and other advanced theories. Part A consists of 100 unsolved conjectures of the author while conjectures in Part B have been recently confirmed. We hope that this material will interest number theorists and stimulate further research. Number theorists are welcome to work on those open conjectures; for some of them we offer prizes for the first correct proofs.Comment: 88 pages. For new additions, see Conj. A76-8