Related papers: Some history about Twin Prime Conjecture
This paper introduces a new method to find the next prime number after a given prime ${P}$. The proposed method is used to derive a system of inequalities, that serve as constraints which should be satisfied by all primes whose successor is…
We study pairs of consecutive odd numbers through a straightforward indexing. We focus in particular on twin primes and their distribution. With a counting argument, we calculate the limit of an alternating sum that is equal to 1 which…
The article provides a counterexample to a conjecture by Blocki-Zwonek.
Under the prime-tuple hypothesis, the set of signed primes is a sumset.
This article offers different proofs of ten inequalities from those already published. So that the readers can see for themselves, the tasks specified in the condition of the source and classical inequalities which used in previously…
The prime number problem falls within the realm of number theory, specifically elementary number theory. Current research approaches have unnecessarily complicated this matter. In contrast to more advanced mathematical tools, the methods of…
Herein we present one hundred inequalities culled from various corners of the probability, statistics, and combinatorics literature. We welcome new suggestions.
The article attempts to demonstrate the rich history of one truly remarkable problem situated at the confluence of probability theory and theory of numbers - finding the probability of co-primality of two randomly selected natural numbers.…
An s-tuple of positive integers are k-wise relatively prime if any k of them are relatively prime. Exact formula is obtained for the probability that s positive integers are k-wise relatively prime.
We survey recent developments on the Restriction conjecture.
I review the classical theory of likelihood based inference and consider how it is being extended and developed for use in complex models and sampling schemes.
We address the question of the infinitude of twin and cousin prime pairs from a probabilistic perspective. Our approach partitions the set of integer numbers greater than $2$ in finite intervals of the form $[p_{n-1}^2,p_n^2)$, $p_{n-1}$…
The results of several papers concerning the \v{C}ern\'y conjecture are deduced as consequences of a simple idea that I call the averaging trick. This idea is implicitly used in the literature, but no attempt was made to formalize the proof…
A short, fairly self-contained proof is given of the Poincar\'e Conjecture. In the previous version there was an error on Page 8. This gap has now been filled.
This is an article for a general mathematical audience on the author's work, joint with Terence Tao, establishing that there are arbitrarily long arithmetic progressions of primes. It is based on several one hour lectures, chiefly given at…
In this small article, unified first law has been analyzed and some results have been deduced from it. The results have been presented in the form of lemmas and some conclusions have been drawn from them.
Recently we discovered (cond-mat/0212043) that the majority of scientific citations are copied from the lists of references used in other papers. Here we show that a model, in which a scientist picks three random papers, cites them,and also…
In this paper, using the well known fact that the series of reciprocals of primes diverges, we obtain a general inequality for gaps of consecutive primes that holds for infinitely many primes. As it is shown the key ingredient for this…
A popular scientific contribution should not contradict any established facts and ought to be understandable. I complied with both these requirements and am offering a sufficiently full introduction to probability theory. Furthermore, I…
In this article the idea of random variables over the set theoretic universe is investigated. We explore what it can mean for a random set to have a specific probability of belonging to an antecedently given class of sets.