相关论文: Are There Infinitely Many Primes?
In the last decade, major efforts have been made to promote inquiry-based mathematics learning at the tertiary level. The Inquiry-Based Mathematics Education (IBME) movement has gained strong momentum among some mathematicians, attracting…
We give an estimation of the existence density for the $2d$ different primes by using a new and simple algorithm for getting the $2d$ different primes. The algorithm is a kind of the sieve method, but the remainders are the central numbers…
The author proves that for $0.9985 < \gamma < 1$, there exist infinitely many primes $p$ such that $[p^{1/\gamma}]$ has at most 5 prime factors counted with multiplicity. This gives an improvement upon the previous results of…
The Prime Number Theorem states that the number of primes in $\{1,\ldots,x\}$, denoted $\pi(x)$, is approximately $\frac{x}{\ln(x)}$. In this paper, we investigate the distribution of primes for domains other than $\N$. First we look at…
We consider incentivized exploration: a version of multi-armed bandits where the choice of arms is controlled by self-interested agents, and the algorithm can only issue recommendations. The algorithm controls the flow of information, and…
PROMETHEE methods are widely used in Multiple Criteria Decision Aiding (MCDA) to deal with real decision making problems. A crucial aspect of the classical PROMETHEE methods is the choice of criteria weights. In this paper, we propose to…
The original formulation of BEAMS - Bayesian Estimation Applied to Multiple Species - showed how to use a dataset contaminated by points of multiple underlying types to perform unbiased parameter estimation. An example is cosmological…
One challenge (or opportunity!) that many instructors face is how varied the backgrounds, abilities, and interests of students are. In order to simultaneously instill confidence in those with weaker preparations and still challenge those…
We create a simple test for distinguishing between sets of primes and random numbers using just the sum-of-digits function. We find that the sum-of-the-digits of prime numbers does not have an equal probability of being odd or even. The…
This note discusses the existence of prime numbers in short intervals. An unconditional elementary argument seems to prove the existence of primes in the short intervals [x, x + y], where y >= x^(1/2)(log x)^e, e > 0, and a sufficiently…
This is the written version of a series of lectures reviewing the basics of duality as applied to p-forms and sigma-models. The ideas are introduced by way of worked examples, often quite detailed. Our approach is very pedestrian and the…
Prime numbers have fascinated mathematicians since antiquity, with ongoing efforts to uncover both their properties and ever-larger examples. While giant primes rarely aid cryptography, they find use in areas such as locally decodable…
The paper considers estimates for some sums and products of functions of prime numbers. Several assertions on this topic have been proven. We also study extremal estimates for strongly additive and strongly multiplicative arithmetic…
This paper describes some of the ideas used in the development of our work on small gaps between primes.
In this paper we continue our study, begun in part I, of the exceptional set of integers, not restricted by elementary congruence conditions, which cannot be represented as sums of three or four squares of primes. We correct a serious…
A classical problem in analytic number theory is to study the distribution of $\alpha p$ modulo 1, where $\alpha$ is irrational and $p$ runs over the set of primes. We consider the subsequence generated by the primes $p$ such that $p+2$ is…
Works, briefly surveyed here, are concerned with two basic methods: Maximum Probability and Bayesian Maximum Probability; as well as with their asymptotic instances: Relative Entropy Maximization and Maximum Non-parametric Likelihood.…
We introduce and investigate a new model of a finite number of particles jumping forward on the real line. The jump lengths are independent of everything, but the jump rate of each particle depends on the relative position of the particle…
{\bf In the fourth extended version of this article, we provide a comprehensive historical survey of 200 different proofs of famous Euclid's theorem on the infinitude of prime numbers (300 {\small B.C.}--2022)}. The author is trying to…
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.…