相关论文: Are There Infinitely Many Primes?
Let $S_{(x,y]} = \left\{\frac{p_n}{p_{n+1}-2} :~ n\in I \right\}$, where $I = \left\{n :~ x<p_n \le y \right\}$, $p_n$ is the $n$-th prime and $x, y \in \mathbb{R}_{>0}$. If $M_\alpha(x,y)$ denotes the $\alpha$-power mean of the elements of…
A learning algorithm based on primary school teaching and learning is presented. The methodology is to continuously evaluate a student and to give them training on the examples for which they repeatedly fail, until, they can correctly…
We propose a semester-long Bayesian statistics course for undergraduate students with calculus and probability background. We cultivate students' Bayesian thinking with Bayesian methods applied to real data problems. We leverage modern…
A sieve is constructed for ordinary twin primes of the form 6m+/-1 that are characterized by their twin rank m. It has no parity problem. Non-rank numbers are identified and counted using odd primes p>=5. Twin- and non-ranks make up the set…
A binary string representation of prime occurrences is a sequence of bits, where $1$ entries encode positions of prime numbers. This is a convenient representation for analysis of prime distribution, since it allows for application of a…
By using Beta Dirichlet series and then Eisenstein series we ca represent primes with first a good approximation and an exact expression. This can be done with arbitrary prime (up to 10^101).
The primes or prime polynomials (over finite fields) are supposed to be distributed `irregularly' , despite nice asymptotic or average behavior. We provide some conjectures/guesses/hypotheses with `evidence' of surprising symmetries in…
In the paper, there are new found methods to determine the range of every exceptional element in exceptional set, we can solve Twin primes problem and Goldbach Conjecture problem basically.
We adopt a physically motivated empirical approach to the characterisation of the distributions of twin and triplet primes within the set of primes, rather than in the set of all natural numbers. Remarkably, the occurrences of twins or…
The computational costs of inference and planning have confined Bayesian model-based reinforcement learning to one of two dismal fates: powerful Bayes-adaptive planning but only for simplistic models, or powerful, Bayesian non-parametric…
We continue investigations on the average number of representations of a large positive integer as a sum of given powers of prime numbers. The average is taken over a short interval, whose admissible length depends on whether or not we…
Definition of the number of prime numbers in the given interval
Recently we have introduced a novel characterisation of the distribution of twin primes that consists of three essential elements. These are: that the twins are most naturally viewed as a subsequence of the primes themselves, that the…
This note explores probabilistic sampling weighted by uncertainty in active learning. This method has been previously used and authors have tangentially remarked on its efficacy. The scheme has several benefits: (1) it is computationally…
Building on the concept of pretentious multiplicative functions, we give a new and largely elementary proof of the best result known on the counting function of primes in arithmetic progressions.
Additive models (AMs) have sparked a lot of interest in machine learning recently, allowing the incorporation of interpretable structures into a wide range of model classes. Many commonly used approaches to fit a wide variety of potentially…
We study a special set of constellations of primes generated by twin primes.
The possibility to study intermittency in a single event of high multiplicity is investigated in the framework of the $\alpha-$model. It is found that, for cascade long enough, the dispersion of intermittency exponents obtained from…
We prove that the primes of the form $x^2+y^2+1$ contain arbitrarily long non-trivial arithmetic progressions.
Departing from a class of infinite series with central binomial coefficients in the numerator and depending on a positive integer parameter, we first extend known identities to all complex parameters. Then we use various methods, including…