Related papers: Three Euler's Sieves and a Fast Prime Generator (F…
For a set of primes $\mathcal{P}$, let $\Psi(x, \mathcal{P})$ be the number of positive integers $n \leq x$ all of whose prime factors lie in $\mathcal{P}$. In this paper we classify the sets of primes $\mathcal{P}$ such that $\Psi(x,…
We present an algorithm analogous to the sieve of Eratosthenes that produces the list of twin primes. Next, we count the number of twin primes resulting from the construction with two different heuristic arguments. The first method is…
Prime Numbers clearly accumulate on defined spiral graphs,which run through the Square Root Spiral. These spiral graphs can be assigned to different spiral-systems, in which all spiral-graphs have the same direction of rotation and the same…
Prime path coverage is a powerful structural testing criterion, but generating all prime paths in a directed graph remains computationally challenging due to the potentially exponential number of them. Existing approaches typically rely on…
An elementary method for computing various prime sequences using the sequence of Farey sequences is described.
We describe an Euler scheme to approximate solutions of L\'evy driven Stochastic Differential Equations (SDE) where the grid points are random and given by the arrival times of a Poisson process. This result extends a previous work of the…
In this paper we show that a certain subset of the Carmichael numbers contains good Euler pseudoprimes, composite numbers that for many bases survive the Solovay-Strassen primality test. We present a classification of Carmichael numbers,…
In this paper, we study the global existence of steady subsonic Euler flows through infinitely long nozzles without the assumption of irrotationality. It is shown that when the variation of Bernoulli's function in the upstream is…
A prime sieve is an algorithm that finds the primes up to a bound $n$. We say that a prime sieve is incremental, if it can quickly determine if $n+1$ is prime after having found all primes up to $n$. We say a sieve is compact if it uses…
Consider the problem of efficiently evaluating isogenies $\phi: E \to E/H$ of elliptic curves over a finite field $\mathbb{F}_q$, where the kernel $H = \langle G\rangle$ is a cyclic group of odd (prime) order: given $E$, $G$, and a point…
We report the first experimental demonstration of prime number sieve via linear optics. The prime numbers distribution is encoded in the intensity zeros of the far field produced by a spatial light modulator hologram, which comprises a set…
We present a novel approach to the Twin Prime Conjecture, basing on the $6x \pm 1$ representation of primes. By defining so-called twin prime generators $x \in \N$, for which both $6x - 1$ and $6x + 1$ are prime, we reformulate the…
Nanomechanical computers promise a greatly improved energetic efficiency compared to their electrical counterparts. However, progress towards this goal is hindered by a lack of modular components, such as logic gates or transistors, and…
In this work, we present the first efficient and practical algorithm for estimating the number of triangles in a graph stream using predictions. Our algorithm combines waiting room sampling and reservoir sampling with a predictor for the…
In this study, we employ Euler's method and Richardson's extrapolation to solve a triple integral, which is then transformed into a third-order initial value problem. Our objective is to resolve the computational challenges associated with…
Fully Homomorphic Encryption (FHE) enables the evaluation of programs directly on encrypted data. However, because only basic operations can be performed on ciphertexts, programs must be expressed as boolean or arithmetic circuits. This…
We fix a gap in our proof of an upper bound for the number of positive integers $n\le x$ for which the Euler function $\varphi(n)$ has all prime factors at most $y$. While doing this we obtain a stronger, likely best-possible result.
Up to 150000 asteroids will be visible in the images of the ESA Euclid space telescope, and the instruments of Euclid offer multiband visual to near-infrared photometry and slitless spectra of these objects. Most asteroids will appear as…
When $k>1$ and $n$ is the product of the smallest $k$ primes, the $(k+1)$-st smallest prime is the least divisor exceeding $1$ of $n^{n^n}-1$. This variant of Euclid's prime generator is discussed with some of its cousins.
The Cerebras Wafer-Scale Engine (WSE) delivers performance at an unprecedented scale of over 900,000 compute units, all connected via a single-wafer on-chip interconnect. Initially designed for AI, the WSE architecture is also well-suited…