Related papers: Large primes in generalized Pascal triangles
In the paper "An Abelian Loop for Non-Composites" (arXiv:110.14716), we introduced a group-like structure consisting of odd prime numbers and 1, with properties that allowed us to prove analogous results to well known theorems in Number…
We report on the implementation of an algorithm for computing the set of all regular triangulations of finitely many points in Euclidean space. This algorithm, which we call down-flip reverse search, can be restricted, e.g., to computing…
We define a family {$\gamma(P)$} of generalized Euler constants indexed by finite sets of primes $P$ and study their distribution. These arise from partial sums of reciprocals of integers not divisible by any prime in $P$. An apparent…
A random matrix is likely to be well conditioned, and motivated by this well known property we employ random matrix multipliers to advance some fundamental matrix computations. This includes numerical stabilization of Gaussian elimination…
In the proposed matrix primes, through which one can readily generate a sequence of primes. The paper also proposes a number of theorems proved by which an infinite number of prime numbers twins
The goal of this paper is to analyze the discrete Lambert map x to xg^x modulo a power of a prime p which is important for security and verification of the ElGamal digital signature scheme. We use p-adic methods (p-adic interpolation and…
The pursuit of quantum advantage in simulating many-body quantum systems on quantum computers has gained momentum with advancements in quantum hardware. This work focuses on leveraging the symmetry properties of these systems, particularly…
The $p$-set, which is in a simple analytic form, is well distributed in unit cubes. The well-known Weil's exponential sum theorem presents an upper bound of the exponential sum over the $p$-set. Based on the result, one shows that the…
We show that an elation generalised quadrangle which has p+1 lines on each point, for some prime p, is classical or arises from a flock of a quadratic cone (i.e., is a flock quadrangle).
Universally decodable matrices can be used for coding purposes when transmitting over slow fading channels. These matrices are parameterized by positive integers $L$ and $n$ and a prime power $q$. Based on Pascal's triangle we give an…
This paper reviews generalized Pareto copulas (GPC), which turn out to be a key to multivariate extreme value theory. Any GPC can be represented in an easy analytic way using a particular type of norm on $\mathbb{R}^d$, called $D$-norm. The…
{\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…
Currently there is no known efficient formula for primes. Besides that, prime numbers have great importance in e.g., information technology such as public-key cryptography, and their position and possible or impossible functional generation…
We present effective upper bounds on the symmetric bilinear complexity of multiplication in extensions of a base finite field Fp2 of prime square order, obtained by combining estimates on gaps between prime numbers together with an optimal…
This paper provides an accurate method to obtain the bidiagonal factorization of many generalized Pascal matrices, which in turn can be used to compute with high relative accuracy the eigenvalues, singular values and inverses of these…
We consider a class of generalized binomials emerging in fractional calculus. After establishing some general properties, we focus on a particular yet relevant case, for which we provide several ready-for-use combinatorial identities,…
We derived the sum identities for generalized harmonic and corresponding oscillatory numbers for which a sieve procedure can be applied. The obtained results enable us to understand better the properties of these numbers and their…
We prove general results which include classical facts about 60 Pascal's lines as special cases. Along similar lines we establish analogous results about configurations of 2520 conics arising from Mystic Octagon. We offer a more…
Translated from the Latin original "Facillima methodus plurimos numeros primos praemagnos inveniendi" (1778). E718 in the Enestrom index. If m is a number of the form 4k+1 and is a sum of two relatively prime squares, then it is prime if…
We describe arithmetic algorithms on a canonical number representation based on the Catalan family of combinatorial objects specified as a Haskell type class. Our algorithms work on a {\em generic} representation that we illustrate on…