Parallel Numerical Algorithms

This paper introduces Harding Umbrella Primes (HUP), a structured prime configuration based on a balanced 7-prime Latin cross. The structure is extended into a full Prime Umbrella Field, where primes form a geometric canopy. A canonical example (HCC-877) is presented and verified. The system blends symmetry, geometry, and additive number theory.

We analyze the distribution of Mersenne primes M p = 2 p-1 using the Ducci Unified Spectral Theory (DUST) framework. By mapping Mersenne prime exponents onto the prime lattice and analyzing their spectral properties, we discover that they follow an exponential growth pattern with predictable spacing. This allows us to make falsifiable predictions for the next undiscovered Mersenne primes. Our analysis suggests the next Mersenne prime exponent is approximately p ≈ 93,921,462, corresponding to a number with over 28 million digits.

We introduce a deterministic sieve algorithm designed to efficiently generate all prime numbers greater than or equal to 3 within a given interval [ , ]. Our method distinguishes itself by directly constructing the set of all odd composite numbers through a unique geometric characterization, identifying them as integer points within a region bounded by two rectangular hyperbolas. This approach fundamentally eliminates the reliance on factorization or prior knowledge of primes, a key advantage for segmented sieving. The algorithm's time complexity is ((-) ln) for a general interval [ , ], and specifically (√) for obtaining primes within fixed-size segments (=-), achieving this efficiency without requiring pre-computed primes up to √. Beyond its primary sieving capability, this robust geometric framework provides a unified and mathematically rigorous approach to fundamental problems in number theory. As a direct consequence, we derive a novel function, (), which acts as a precise, deterministic indicator for twin prime pairs. This powerful reformulation transforms the Twin Prime Conjecture into demonstrating that () vanishes for infinitely many values of. We also explicitly show how this model directly supports efficient primality testing of individual numbers and offers a novel combinatorial and analytic derivation relating to the Prime Number Theorem. Our approach, including the derivation of odd-valued floor and ceiling functions and their Fourier series representations, provides a fresh analytical lens for these classical problems.

Various bounds on p, such as Bertrand's Postulate and Legendre's Conjecture, propose regions around n that have at at least one prime within them. Using Prime Generator Theory, I show more precise symmetric bounds on p, such that for n a prime exists symmetrically within a distance of n 1/2 below and above it. That is to say, a prime exists for: n-n^(1/2) < p < n and n < p < n + n^(1/2) .

We formalize a prime generation method-the XOR-wheel sieve-that is provably equivalent to the classical Sieve of Eratosthenes. The method (i) prefilters candidates to {2, 3} ∪ {n : n ≡ ±1 (mod 6)} ("hex spokes") and (ii) clears (sets to false) multiples starting at p 2 for every surviving prime p. We give a succinct correctness proof, an impossibility result for XOR-toggle variants, complexity notes, and a documented stress test up to N = 200,000 with zero mismatches against a plain sieve. Our formulation preserves the intuitive "first-cause collapse" narrative while remaining mathematically orthodox.

This paper introduces a new structural and deterministic method for generating and analyzing prime numbers, named the Prime Hole Table Method. Inspired by the concept that primes act like atomic building blocks, this framework constructs primes through systematic and observable counting mechanisms. It explores key structural equations such as C(n) = S-I(n), P(n) = C(n) + 1, and I(n) + P(n) = P_max.

In the article, the author shows the transition from the ternary Goldbach problem to the binary and then to the solution of the problem of the infinity of twins. This article is the final one, in which the errors and shortcomings of his previous articles on this topic are corrected.

In this paper I present the distribution of prime numbers which was treated in many
researches by studying the function of Riemann; because it has a remarkable property; its
non trivial zeros are prime numbers; but in this work I will show that we can find the
distribution of prime numbers on remaining in natural numbers onl

This paper introduces a novel visualization method for prime number distribution, based on a modified version of the sieve of Eratosthenes combined with a coordinate system. Each cell in this coordinate system represents a specific number, determined by combining the row and column indices of the grid. The multiples of known primes are marked to exclude composite numbers, leaving only prime candidates unmarked. This visualization method provides an accessible and intuitive framework for analysing prime number patterns, making prime candidates easy to identify even in larger numerical ranges. By organizing numbers in a structured way, the coordinate system reveals patterns that provide insights into the relationships between primes and composite numbers.

the author has improved the published articles. about this theme. In turn, the author strives to convey to reader important conclusions in his opinion, which in this article, namely that a transition from the ternary problem is possible to binary and further to solving the problem of infinity of numbers of twins.

In this paper proof of the Polignac&#39;s Conjecture for gap equal to eight is going to be presented. It will be shown that consecutive primes with gap eight could be obtained through two stage sieve process, and that will be used to prove that infinitely many primes with gap eight exist. The proof represents an simple extension of the recently presented proof that infinitely many sexy prime exist. The major contribution of this paper is presentation of all elementary modules that are necessary for the proof of Polgnac&#39;s conjecture in general case.

This paper describes the mathematical foundation of Prime Generators and their use in creating a fast Twin Primes Segmented Sieve of Zakiya (SSoZ), and also their applications to Number Theory, including Mersenne Primes, creating an exact Prime-counting Function, and implications for the Riemann Hypothesis.

Abstract. This paper presents a new distributed approach for generating all prime numbers in a given interval of integers. From Eratosthenes, who elaborated the first prime sieve (more than 2000 years ago), to the current generation of parallel computers, which have permitted to reach larger bounds on the interval or to obtain previous results in a shorter time, prime numbers generation still represents an attractive domain of research and plays a central role in cryptography. We propose a fully distributed algorithm for finding all primes in the interval [2..., n], based on the wheel sieve and the SMER (Scheduling by Multiple Edge Reversal) multigraph dynamics. Given a multigraph M of arbitrary topology, having N nodes, an SMER-driven system is defined by the number of directed edges (arcs) between any two nodes of M, and by the global period length of all “arc reversals ” in M. The new prime number generation method inherits the distributed and parallel nature of SMER and requires...

Prime hunting" can be considered as a research area of computational number theory. Its goal is to find special combinations of integers and prove their primality. Four research groups, established by A. Járai between 1992 and 2014, published numerous world class scientific results. In this period, due to Járai's arithmetic routines fastest in the world, they reached the world record 19 times, namely found the largest known twin primes 9 times, Sophie Germain primes 7 times, a prime of the form n 4 + 1, a number which is simultaneously twin and Sophie Germain prime and the three largest known primes forming a Cunningham chain of length 3 of the first kind. When A. Járai retired, the investigations of this area were suspended. In the beginning of 2020 the research was reopened by G. Farkas at the newly-founded campus (Szombathely) of ELTE. The first signal success came in the end of May 2020. They proved the primality of the numbers which form the largest known Cunningham chains of length 2 of the 2nd kind. In this paper we report on a newly started prime hunting project with the aim of increasing our students' research activity.

This paper investigates the problem of finding large prime gaps (the difference between two consecutive prime numbers, pi+1pi) and on the development of a small, efficient program for generating such large prime gaps for a single computer, a laptop or a workstation. In Wikipedia [1], one can find a table of all known record prime gaps less than 2 64 , the record is a 20 decimal digit number. We wanted to go beyond 64 bit numbers and demonstrate algorithms that do not needed a huge number of computers in a grid to produce useful results. After some preliminary tests, we found that the Sieve of Eratosthenes, SE, from the year 250 BC was the fastest for finding prime numbers and it could also be made space efficient. Each odd number is represented by one bit and when storing 8 odd numbers in a single byte (representing 16 consecutive numbers ignoring the even numbers), we found that we should not make one long SE table, but instead divide the SE table into segments (called SE segments), each of length 10 8 or 10 9 and dynamically generate the necessary SE segments as to find prime numbers. First, we made a basic segment of all prime numbers < 10 8 (in less than a second). We also relied heavily on the old observation [2] that when using SE to find all prime numbers ≤ , we cross out all numbers using the prime numbers ≤ √ 2 , and that the first number crossed off when crossing out for prime number p is 2. When we want to find prime gaps, we first create one or more consecutive SE in that range, say starting on 2 74 and ending with the value Minitially these big segments are crossed out by our first basic set of primes < 10 8 , To find all prime number in these big segments, we next need the rest of prime numbers ≤ √ 2. These can be all be constructed by using our first set of prime numbers to generate segments of consecutive SE from 10 8. The primes in these segments are used to cross out in the big SE segment and can then be discarded (each prime used only once). Our most significant algorithm was to find a simple formula for using primes from a range 3-2 36 to cross out the non-primes in any SE segment without crossing out in all the numbers between 2 36 and 2 72. This leads to an exponential saving in both space and execution time. In addition to this, we created a small package Int3 to represent numbers > 2 64 by storing 8 decimal values in each of 3 integer variables together with the necessary mathematical operations. The Int3 package can handle numbers up to 24 decimal digits and is significantly faster than the BigInteger package in the Java library. We also created a faster algorithm for finding all record prime gaps. The results presented in this paper are some tables of prime gaps for primes significantly larger than 2 64 and data supporting an observation that big prime gaps in these segments are much more frequent than the ones we find in the Wikipedia table where the search starts at prime number 3. Our combined set of algorithms is also sufficiently fast to test every entry in the Wikipedia table in less than 5 minutes. We conclude by reflecting on the use of brute force (more computers) versus smarter algorithms.

This paper presents a new distributed approach for generating all prime numbers in a given interval of integers. From Eratosthenes, who elaborated the first prime sieve (more than 2000 years ago), to the current generation of parallel computers, which have permitted to reach larger bounds on the interval or to obtain previous results in a shorter time, prime numbers generation still represents an attractive domain of research and plays a central role in cryptography. We propose a fully distributed algorithm for finding all primes in the interval [2. .. , n], based on the wheel sieve and the SMER (Scheduling by Multiple Edge Reversal) multigraph dynamics. Given a multigraph M of arbitrary topology, having N nodes, an SMER-driven system is defined by the number of directed edges (arcs) between any two nodes of M, and by the global period length of all "arc reversals" in M. The new prime number generation method inherits the distributed and parallel nature of SMER and requires at most n + √ n time steps.

In this article, we propose a fully distributed algorithm for finding all primes in an given interval [2..n] (or (L, R), more generally), based on the SMER -Scheduling by Multiple Edge Reversal -multigraph dynamics. Given a multigraph M of arbitrary topology, having N nodes, the SMER-driven system is defined by the number of directed edges (arcs) between any two nodes of M, and by the global period length of all "arc reversals" in M. In the domain of prime numbers generation, such a graph method shows quite elegant, and it also yields a totally new kind of distributed prime sieving algorithms of an entirely original design. The maximum number of steps required by the algorithm is at most n + √ n. Although far beyond the O(n/ log log n) steps required by the improved sequential "wheel sieve" algorithms, our SMER-based algorithm is fully distributed and of linear (step) complexity. The message complexity of the algorithm is at most n∆N + √ n∆N , where ∆N denotes the maximum "multidegree" of the arbitrary multigraph M, and the space required per process is linear.

The performance and use of parallel computing in the field of differential calculus is increased tremendously opening up new avenues for applying these in the field of numerical computation for high speed performance. The computation time required to find analytical as well as numerical solution is tested and compared. In this work we have harnessed this property of GPU to accelerate the grid point calculations for numerical calculations and the performance of numerical method using CPU and GPU is compared. The numerical Methods for integer order PDE are studied, analyzed and implemented on GPU using parallel computing toolbox of MATLAB. The finite difference methods of PDE like explicit, implicit method are tested for the results, for parabolic, hyperbolic and elliptical type of PDE’s. The positive speed up is achieved for elliptical type of PDE. The verification of results with the analytical solution is made by the mean square error.

This paper describes the mathematical foundation of Prime Generators and their use in creating a fast Twin Primes Segmented Sieve of Zakiya (SSoZ), and also their applications to Number Theory, including Mersenne Primes, creating an exact Prime-counting Function, and implications for the Riemann Hypothesis.

Benchmark results for the Numerical Aerodynamic Simulation (NAS) Program at NASA Ames Research Center, which is dedicated to advancing the science of computational aerodynamics are presented. The benchmark performance results are for the Y-MP, Y-MO EL, and C-90 systems from Cray Research; the TC2000 from Bolt Baranek and Newman; the Gamma iPSC/860 from Intel; the CM-2, CM-200, and CM-5 from Thinking Machines; the CS-1 from Meiko Scientific; the MP-1 and MP-2 from MasPar Computer; ...

In this paper, we propose a new primality test, and then we employ this test to find a formula for {\pi} that computes the number of primes within any interval. We finally propose a new formula that computes the nth prime number as well as the next prime for any given number

In this paper, we propose a new primality test, and then we employ this test to find a formula for {\pi} that computes the number of primes within any interval. We finally propose a new formula that computes the nth prime number as well as the next prime for any given number

In this paper, we propose a new primality test, and then we employ this test to find a formula for {\pi} that computes the number of primes within any interval. We finally propose a new formula that computes the nth prime number as well as the next prime for any given number

Most of the problems in engineering science can be modelled using differential equations. However quite often, the differential equations are not amenable to direct analytical solutions because of complex geometries and boundary conditions. Thus recourse is taken to approximate numerical solution techniques. Depending on the discretization of the domain, the numerical model of the problem at hand could become very large-sometimes running into thousands/lakhs of degrees of freedom. Solution of such a large set of equations is a daunting task computationally and hence efforts are directed towards development of efficient strategies and use of High Performance Computing (HPC). The present work aims to implement an efficient solver for system of linear equations with sparse, symmetric and positive-definite matrix of coefficients using the most prominent Conjugate Gradient iterative method and study the performance improvement obtained in parallel computing framework using MPI and GPU enabled CUDA technologies considering a suitable example problem.

This paper describes the mathematical foundation of Prime Generators and their use in creating a fast Twin Primes  Segmented Sieve of Zakiya (SSoZ), and also their applications to Number Theory, including Mersenne Primes, creating an exact Prime-counting Function, and implications for the Riemann Hypotheses