DOCSLIB.ORG

Modulo Primality Test.Pdf

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Modulo Primality Test.Pdf Modulo primality test. 1. Abstract. Verifying large prime numbers is a time-consuming proces, even with modern day computers. This article describes the "modulo primality test" that combines the segmented prime spiral, modular arithmetic and the primorial sieve. The "modulo primality test" can be used to factorize RSA-numbers. In the segmented prime spiral with one segment each integer is identified as an unique member of the Eastward family of quadratic polynomials with only two terms. By way of modular arithmetic the number of digits is reduced when checking a large integer for a possible divisor. In addition the primorial sieve can be used to deselect sets of divisors. The modulo primality test can also be used as another methode to check the correctness of computercalculations. 2. The prime spiral with one segment. The modulo primality test is based on the prime spiral with one segment. This prime spiral is derived from the Ulam spiral with startvalue 0 when placed in a Cartesian coordinate system (appendix A). There is an infinite set of families of segmented prime spirals. The Ulam spiral has four segments. The segmented prime spirals are unusual since integers on the seam appear twice. These doubled integers disappear behind the overlap when putting a prime spiral together, like when folding the prime spiral with one segment into a cone (Fig. 1), or the SE main diagnonal in the Ulam spiral (appendix A). Each integer in the prime spiral with one segment is member of the Eastward family of functions 2 2 f1, c(nE) = 1 n + 0 n + c = n + c with −n < c ≤ n which has just two terms (Fig. 1). For instance the integer g = 90 with n = ⌊⌊⌊ √g ⌉⌉⌉ has the parameters n = 9 and c = 9 and is thus found on the NE main diagonal at the location (x, y) = ( n, c) = (9, 9). The location (10, −10) of g = 90 on the SE main diagonal does not comply with the definition of the families of functions (appendix A). 2 2 Note that the integer g = 90 also belongs to f1,0 (nNE ) = 1 n + 1 n + 0 and f1,18 (nSE ) = 1 n − 1 n + 18 with n = ⌊⌊⌊ √g ⌉⌉⌉ Fig. 1: Partial prime spiral with one segment based on the Ulam spiral. Hans Dicker \ _Modulo primality test.doc 1/13 3. The modulo primality test. The simplest primality test to verify if an integer g is a prime number is trial division. Find a prime integer d from 2 to √g that evenly divides g (the division leaves no remainder). As sone as g is evenly divisible by d then g is composite, otherwise g is a prime number. In the prime spiral with one segment each integer g is a unique member of the family of quadratic functions 2 2 fa,b,c(n) = f1, b,c(n) = f1, c(nE) = 1 n + 0 n + c = n + c with −n < c ≤ n. This specific function has only two terms. So, each integer g is defined as g = n2 + c with −n < c ≤ n and n = ⌊⌊⌊ √g ⌉⌉⌉. The modulo primality test reduces the integer g = n2 + c into g' = n' · n' + c with n' = n (mod d). When there is a divisor d in 2 ≤ d ≤ √g with g' (mod d) = 0 then g is composite, otherwise g is a prime number. The modulo primality test takes more operations than the straightforward division of the integer g by the divisors (appendix C: "Modular arithmetic"). The strength of the modulo primality test lies in the reduction of the size of the integer g (Fig. 2). Fig. 2: The reduction of a integer as function of the divisor d via modular arithmetic. 2 g (mod d) 1 d 1 Fig. 2 shows that the reduction of bounces back to at every = with m ∈ { ..., 5, 4, 3, 2}. g m n m 1 From 1 towards the value of n' = n (mod d) slowly approaches zero via a quadratic function. m + 1 m Further reduction of g' (mod d) / g can be obtained by selective changing n2 = n · n ≡ n' · n' (mod d) into n2 = n · n ≡ n' · (n' − d) (mod d). Hans Dicker \ _Modulo primality test.doc 2/13 4. The modulo primality test to find verify large primes. In the prime number theorem the prime-counting function π(g) for the positive integers up to g is defined as: g π g)( ≈ with log ( g) the natural logarithm of g. log g)( For large enough g, the probability that a random integer not greater than g is prime is close to 1 / log ( g). Among the positive integers g ≈ 10 9 about one in 21 is prime, since log(10 9) ≈ 20.7. For prime numbers ≈ 10^10^8, about one in 2.3 · 10 8 is prime, since log(10^10^8) = 10 8 · log (10) ≈ 2.3 · 10 8. The modulo primality test can be deployed to search for ever larger prime numbers, like the first prime number with at least one hundred million digits when written in base 10. Below is a describtion of how to find the first prime number after integer gs with the modulo primality test. Imagine trying to find the first prime number after the integer gs = 10^10^8 − 1 2 1. The prime spiral with one segment defines the integer gs as member of f1, c(nE) = n + c with −n < c ≤ n. Calculate both n = ⌊⌊⌊ √g ⌉⌉⌉ and c = g − n2. 2. Take a list of prime numbers up to for instance pend = p238 . Define pi ∈ {p2, ..., pend } thus pi ∈ {3, ..., 1499}. Other options are: The P6#−sieve gives a list of all π(p6#) = 3,248 prime numbers < p6# with p6# = 30,030. 9 The extended P9#−sieve supplies a list of all prime numbers up to 10 . ' ' ' 3. Calculate gs (mod pi) for every given pi via gs'(pi) = ( n · n + c ) (mod pi) with n = n (mod pi). Store the calculations of gs'(pi) in an array. 4. Select the next odd integer gv, with ∆g = gv − gs the distance between gs and gv. Calculate gv'(pi) = ( gs'(pi) + ∆g ) (mod pi) for every given pi up to pend . Overwrite gs'( pi) with gv'( pi). As sone as gv' = 0 then gv is not a prime number. Repeat step 4. 5. When gv' ≠ 0 for every given pi up to pend then gv could be a prime number. Use modular arithmetic (appendix C) to check gv for primality. For instance with the P4#−sieve the division by the prime divisors pend < d < √gv can be approximated by d ∈ { S(p4#) j + k • p4# │ 1 ≤ j ≤ φ(p4#) ∧ k ∈ N0 } (see Appendix B). Hans Dicker \ _Modulo primality test.doc 3/13 5. The modulo primality test and RSA cryptography. RSA cryptography is based on two large prime numbers gA and gB to generate a composite number g = gA · gB. Multiplying the two large numbers gA and gB is easy. Factoring the large number g is very difficult. For example, the RSA-100 number is defined as the semi-prime g = 0.15226... · 10 100 , the product of the prime 50 50 numbers gA = 0.37975... · 10 and gB = 0.40094... · 10 . For demonstration purposes the RSA-100 number is replaced by the semi-prime g = 1,523,012,791 = 0.15230... · 10 10 5 5 and the prime numbers gA = 0.37987 · 10 and gB = 0.40093 · 10 . Out of the infinite set of primorial sieves, the P4#−sieve is implemented with p4# = 210 and φ(p4#) = 48. The integer g = 1,523,012,791 ≡ 181 (mod p4#) could be prime since 181 = S(p4#) 41 (see Appendix B). The segmented prime spiral with one segment splits g into the two terms of the Eastward quadratic polynomial. 2 The function f1, c(nE) = n + c with −n < c ≤ n gives n = ⌊⌊⌊ √g ⌉⌉⌉ = 39,026 and c = −15,885. Find d │ g via g = n · n − 15,885 ≡ n' · n' − 15,885 (mod d) with n' ≡ n (mod d). Possible divisors p4 < d < √g are d ∈ { S(p4#) j + k · p4# │ 1 ≤ j ≤ φ(p4#) ∧ k ∈ N0 }, based on the fourth double primorial sieve. Start at the end and work backwards, since the principles of RSA crytograhpy define g = gA · gB with gA ≈ gB ≈ √g ≈ n. 2 d ≤ n S(p4#) j n = 39,026 f1,−15885 (nE) = n − 15,885 = 1,523,012,791 Comment n' = n (mod d) f1,0,−15885 (39,026) ≡ n' · n' − 15,885 (mod d) about g 39,023 173 3 32 − 15,885 ≡ −15,876 − −1 · d ≡ 23,147 possible prime 39,019 169 7 72 − 15,885 ≡ −15,836 − − 1 · d ≡ 23,183 possible prime 39,017 167 9 92 − 15,885 ≡ −15,804 − − 1 · d ≡ 23,213 possible prime . 38,027 17 999 999 2 − 15,885 ≡ 982,116 − 25 · d ≡ 31,441 possible prime 38,023 13 1,003 1,003 2 − 15,885 ≡ 990,124 − 26 · d ≡ 1,526 possible prime . 37,997 197 1,029 1,029 2 − 15,885 ≡ 1,042,956 − 27 · d ≡ 17,037 possible prime 37,993 193 1,033 1,033 2 − 15,885 ≡ 1,051,204 − 27 · d ≡ 25,393 possible prime 37,991 191 1,035 1,031 2 − 15,885 ≡ 1,055,340 − 27 · d ≡ 29,583 possible prime 37,987 187 1,039 1,039 2 − 15,885 ≡ 1,063,636 − 28 · d ≡ 0 ◄ NOT prime The "modulo primality test" claims: Divisions? Who needs divisions! Based on the modulo primality test the RSA-100 number is reduced to maximum 0.001 of its original size.
Load more

Recommended publications
  • On the First Occurrences of Gaps Between Primes in a Residue Class
    On the First Occurrences of Gaps Between Primes in a Residue Class Alexei Kourbatov JavaScripter.net Redmond, WA 98052 USA akourbatov@gmail.com Marek Wolf Faculty of Mathematics and Natural Sciences Cardinal Stefan Wyszy´nski University Warsaw, PL-01-938 Poland m.wolf@uksw.edu.pl To the memory of Professor Thomas R. Nicely (1943–2019) Abstract We study the first occurrences of gaps between primes in the arithmetic progression (P): r, r + q, r + 2q, r + 3q,..., where q and r are coprime integers, q > r 1. ≥ The growth trend and distribution of the first-occurrence gap sizes are similar to those arXiv:2002.02115v4 [math.NT] 20 Oct 2020 of maximal gaps between primes in (P). The histograms of first-occurrence gap sizes, after appropriate rescaling, are well approximated by the Gumbel extreme value dis- tribution. Computations suggest that first-occurrence gaps are much more numerous than maximal gaps: there are O(log2 x) first-occurrence gaps between primes in (P) below x, while the number of maximal gaps is only O(log x). We explore the con- nection between the asymptotic density of gaps of a given size and the corresponding generalization of Brun’s constant. For the first occurrence of gap d in (P), we expect the end-of-gap prime p √d exp( d/ϕ(q)) infinitely often. Finally, we study the gap ≍ size as a function of its index in thep sequence of first-occurrence gaps. 1 1 Introduction Let pn be the n-th prime number, and consider the difference between successive primes, called a prime gap: pn+1 pn.
    [Show full text]
  • Ramanujan, Robin, Highly Composite Numbers, and the Riemann Hypothesis
    Contemporary Mathematics Volume 627, 2014 http://dx.doi.org/10.1090/conm/627/12539 Ramanujan, Robin, highly composite numbers, and the Riemann Hypothesis Jean-Louis Nicolas and Jonathan Sondow Abstract. We provide an historical account of equivalent conditions for the Riemann Hypothesis arising from the work of Ramanujan and, later, Guy Robin on generalized highly composite numbers. The first part of the paper is on the mathematical background of our subject. The second part is on its history, which includes several surprises. 1. Mathematical Background Definition. The sum-of-divisors function σ is defined by 1 σ(n):= d = n . d d|n d|n In 1913, Gr¨onwall found the maximal order of σ. Gr¨onwall’s Theorem [8]. The function σ(n) G(n):= (n>1) n log log n satisfies lim sup G(n)=eγ =1.78107 ... , n→∞ where 1 1 γ := lim 1+ + ···+ − log n =0.57721 ... n→∞ 2 n is the Euler-Mascheroni constant. Gr¨onwall’s proof uses: Mertens’s Theorem [10]. If p denotes a prime, then − 1 1 1 lim 1 − = eγ . x→∞ log x p p≤x 2010 Mathematics Subject Classification. Primary 01A60, 11M26, 11A25. Key words and phrases. Riemann Hypothesis, Ramanujan’s Theorem, Robin’s Theorem, sum-of-divisors function, highly composite number, superabundant, colossally abundant, Euler phi function. ©2014 American Mathematical Society 145 This is a free offprint provided to the author by the publisher. Copyright restrictions may apply. 146 JEAN-LOUIS NICOLAS AND JONATHAN SONDOW Figure 1. Thomas Hakon GRONWALL¨ (1877–1932) Figure 2. Franz MERTENS (1840–1927) Nowwecometo: Ramanujan’s Theorem [2, 15, 16].
    [Show full text]
  • The History of the Primality of One—A Selection of Sources
    THE HISTORY OF THE PRIMALITY OF ONE|A SELECTION OF SOURCES ANGELA REDDICK, YENG XIONG, AND CHRIS K. CALDWELLy This document, in an altered and updated form, is available online: https:// cs.uwaterloo.ca/journals/JIS/VOL15/Caldwell2/cald6.html. Please use that document instead. The table below is a selection of sources which address the question \is the number one a prime number?" Whether or not one is prime is simply a matter of definition, but definitions follow use, context and tradition. Modern usage dictates that the number one be called a unit and not a prime. We choose sources which made the author's view clear. This is often difficult because of language and typographical barriers (which, when possible, we tried to reproduce for the primary sources below so that the reader could better understand the context). It is also difficult because few addressed the question explicitly. For example, Gauss does not even define prime in his pivotal Disquisitiones Arithmeticae [45], but his statement of the fundamental theorem of arithmetic makes his stand clear. Some (see, for example, V. A. Lebesgue and G. H. Hardy below) seemed ambivalent (or allowed it to depend on the context?)1 The first column, titled `prime,' is yes when the author defined one to be prime. This is just a raw list of sources; for an evaluation of the history see our articles [17] and [110]. Any date before 1200 is an approximation. We would be glad to hear of significant additions or corrections to this list. Date: May 17, 2016. 2010 Mathematics Subject Classification.
    [Show full text]
  • ON the PRIMALITY of N! ± 1 and 2 × 3 × 5 ×···× P
    MATHEMATICS OF COMPUTATION Volume 71, Number 237, Pages 441{448 S 0025-5718(01)01315-1 Article electronically published on May 11, 2001 ON THE PRIMALITY OF n! 1 AND 2 × 3 × 5 ×···×p 1 CHRIS K. CALDWELL AND YVES GALLOT Abstract. For each prime p,letp# be the product of the primes less than or equal to p. We have greatly extended the range for which the primality of n! 1andp# 1 are known and have found two new primes of the first form (6380! + 1; 6917! − 1) and one of the second (42209# + 1). We supply heuristic estimates on the expected number of such primes and compare these estimates to the number actually found. 1. Introduction For each prime p,letp# be the product of the primes less than or equal to p. About 350 BC Euclid proved that there are infinitely many primes by first assuming they are only finitely many, say 2; 3;:::;p, and then considering the factorization of p#+1: Since then amateurs have expected many (if not all) of the values of p# 1 and n! 1 to be prime. Careful checks over the last half-century have turned up relatively few such primes [5, 7, 13, 14, 19, 25, 32, 33]. Using a program written by the second author, we greatly extended the previous search limits [8] from n ≤ 4580 for n! 1ton ≤ 10000, and from p ≤ 35000 for p# 1top ≤ 120000. This search took over a year of CPU time and has yielded three new primes: 6380!+1, 6917!−1 and 42209# + 1.
    [Show full text]
  • Sum of Divisors of the Primorial and Sum of Squarefree Parts
    International Mathematical Forum, Vol. 12, 2017, no. 7, 331 - 338 HIKARI Ltd, www.m-hikari.com https://doi.org/10.12988/imf.2017.7113 Two Topics in Number Theory: Sum of Divisors of the Primorial and Sum of Squarefree Parts Rafael Jakimczuk Divisi´onMatem´atica,Universidad Nacional de Luj´an Buenos Aires, Argentina Copyright c 2017 Rafael Jakimczuk. This article is distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduc- tion in any medium, provided the original work is properly cited. Abstract Let pn be the n-th prime and σ(n) the sum of the positive divisors of n. Let us consider the primorial Pn = p1:p2 : : : pn, the sum of its Qn positive divisors is σ(Pn) = i=1(1 + pi). In the first section we prove the following asymptotic formula n 6 σ(P ) = Y(1 + p ) = eγ P log p + O (P ) : n i π2 n n n i=1 Let a(k) be the squarefree part of k, in the second section we prove the formula π2 X a(k) = x2 + o(x2): 30 1≤k≤x We also study integers with restricted squarefree parts and generalize these results to s-th free parts. Mathematics Subject Classification: 11A99, 11B99 Keywords: Primorial, divisors, squarefree parts, s-th free parts, average of arithmetical functions 332 Rafael Jakimczuk 1 Sum of Divisors of the Primorial In this section p denotes a positive prime and pn denotes the n-th prime. The following Mertens's formulae are well-known (see [5, Chapter VI]) 1 1 ! X = log log x + M + O ; (1) p≤x p log x where M is called Mertens's constant.
    [Show full text]
  • Various Arithmetic Functions and Their Applications
    University of New Mexico UNM Digital Repository Mathematics and Statistics Faculty and Staff Publications Academic Department Resources 2016 Various Arithmetic Functions and their Applications Florentin Smarandache University of New Mexico, smarand@unm.edu Octavian Cira Follow this and additional works at: https://digitalrepository.unm.edu/math_fsp Part of the Algebra Commons, Applied Mathematics Commons, Logic and Foundations Commons, Number Theory Commons, and the Set Theory Commons Recommended Citation Smarandache, Florentin and Octavian Cira. "Various Arithmetic Functions and their Applications." (2016). https://digitalrepository.unm.edu/math_fsp/256 This Book is brought to you for free and open access by the Academic Department Resources at UNM Digital Repository. It has been accepted for inclusion in Mathematics and Statistics Faculty and Staff Publications by an authorized administrator of UNM Digital Repository. For more information, please contact amywinter@unm.edu, lsloane@salud.unm.edu, sarahrk@unm.edu. Octavian Cira Florentin Smarandache Octavian Cira and Florentin Smarandache Various Arithmetic Functions and their Applications Peer reviewers: Nassim Abbas, Youcef Chibani, Bilal Hadjadji and Zayen Azzouz Omar Communicating and Intelligent System Engineering Laboratory, Faculty of Electronics and Computer Science University of Science and Technology Houari Boumediene 32, El Alia, Bab Ezzouar, 16111, Algiers, Algeria Octavian Cira Florentin Smarandache Various Arithmetic Functions and their Applications PONS asbl Bruxelles, 2016 © 2016 Octavian Cira, Florentin Smarandache & Pons. All rights reserved. This book is protected by copyright. No part of this book may be reproduced in any form or by any means, including photocopying or using any information storage and retrieval system without written permission from the copyright owners Pons asbl Quai du Batelage no.
    [Show full text]
  • Prime Harmonics and Twin Prime Distribution
    Prime Harmonics and Twin Prime Distribution Serge Dolgikh National Aviation University, Kyiv 02000 Ukraine, sdolgikh@nau.edu.ua Abstract: Distribution of twin primes is a long stand- in the form: 0; p−1;:::;2;1 with the value of 0 the highest ing problem in the number theory. As of present, it is in the cycle of length p. Trivially, the positions with the not known if the set of twin primes is finite, the problem same value of the modulo p are separated by the minimum known as the twin primes conjecture. An analysis of prime of p odd steps. modulo cycles, or prime harmonics in this work allowed For a prime p, the prime harmonic function hp(x) can be to define approaches in estimation of twin prime distri- defined on IN1 as the modulo of x by p in the above format: butions with good accuracy of approximation and estab- lish constraints on gaps between consecutive twin prime Definition 1. A single prime harmonic hp(x) is defined for pairs. With technical effort, the approach and the bounds an odd integer x ≥ 1 as: obtained in this work can prove sufficient to establish that hp(1) := (p − 1)=2 the next twin prime exists within the estimated distance, hp(n + 1) := hp(n) - 1, hp(n) > 0 leading to the conclusion that the set of twin primes is un- hp(n + 1) := p − 1, hp(n) = 0 limited and reducing the infinitely repeating distance be- Clearly, h (x) = 0 ≡ p j x. tween consecutive primes to two.
    [Show full text]
  • Multiplication Modulo N Along the Primorials with Its Differences And
    Multiplication Modulo n Along The Primorials With Its Differences And Variations Applied To The Study Of The Distributions Of Prime Number Gaps A.K.A. Introduction To The S Model Russell Letkeman r. letkeman@ gmail. com Dedicated to my son Panha May 19, 2013 Abstract The sequence of sets of Zn on multiplication where n is a primorial gives us a surprisingly simple and elegant tool to investigate many properties of the prime numbers and their distributions through analysis of their gaps. A natural reason to study multiplication on these boundaries is a construction exists which evolves these sets from one primorial boundary to the next, via the sieve of Eratosthenes, giving us Just In Time prime sieving. To this we add a parallel study of gap sets of various lengths and their evolution all of which together informs what we call the S model. We show by construction there exists for each prime number P a local finite probability distribution and it is surprisingly well behaved. That is we show the vacuum; ie the gaps, has deep structure. We use this framework to prove conjectured distributional properties of the prime numbers by Legendre, Hardy and Littlewood and others. We also demonstrate a novel proof of the Green-Tao theorem. Furthermore we prove the Riemann hypoth- esis and show the results are perhaps surprising. We go on to use the S model to predict novel structure within the prime gaps which leads to a new Chebyshev type bias we honorifically name the Chebyshev gap bias. We also probe deeper behavior of the distribution of prime numbers via ultra long scale oscillations about the scale of numbers known as Skewes numbers∗.
    [Show full text]
  • On Prime Numbers and Related Applications
    International Journal of Innovative Technology and Exploring Engineering (IJITEE) ISSN: 2278-3075, Volume-8 Issue-12, October 2019 On Prime Numbers and Related Applications V Yegnanarayanan, Veena Narayanan, R Srikanth Abstract: In this paper we probed some interesting aspects of Our motivating factor for this work stems from the classic primorial and factorial primes. We did some numerical analysis result of called Prime Number Theorem which beautifully about the distribution of prime numbers and tabulated our models the statistical behavioral pattern of huge primes viz., findings. Also, we pointed out certain interesting facts about the utility value of the study of prime numbers and their distributions chance for an arbitrarily picked 푛휖푁 to be a prime number is in control engineering and Brain networks. inverse in proportion to log(푛). Keywords: Factorial primes, Primorial primes, Prime numbers. II. RESULTS AND DISCUSSIONS I. INTRODUCTION A. Discussion on Primorial and Factorial Primes The irregular distribution of prime numbers makes it Our journey is driven by fascinating characteristics of difficult for us to presume and discover. Euclid about 2300 primordial primes and factorial primes. A primordial years before established that primes are infinitely many. Till function denoted 푃# is the product of all primes till P and date we are not aware of a formula for finding the nth 푛 includes P. The factorial function is 푛! = 푖=1 푖. A natural prime. It appears as if they are randomly embedded in N and curiosity pushes us to probe large primes. In order to do this, may be because of this that sequences of primes do not tend we need to get an idea by analyzing numbers such as to emanate from naturally happening processes.
    [Show full text]
  • An Amazing Prime Heuristic
    AN AMAZING PRIME HEURISTIC CHRIS K. CALDWELL 1. Introduction The record for the largest known twin prime is constantly changing. For example, in October of 2000, David Underbakke found the record primes: 83475759 · 264955 ± 1: The very next day Giovanni La Barbera found the new record primes: 1693965 · 266443 ± 1: The fact that the size of these records are close is no coincidence! Before we seek a record like this, we usually try to estimate how long the search might take, and use this information to determine our search parameters. To do this we need to know how common twin primes are. It has been conjectured that the number of twin primes less than or equal to N is asymptotic to Z N dx 2C2N 2C2 2 ∼ 2 2 (log x) (log N) where C2, called the twin prime constant, is approximately 0:6601618. Using this we can estimate how many numbers we will need to try before we find a prime. In the case of Underbakke and La Barbera, they were both using the same sieving software (NewPGen1 by Paul Jobling) and the same primality proving software (Proth.exe2 by Yves Gallot) on similar hardware{so of course they choose similar ranges to search. But where does this conjecture come from? In this chapter we will discuss a general method to form conjectures similar to the twin prime conjecture above. We will then apply it to a number of different forms of primes such as Sophie Germain primes, primes in arithmetic progressions, primorial primes and even the Goldbach conjecture.
    [Show full text]
  • The First 50 Million Prime Numbers*
    7 sin ~F(s) The First 50 Million I - s ; sin -- Prime Numbers* s Don Zagier for the algebraist it is "the characteristic of a finite field" or "a point in Spec ~" or "a non-archimedean valuation"; To my parents the combinatorist defines the prime num- bers inductively by the recursion (I) 1 Pn+1 = [I - log2( ~ + n (-1)r )] r~1 ISil<'~'<irSn 2 pi1"''pir - I ([x] = biggest integer S x); and, finally, the logicians have recent- iv been defininq the primes as the posi- tive values of the polynomial (2) F(a,b,c,d,e,f,g,h,i,j,k,l,m,n,o,p,q, r,s,t,u,v,w,x,y,z) = [k + 2] [1 - (wz+h+j-q) 2 - (2n+p+q+z-e) 2 - (a2y2-y2+1-x2)2 - ({e4+2e3}{a+1}2+1-o2) 2 - (16{k+1}3{k+2}{n+1}2+1-f2)2 - ({(a+u~-u2a) 2-1}{n+4dy}2+1-{x+cu}2) 2 - (ai+k+1-l-i) 2 - I would like to tell you today about a ({gk+2g+k+1}{h+j}+h-z) 2 - subject which, although I have not worked (16r2y~{a2-1}+1-u2) 2 - in it myself, has always extraordinarily (p-m+l{a-n-1}+b{2an+2a-n2-2n-2}) 2 - captivated me, and which has fascinated (z-pm+pla-p21+t{2ap-p2-1}) 2 - mathematicians from the earliest times (qTx+y{a-p-1}+s{2ap+2a-p2-2~-2})2 - until the present - namely, the question (a~12-12+1-m2) 2 - (n+l+v-y) ].
    [Show full text]
  • Prime Numbers and the (Double) Primorial Sieve
    Prime numbers and the (double) primorial sieve. 1. The quest. This research into the distribution of prime numbers started by placing a prime number on the short leg of a Pythagorean triangle. The properties of the Pythagorean triples imply that prime numbers > p4 (with p4 = 7) are not divisible by pi ∈ {2, 3, 5, 7}. These divisors have a combined repeating pattern based on the primorial p4#, the product of the first four prime numbers. This pattern defines the fourth (double) primorial sieve. The P4#−sieve is the first sieve with all the properties of the infinite set of primorial sieves. It also gives an explanation for the higher occurrence of the (9, 1) last digit gap among prime numbers. The (double) primorial sieve has two main functions. The primorial sieve is a method to generate a consecutive list of prime numbers, since the base of the n-th primorial sieve contains all prime numbers < pn# . The double primorial sieve offers preliminary filtering of natural numbers when they are sequential stacked above the base of the sieve. Possible prime numbers are only found in the columns supported by the φ(pn#) struts. Each strut is relative prime to pn# and is like a support beam under a column. Fig. 1: Examples of imaginary struts to support a specific column. Hans Dicker \ _Primorial_sieve En.doc pag 1 / 11 2. The principal of the (double) primorial sieve. The width of the primorial sieve Pn#−sieve is determined by the primorial pn#, the product of the first n prime numbers. All natural numbers g > pn# form a matrix of infinite height when stacked sequential on top of the base of the sieve (Fig.
    [Show full text]