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On the Prime Number Subset of the Fibonacci Numbers
Introduction to Sieves Fibonacci Numbers Fibonacci sequence mod a prime Probability of Fibonacci Primes Matrix Approach On the Prime Number Subset of the Fibonacci Numbers Lacey Fish1 Brandon Reid2 Argen West3 1Department of Mathematics Louisiana State University Baton Rouge, LA 2Department of Mathematics University of Alabama Tuscaloosa, AL 3Department of Mathematics University of Louisiana Lafayette Lafayette, LA Lacey Fish, Brandon Reid, Argen West SMILE Presentations, 2010 LSU, UA, ULL Sieve Theory Introduction to Sieves Fibonacci Numbers Fibonacci sequence mod a prime Probability of Fibonacci Primes Matrix Approach Basic Definitions What is a sieve? What is a sieve? A sieve is a method to count or estimate the size of “sifted sets” of integers. Well, what is a sifted set? A sifted set is made of the remaining numbers after filtering. Lacey Fish, Brandon Reid, Argen West LSU, UA, ULL Sieve Theory Introduction to Sieves Fibonacci Numbers Fibonacci sequence mod a prime Probability of Fibonacci Primes Matrix Approach Basic Definitions What is a sieve? What is a sieve? A sieve is a method to count or estimate the size of “sifted sets” of integers. Well, what is a sifted set? A sifted set is made of the remaining numbers after filtering. Lacey Fish, Brandon Reid, Argen West LSU, UA, ULL Sieve Theory Introduction to Sieves Fibonacci Numbers Fibonacci sequence mod a prime Probability of Fibonacci Primes Matrix Approach Basic Definitions History Two Famous and Useful Sieves Sieve of Eratosthenes Brun’s Sieve Lacey Fish, Brandon Reid, Argen West -
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. -
There Are Infinitely Many Mersenne Primes
Scan to know paper details and author's profile There are Infinitely Many Mersenne Primes Fengsui Liu ZheJiang University Alumnus ABSTRACT From the entire set of natural numbers successively deleting the residue class 0 mod a prime, we retain this prime and possibly delete another one prime retained. Then we invent a recursive sieve method for exponents of Mersenne primes. This is a novel algorithm on sets of natural numbers. The algorithm mechanically yields a sequence of sets of exponents of almost Mersenne primes, which converge to the set of exponents of all Mersenne primes. The corresponding cardinal sequence is strictly increasing. We capture a particular order topological structure of the set of exponents of all Mersenne primes. The existing theory of this structure allows us to prove that the set of exponents of all Mersenne primes is an infnite set . Keywords and phrases: Mersenne prime conjecture, modulo algorithm, recursive sieve method, limit of sequence of sets. Classification: FOR Code: 010299 Language: English LJP Copyright ID: 925622 Print ISSN: 2631-8490 Online ISSN: 2631-8504 London Journal of Research in Science: Natural and Formal 465U Volume 20 | Issue 3 | Compilation 1.0 © 2020. Fengsui Liu. This is a research/review paper, distributed under the terms of the Creative Commons Attribution-Noncom-mercial 4.0 Unported License http://creativecommons.org/licenses/by-nc/4.0/), permitting all noncommercial use, distribution, and reproduction in any medium, provided the original work is properly cited. There are Infinitely Many Mersenne Primes Fengsui Liu ____________________________________________ ABSTRACT From the entire set of natural numbers successively deleting the residue class 0 mod a prime, we retain this prime and possibly delete another one prime retained. -
Integer Sequences
UHX6PF65ITVK Book > Integer sequences Integer sequences Filesize: 5.04 MB Reviews A very wonderful book with lucid and perfect answers. It is probably the most incredible book i have study. Its been designed in an exceptionally simple way and is particularly just after i finished reading through this publication by which in fact transformed me, alter the way in my opinion. (Macey Schneider) DISCLAIMER | DMCA 4VUBA9SJ1UP6 PDF > Integer sequences INTEGER SEQUENCES Reference Series Books LLC Dez 2011, 2011. Taschenbuch. Book Condition: Neu. 247x192x7 mm. This item is printed on demand - Print on Demand Neuware - Source: Wikipedia. Pages: 141. Chapters: Prime number, Factorial, Binomial coeicient, Perfect number, Carmichael number, Integer sequence, Mersenne prime, Bernoulli number, Euler numbers, Fermat number, Square-free integer, Amicable number, Stirling number, Partition, Lah number, Super-Poulet number, Arithmetic progression, Derangement, Composite number, On-Line Encyclopedia of Integer Sequences, Catalan number, Pell number, Power of two, Sylvester's sequence, Regular number, Polite number, Ménage problem, Greedy algorithm for Egyptian fractions, Practical number, Bell number, Dedekind number, Hofstadter sequence, Beatty sequence, Hyperperfect number, Elliptic divisibility sequence, Powerful number, Znám's problem, Eulerian number, Singly and doubly even, Highly composite number, Strict weak ordering, Calkin Wilf tree, Lucas sequence, Padovan sequence, Triangular number, Squared triangular number, Figurate number, Cube, Square triangular -
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. -
There Are Infinitely Many Fibonacci Primes
Global Journal of Science Frontier Research: F Mathematics & Decision Science Volume 20 Issue 5 Version 1.0 Year 2020 Type : Double Blind Peer Reviewed International Research Journal Publisher: Global Journals Online ISSN: 2249-4626 & Print ISSN: 0975-5896 There are Infinitely Many Fibonacci Primes By Fengsui Liu Zhe Jiang University Abstract- We invent a novel algorithm and solve the Fibonacci prime conjecture by an interaction between proof and algorithm. From the entire set of natural numbers successively deleting the residue class 0 mod a prime, we retain this prime and possibly delete another one prime retained, then we invent a recursive sieve method, a modulo algorithm on finite sets of natural numbers, for indices of Fibonacci primes. The sifting process mechanically yields a sequence of sets of natural numbers, which converges to the index set of all Fibonacci primes. The corresponding cardinal sequence is strictly increasing. The algorithm reveals a structure of particular order topology of the index set of all Fibonacci primes, then we readily prove that the index set of all Fibonacci primes is an infinite set based on the existing theory of the structure. Some mysteries of primes are hidden in second order arithmetics. Keywords and phrases: Fibonacci prime conjecture, recursive sieve method, order topology, limit of a sequence of sets. GJSFR-F Classification: MSC 2010: 11N35, 11N32, 11U09, 11Y16, 11B37, 11B50 Ther eareInfinitelyManyFibonacciPrimes Strictly as per the compliance and regulations of: © 2020. Fengsui Liu. This is a research/review paper, distributed under the terms of the Creative Commons Attribution- Noncommercial 3.0 Unported License http://creativecommons.org/licenses/by-nc/3.0/), permitting all non commercial use, distribution, and reproduction in any medium, provided the original work is properly cited. -
New Congruences and Finite Difference Equations For
New Congruences and Finite Difference Equations for Generalized Factorial Functions Maxie D. Schmidt University of Washington Department of Mathematics Padelford Hall Seattle, WA 98195 USA [email protected] Abstract th We use the rationality of the generalized h convergent functions, Convh(α, R; z), to the infinite J-fraction expansions enumerating the generalized factorial product se- quences, pn(α, R)= R(R + α) · · · (R + (n − 1)α), defined in the references to construct new congruences and h-order finite difference equations for generalized factorial func- tions modulo hαt for any primes or odd integers h ≥ 2 and integers 0 ≤ t ≤ h. Special cases of the results we consider within the article include applications to new congru- ences and exact formulas for the α-factorial functions, n!(α). Applications of the new results we consider within the article include new finite sums for the α-factorial func- tions, restatements of classical necessary and sufficient conditions of the primality of special integer subsequences and tuples, and new finite sums for the single and double factorial functions modulo integers h ≥ 2. 1 Notation and other conventions in the article 1.1 Notation and special sequences arXiv:1701.04741v1 [math.CO] 17 Jan 2017 Most of the conventions in the article are consistent with the notation employed within the Concrete Mathematics reference, and the conventions defined in the introduction to the first articles [11, 12]. These conventions include the following particular notational variants: ◮ Extraction of formal power series coefficients. The special notation for formal n k power series coefficient extraction, [z ] k fkz :7→ fn; ◮ Iverson’s convention. -
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. -
P ± 1 Table 1. Factorial and Primorial Primes
mathematics of computation volume 64,number 210 april 1995,pages 889-890 ON THE PRIMALITY OF «! ± 1 AND 2 • 3 • 5 -p ± 1 CHRIS K. CALDWELL Abstract. For each prime p let p# be the product of the primes less than or equal to p . Using a new type of microcomputer coprocessor, we have found five new primes of the form n\ - 1 , two new primes of the form p# + 1 , seven new primes of the form p# - 1 , and greatly extended the search bounds for primes of the form n\ ± 1 and p# ± 1 . For each prime p let p# be the product of the primes less than or equal to p . Since Euclid's proof of the infinitude of primes, the numbers «! ± 1 and /?# ± 1 have been frequently checked for primality, most recently in [ 1, 2, 4, 5, and 7] using the classic TV2- 1 primality tests of [3]. Using a new type of microcomputer coprocessor board designed for doing huge integer arithmetic, we have greatly expanded the range over which these numbers have been tested and found the fourteen new primes marked by an asterisk in Table 1. Table 1. Factorial and primorial primes form values for which the form is prime search limit n!+l 1, 2, 3, 11, 27, 37, 41, 73, 77, 116, 154, 320, 340, 399, 427, 4580 872 and 1477 (4042 digits)_ «!- 1 3, 4, 6, 7, 12, 14, 30, 32, 33, 38, 94, 166, 324, 379 and 469, 4580 546*, 974*, 1963*,3507* and 3610* (11277 digits) />#+! 2, 3, 5, 7, 11, 31, 379, 1019, 1021, 2657, 3229, 4547, 4787, 35000 11549, 13649, 18523, 23801* and 24029* (10387 digits) p#- 1 3, 5, 11, 41, 89, 317, 337, 991, 1873, 2053, 2377*,4093*, 35000 4297*, 4583*, 6569*, 13033*and 15877*(6845 digits) *Newly shown prime. -
Fibonacci's Answer to Primality Testing?
FIBONACCI'S ANSWER TO PRIMALITY TESTING? JULIAN BEAUCHAMP Abstract. In this paper, we consider various approaches to primality testing and then ask whether an effective deterministic test for prime numbers can be found in the Fibonacci numbers. Introduction Prime numbers play an important role in encryption and cyber-security. They protect digitial data, including personal data and bank details. Testing whether a number is prime or not, is therefore becoming increasingly important for mathe- maticians and for those working in digitial security. The study of prime numbers and their properties go back to the ancient Greek mathematician Pythagoras (570-495 BC) who understood the idea of primality. But a primality test is a test to determine whether or not a number is prime. This is different from finding a number's constituent prime factors (also known as prime factorization). A number is said to be prime if it is divisible only by 1 and itself. Otherwise it is composite. When the numbers are small, it is relatively easy to determine whether a number is prime. But as they get exponentially larger they get harder to determine. So the pressing question is what makes an efficient algorithm? The following characteristics make an efficient algorithm - general, deterministic, unconditional, and polynomial:1 General. An algorithm that is general works for all numbers. Algorithms that are not general only work on certain numbers (e.g. the Lucas-Lehmer test for Mersenne numbers). Deterministic.A deterministic test (e.g. the Lucas-Lehmer Test) will tell us with absolute certainty whether a number is prime or not every time it is run. -
Elementary Number Theory and Its Applications
Elementary Number Theory andlts Applications KennethH. Rosen AT&T Informotion SystemsLaboratories (formerly part of Bell Laborotories) A YY ADDISON-WESLEY PUBLISHING COMPANY Read ing, Massachusetts Menlo Park, California London Amsterdam Don Mills, Ontario Sydney Cover: The iteration of the transformation n/2 if n T(n) : \ is even l Qn + l)/2 if n is odd is depicted.The Collatz conjectureasserts that with any starting point, the iteration of ?"eventuallyreaches the integer one. (SeeProblem 33 of Section l.2of the text.) Library of Congress Cataloging in Publication Data Rosen, Kenneth H. Elementary number theory and its applications. Bibliography: p. Includes index. l. Numbers, Theory of. I. Title. QA24l.R67 1984 512',.72 83-l1804 rsBN 0-201-06561-4 Reprinted with corrections, June | 986 Copyright O 1984 by Bell Telephone Laboratories and Kenneth H. Rosen. All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical,photocopying, recording, or otherwise,without prior written permission of the publisher. printed in the United States of America. Published simultaneously in Canada. DEFGHIJ_MA_8987 Preface Number theory has long been a favorite subject for studentsand teachersof mathematics. It is a classical subject and has a reputation for being the "purest" part of mathematics, yet recent developments in cryptology and computer science are based on elementary number theory. This book is the first text to integrate these important applications of elementary number theory with the traditional topics covered in an introductory number theory course. This book is suitable as a text in an undergraduatenumber theory courseat any level. -
Numbers 1 to 100
Numbers 1 to 100 PDF generated using the open source mwlib toolkit. See http://code.pediapress.com/ for more information. PDF generated at: Tue, 30 Nov 2010 02:36:24 UTC Contents Articles −1 (number) 1 0 (number) 3 1 (number) 12 2 (number) 17 3 (number) 23 4 (number) 32 5 (number) 42 6 (number) 50 7 (number) 58 8 (number) 73 9 (number) 77 10 (number) 82 11 (number) 88 12 (number) 94 13 (number) 102 14 (number) 107 15 (number) 111 16 (number) 114 17 (number) 118 18 (number) 124 19 (number) 127 20 (number) 132 21 (number) 136 22 (number) 140 23 (number) 144 24 (number) 148 25 (number) 152 26 (number) 155 27 (number) 158 28 (number) 162 29 (number) 165 30 (number) 168 31 (number) 172 32 (number) 175 33 (number) 179 34 (number) 182 35 (number) 185 36 (number) 188 37 (number) 191 38 (number) 193 39 (number) 196 40 (number) 199 41 (number) 204 42 (number) 207 43 (number) 214 44 (number) 217 45 (number) 220 46 (number) 222 47 (number) 225 48 (number) 229 49 (number) 232 50 (number) 235 51 (number) 238 52 (number) 241 53 (number) 243 54 (number) 246 55 (number) 248 56 (number) 251 57 (number) 255 58 (number) 258 59 (number) 260 60 (number) 263 61 (number) 267 62 (number) 270 63 (number) 272 64 (number) 274 66 (number) 277 67 (number) 280 68 (number) 282 69 (number) 284 70 (number) 286 71 (number) 289 72 (number) 292 73 (number) 296 74 (number) 298 75 (number) 301 77 (number) 302 78 (number) 305 79 (number) 307 80 (number) 309 81 (number) 311 82 (number) 313 83 (number) 315 84 (number) 318 85 (number) 320 86 (number) 323 87 (number) 326 88 (number)