Karl-Heinz Kuhl PRIME NUMBERS– THINGS LONG-KNOWN AND THINGS NEW- FOUND A JOURNEY THROUGH THE LANDSCAPE OF THE PRIME NUMBERS Amazing properties and insights – not from the perspective of a mathematician, but from that of a voyager who, pausing here and there in the landscape of the prime numbers, approaches their secrets in a spirit of playful adventure, eager to experiment and share their fascination with others who may be interested. Third, revised and updated edition (2020) 0 Prime Numbers – things long- known and things new-found A journey through the landscape of the prime numbers Amazing properties and insights – not from the perspective of a mathematician, but from that of a voyager who, pausing here and there in the landscape of the prime numbers, approaches their secrets in a spirit of playful adventure, eager to experiment and share their fascination with others who may be interested. Dipl.-Phys. Karl-Heinz Kuhl Parkstein, December 2020 1 1 + 2 + 3 + 4 + ⋯ = − 12 (Ramanujan) Web: https://yapps-arrgh.de (Yet another promising prime number source: amazing recent results from a guerrilla hobbyist) Link to the latest online version https://yapps-arrgh.de/primes_Online.pdf Some of the text and Mathematica programs have been removed from the free online version. The printed and e-book versions, however, contain both the text and the programs in their entirety. Recent supple- ments to the book can be found here: https://yapps-arrgh.de/data/Primenumbers_supplement.pdf Please feel free to contact the author if you would like a deeper insight into the many Mathematica programs. Contact: [email protected] 1 For Michèle ISBN 978-3-939247-93-7 Publishing house: Eckhard Bodner, Pressath, Germany - 2017 Third, revised edition – December 2020 Translation: Ewan Whyte The illustration on the title page shows the graphic from Figure 82, Chapter 9.2. Cover design: Karl-Heinz Kuhl Copyright: this work and all embedded illustrations and computer programs are copyright protected. Any commercial use that has not been expressly authorized by the author is prohibited. The new algorithms and methods described in this book are protected by notarization (with an indication of the date). The contents of this book (and of the free online version available for download) including all related files may be used, distributed, published on the Internet and referred to by readers in their own publications in each case for private and non-commercial purposes only and provided all contents are quoted correctly, in full and in an unaltered form, accompanied by a description of the book, the name of the author and a link to the website above. This applies to all texts, graphics and computer programs as well as other files. Citations in particular from passages printed in blue should be accompanied by an indication that the material in question is considered ‘new’. Liability: the author is not responsible for damages of any kind that may result from use of the computer program listings (whether in the Appendix, on the accompanying CD or in the body of the text). Furthermore, the author gives no warranty that all programs are free from errors or that they will run in all operating system environments. 2 1 Table of Contents 2 Introduction ................................................................................................................................................. 8 2.1 Mathematical notation used in this book .............................................................................. 10 3 Basics of prime numbers ...................................................................................................................... 14 3.1 Quick start: what do we know for certain? .......................................................................... 16 3.2 Quick start: what are our (unproven) conjectures? ......................................................... 17 3.3 Quick start: what is still unsolved? .......................................................................................... 18 3.4 Quick start: what is new? ............................................................................................................. 19 4 Special kinds of prime numbers ........................................................................................................ 20 4.1 Twin primes ...................................................................................................................................... 20 4.2 Prime triplets and quadruplets ................................................................................................. 23 4.3 Prime n-tuplets ............................................................................................................................... 25 4.4 Correlations of the last digits in the prime number sequence ..................................... 32 4.5 Mersenne prime numbers ........................................................................................................... 34 4.5.1 GIMPS – the Great Internet Mersenne Prime Search ................................................ 39 4.6 Fermat prime numbers................................................................................................................. 40 4.7 Lucky primes ..................................................................................................................................... 42 4.8 Perfect numbers .............................................................................................................................. 44 4.8.1 General issues and definition ............................................................................................. 44 4.8.2 Properties ................................................................................................................................... 45 4.9 Sophie Germain prime numbers ............................................................................................... 47 4.9.1 Computation and properties .............................................................................................. 48 4.10 Fibonacci numbers and other recursive sequences ......................................................... 49 4.10.1 Linear recursion: a mighty instrument .......................................................................... 52 4.10.2 Fibonacci prime and pseudoprime numbers ............................................................... 61 4.10.3 Meta-Fibonacci sequences ................................................................................................... 63 4.11 Carmichael and Knödel numbers ............................................................................................. 64 4.12 Emirp numbers ................................................................................................................................ 65 4.13 Wagstaff prime numbers ............................................................................................................. 65 4.14 Wieferich prime numbers ........................................................................................................... 67 4.15 Wilson prime numbers ................................................................................................................. 70 4.16 Wolstenholme prime numbers.................................................................................................. 71 4.17 RG numbers (= recursive Gödelized) ...................................................................................... 72 4.17.1 GOCRON type 6 (‘prime OCRONs‘) ................................................................................... 72 3 4.17.2 GOCRON type 4 (with the symbols ‚‘2’,‘*’,‘P’ and ‘^’) ................................................ 76 5 Digression: Riemann’s zeta function 휁(푠) ..................................................................................... 79 5.1 General ................................................................................................................................................ 79 5.2 The different representations of 휁(푠) ..................................................................................... 85 5.3 Product representation of 휁(푠) in the complex domain ................................................. 87 5.4 An unexpected product representation of a slightly different 휁(푠) ........................... 93 5.5 A counting function for the number of zeros ...................................................................... 96 5.6 The zeta function and quantum chaos: a gangway to physics...................................... 99 6 Digression: the Riemann function 푅(푠) ...................................................................................... 103 7 A few important arithmetical functions ...................................................................................... 104 7.1 Omega functions: number of prime factors ...................................................................... 104 7.2 The Liouville function ................................................................................................................ 106 7.3 The Chebyshev function ............................................................................................................ 108 7.4 The Euler phi function (totient function) ........................................................................... 111 7.4.1 Calculation and graphic representation of the phi function ............................... 111 7.4.2 Properties of