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Proof by Cases Article

Proof by Cases in Formal Logic
Front cover image for Proof by Cases in Formal Logic
Format:
Article
Publication year:
1934
 
|
Peer-reviewed
Journal:
  • Annals of Mathematics v35 n3 (19340701): 529-544
Held by Saint Leo University Cannon Memorial Library
View Description
Staff View:
MARC Record
ISSN:
  • 0003-486X
Unique Identifier:
5545251769
Subjects:
Database:
Arts & Sciences I

Prime Numbers and Cryptography

ebooks

Articles - Web of Science

An infinite primality conjecture for prime-subscripted Fibonacci numbers

By:Leyendekkers, JV (Leyendekkers, J. V.)[ 1 ] ; Shannon, AG (Shannon, A. G.)[ 2,3 ]

NOTES ON NUMBER THEORY AND DISCRETE MATHEMATICS

Volume: 21

Issue: 1

Pages: 51-55

Published: 2015

Document Type:Article

Abstract

The row structures of the prime-subscripted Fibonacci numbers in the modular ring Z(4) show distinction between primes and composites. The class structure of the Fibonacci numbers suggest that these row structures must survive to infinity and hence that Fibonacci primes must too. The functions F-p = K-p +/- 1 and Fp (factors) = k(p) +/- 1 support the structural evidence. The graph of (K/k) versus p displays a Raman-spectra form persisting to infinity: ln(K/k) is linear in p in the composite case while primes lie along the p-axis to infinity.

Author Information

Reprint Address: Leyendekkers, JV (reprint author)


Univ Sydney, Fac Sci, Sydney, NSW 2006, Australia.

 

Addresses:


[ 1 ] Univ Sydney, Fac Sci, Sydney, NSW 2006, Australia

[ 2 ] Univ Technol Sydney, Fac Engn & IT, Sydney, NSW 2007, Australia
       [ 3 ] Campion Coll, Toongabbie East, NSW 2146, Australia

 

E-mail Addresses:t.shannon@warrane.unsw.edu.au; Anthony.Shannon@uts.edu.au

Publisher

BULGARIAN ACAD SCIENCE, CENTRAL LIBRARY 7 NOEMVRI NO 1, SOFIA, 00000, BULGARIA

Categories / Classification

Research Areas:Mathematics

Web of Science Categories:Mathematics

Articles - Science Direct

 

Research paper

Multidimensional scaling and visualization of patterns in prime numbers

 
 
 

Highlights

 

We explore prime numbers as objects in a complex systems.

The dissimilarity information between objects is measured by different metrics.

The data are processed by multidimensional scaling algorithms.

The multidimensional scaling loci are explored using the tools of fractional calculus.

 

Abstract

This paper explores the prime numbers (PN) in the perspective of complex systems (CS) using computational and information visualization resources. The PN are interpreted as features that characterize the outputs of a CS. Four distinct metrics are adopted to assess the differences between such objects, namely the Canberra, Euclidean, Jaccard and Lorentzian distances, and the information is treated with a multidimensional scaling (MDS) algorithm. The MDS produces loci, organized according with the objects’ features, that are analyzed under the light of the emerging patterns. Additionally, these patterns are explored in the Fourier domain under the point of view of fractional calculus. The representations constitute a new philosophy for tackling the challenging topic of PN using advanced scientific visualization.

Keywords

Complex systems
Prime numbers
Information visualization
Distance metrics
Multidimensional scaling