An infinite primality conjecture for prime-subscripted Fibonacci numbers

J. V. Leyendekkers and A. G. Shannon
Notes on Number Theory and Discrete Mathematics, ISSN 1310–5132
Volume 21, 2015, Number 1, Pages 51–55
Full paper (PDF, 129 Kb)

Details

Authors and affiliations

J. V. Leyendekkers
Faculty of Science, The University of Sydney
NSW 2006, Australia

A. G. Shannon 
Faculty of Engineering & IT, University of Technology
Sydney, NSW 2007, Australia, &
Campion College
PO Box 3052, Toongabbie East, NSW 2146, Australia

Abstract

The row structures of the prime-subscripted Fibonacci numbers in the modular ring Z₄ 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 F_p (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.

Keywords

• Fibonacci numbers
• Prime numbers
• Composite numbers
• Modular rings
• Raman spectra

AMS Classification

• 11B39
• 11B50

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