Short remark on Fibonacci-Wieferich primes

Jiří Klaška

Acta Mathematica Universitatis Ostraviensis (2007)

  • Volume: 15, Issue: 1, page 21-25
  • ISSN: 1804-1388

Abstract

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This paper has been inspired by the endeavour of a large number of mathematicians to discover a Fibonacci-Wieferich prime. An exhaustive computer search has not been successful up to the present even though there exists a conjecture that there are infinitely many such primes. This conjecture is based on the assumption that the probability that a prime p is Fibonacci-Wieferich is equal to 1 / p . According to our computational results and some theoretical consideratons, another form of probability can be assumed. This observation leads us to interesting consequences.

How to cite

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Klaška, Jiří. "Short remark on Fibonacci-Wieferich primes." Acta Mathematica Universitatis Ostraviensis 15.1 (2007): 21-25. <http://eudml.org/doc/35169>.

@article{Klaška2007,
abstract = {This paper has been inspired by the endeavour of a large number of mathematicians to discover a Fibonacci-Wieferich prime. An exhaustive computer search has not been successful up to the present even though there exists a conjecture that there are infinitely many such primes. This conjecture is based on the assumption that the probability that a prime $p$ is Fibonacci-Wieferich is equal to $1/p$. According to our computational results and some theoretical consideratons, another form of probability can be assumed. This observation leads us to interesting consequences.},
author = {Klaška, Jiří},
journal = {Acta Mathematica Universitatis Ostraviensis},
keywords = {Fibonacci-Wieferich primes; heuristics on distributions of primes with arithmetic constraints; Fibonacci numbers; Wall-Sun-Sun prime; modular periodicity; periodic sequence; Fibonacci-Wieferich primes; heuristics on distributions of primes with arithmetic constraints},
language = {eng},
number = {1},
pages = {21-25},
publisher = {University of Ostrava},
title = {Short remark on Fibonacci-Wieferich primes},
url = {http://eudml.org/doc/35169},
volume = {15},
year = {2007},
}

TY - JOUR
AU - Klaška, Jiří
TI - Short remark on Fibonacci-Wieferich primes
JO - Acta Mathematica Universitatis Ostraviensis
PY - 2007
PB - University of Ostrava
VL - 15
IS - 1
SP - 21
EP - 25
AB - This paper has been inspired by the endeavour of a large number of mathematicians to discover a Fibonacci-Wieferich prime. An exhaustive computer search has not been successful up to the present even though there exists a conjecture that there are infinitely many such primes. This conjecture is based on the assumption that the probability that a prime $p$ is Fibonacci-Wieferich is equal to $1/p$. According to our computational results and some theoretical consideratons, another form of probability can be assumed. This observation leads us to interesting consequences.
LA - eng
KW - Fibonacci-Wieferich primes; heuristics on distributions of primes with arithmetic constraints; Fibonacci numbers; Wall-Sun-Sun prime; modular periodicity; periodic sequence; Fibonacci-Wieferich primes; heuristics on distributions of primes with arithmetic constraints
UR - http://eudml.org/doc/35169
ER -

References

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  1. R. Crandall K. Dilcher C. Pomerance, A search for Wieferich and Wilson primes, Math. Comp. 66 (1997) 443-449. (1997) MR1372002
  2. H. Davenport, Multiplicative Number Theory, Springer-Verlag New York 3rd ed. (2000). Zbl1002.11001MR1790423
  3. A.-S. Elsenhans J. Jahnel, The Fibonacci sequence modulo p2 - An investigation by computer for p < 10**14, The On-Line Encyclopedia of Integer Sequences (2004) 27 p. 
  4. Hua-Chieh Li, Fibonacci primitive roots and Wall's question, The Fibonacci Quarterly 37 (1999) 77-84. (1999) Zbl0936.11011MR1676707
  5. J. Klaka, Criteria for Testing Wall's Question, preprint (2007). 
  6. R. J. Mcintosh E. L. Roettger, 10.1090/S0025-5718-07-01955-2, Math. Comp. 76 (2007) 2087-2094. MR2336284DOI10.1090/S0025-5718-07-01955-2
  7. L. Skula, A note on some relations among special sums of reciprocals modulo p, to appear in Math. Slovaca (2008). Zbl1164.11001MR2372821
  8. Zhi-Hong Sun, Zhi-Wei Sun, Fibonacci Numbers and Fermat's Last Theorem, Acta Arith. 60 (1992) 371-388. (1992) MR1159353
  9. D. D. Wall, 10.2307/2309169, Amer. Math. Monthly 67 no. 6, (1960) 525-532. (1960) Zbl0101.03201MR0120188DOI10.2307/2309169
  10. H. C Williams, 10.4153/CMB-1982-053-0, Canad. Math. Bull. 25 (1982) 366-370. (1982) MR0668957DOI10.4153/CMB-1982-053-0

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