| Title:
|
Short remark on Fibonacci-Wieferich primes (English) |
| Author:
|
Klaška, Jiří |
| Language:
|
English |
| Journal:
|
Acta Mathematica Universitatis Ostraviensis |
| ISSN:
|
1214-8148 |
| Volume:
|
15 |
| Issue:
|
1 |
| Year:
|
2007 |
| Pages:
|
21-25 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
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. (English) |
| Keyword:
|
Fibonacci-Wieferich primes |
| Keyword:
|
heuristics on distributions of primes with arithmetic constraints |
| Keyword:
|
Fibonacci numbers |
| Keyword:
|
Wall-Sun-Sun prime |
| Keyword:
|
modular periodicity |
| Keyword:
|
periodic sequence |
| MSC:
|
11A07 |
| MSC:
|
11B39 |
| MSC:
|
11B50 |
| MSC:
|
11Y99 |
| idZBL:
|
Zbl 1203.11021 |
| idMR:
|
MR2418779 |
| . |
| Date available:
|
2009-12-29T09:22:25Z |
| Last updated:
|
2015-03-15 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/137492 |
| . |
| Reference:
|
[1] R. Crandall K. Dilcher C. Pomerance: A search for Wieferich and Wilson primes.Math. Comp. 66 (1997) 443-449. MR 1372002 |
| Reference:
|
[2] H. Davenport: Multiplicative Number Theory.Springer-Verlag New York 3rd ed. (2000). Zbl 1002.11001, MR 1790423 |
| Reference:
|
[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. |
| Reference:
|
[4] Hua-Chieh Li: Fibonacci primitive roots and Wall's question.The Fibonacci Quarterly 37 (1999) 77-84. Zbl 0936.11011, MR 1676707 |
| Reference:
|
[5] J. Klaka: Criteria for Testing Wall's Question.preprint (2007). |
| Reference:
|
[6] R. J. Mcintosh E. L. Roettger: A search for Fibonacci-Wieferich and Wolstenholme primes.Math. Comp. 76 (2007) 2087-2094. MR 2336284, 10.1090/S0025-5718-07-01955-2 |
| Reference:
|
[7] L. Skula: A note on some relations among special sums of reciprocals modulo p.to appear in Math. Slovaca (2008). Zbl 1164.11001, MR 2372821 |
| Reference:
|
[8] Zhi-Hong Sun, Zhi-Wei Sun: Fibonacci Numbers and Fermat's Last Theorem.Acta Arith. 60 (1992) 371-388. MR 1159353 |
| Reference:
|
[9] D. D. Wall: Fibonacci Series Modulo m.Amer. Math. Monthly 67 no. 6, (1960) 525-532. Zbl 0101.03201, MR 0120188, 10.2307/2309169 |
| Reference:
|
[10] H. C Williams: A Note on the Fibonacci Quotient Fp-e/p.Canad. Math. Bull. 25 (1982) 366-370. MR 0668957, 10.4153/CMB-1982-053-0 |
| . |
