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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; Fibonacci numbers; Wall-Sun-Sun prime; modular periodicity; periodic sequence
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.
References:
[1] R. Crandall K. Dilcher C. Pomerance: A search for Wieferich and Wilson primes. Math. Comp. 66 (1997) 443-449. MR 1372002
[2] H. Davenport: Multiplicative Number Theory. Springer-Verlag New York 3rd ed. (2000). MR 1790423 | Zbl 1002.11001
[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. MR 1676707 | Zbl 0936.11011
[5] J. Klaka: Criteria for Testing Wall's Question. preprint (2007).
[6] R. J. Mcintosh E. L. Roettger: A search for Fibonacci-Wieferich and Wolstenholme primes. Math. Comp. 76 (2007) 2087-2094. DOI 10.1090/S0025-5718-07-01955-2 | MR 2336284
[7] L. Skula: A note on some relations among special sums of reciprocals modulo p. to appear in Math. Slovaca (2008). MR 2372821 | Zbl 1164.11001
[8] Zhi-Hong Sun, Zhi-Wei Sun: Fibonacci Numbers and Fermat's Last Theorem. Acta Arith. 60 (1992) 371-388. MR 1159353
[9] D. D. Wall: Fibonacci Series Modulo m. Amer. Math. Monthly 67 no. 6, (1960) 525-532. DOI 10.2307/2309169 | MR 0120188 | Zbl 0101.03201
[10] H. C Williams: A Note on the Fibonacci Quotient Fp-e/p. Canad. Math. Bull. 25 (1982) 366-370. DOI 10.4153/CMB-1982-053-0 | MR 0668957
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