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Short remark on Fibonacci-Wieferich primes. (English) Zbl 1203.11021
A Wieferich prime is a prime $$p$$ such that $$2^{p-1}\equiv 1\pmod {p^2}$$. By analogy, a Fibonacci–Wieferich prime is a prime $$p$$ such that $$F_{p-(p|5)}\equiv 0\pmod {p^2}$$, where $$(p|5)$$ is the Legendre symbol of $$p$$ with respect to $$5$$. Only two Wieferich primes are known and no Fibonacci-Wieferich prime is known although it is known that there is no such prime $$p<2\times 10^{14}$$. A side remark in the last section of the paper [C. Crandall, K. Dilcher and C. Pomerance, “A search for Wieferich and Wilson primes,” Math. Comput. 66, No. 217, 433–449 (1997; Zbl 0854.11002)] suggests that there should be roughly $$\log(\log y/\log x)$$ Fibonacci-Wieferich primes $$p$$ in the interval $$[x,y]$$. The prediction is based on the fact that the integer $$F_{p-(p|5)}/p$$ should be uniformly distributed modulo $$p$$, so it should land in the congruence class $$0$$ modulo $$p$$ about $$1/p$$ of the times. Now one sums $$1/p$$ for $$p\in [x,y]$$ and invokes Mertens’ formula to get the prediction. In the paper under review, the author looks at the factorization of $$p$$ in $${\mathbb K}={\mathbb Q}[{\sqrt{5}}]$$ and notes that the residual ring $${\mathcal O}_{\mathbb K}/p^2{\mathcal O}_{\mathbb K}$$ has $$q-1$$ elements $$\zeta$$ of exponent $$q-1$$, where $$q=p$$, or $$p^2$$ according to whether $$(p|5)=1$$, or $$(p|5)=-1$$, respectively. He then deduces (Remark 2.4) that perhaps it is less likely for a prime $$p\equiv \pm 2\pmod 5$$ to be Fibonacci–Wieferich than for a prime $$p\equiv \pm 1\pmod 5$$. His arguments seem to suggest that there should be only finitely many (or none) Fibonacci–Wieferich primes which are congruent to $$\pm 2\pmod 5$$, so that there should be only about $$(1/2)\log(\log y/\log x)$$ Fibonacci–Wieferich primes $$p$$ in $$[x,y]$$, all but finitely many of them being primes which are $$\pm 1\pmod 5$$.
Most of the algebraic number theory in $${\mathbb Q}({\sqrt{5}})$$ appearing in the paper is cited from [Czech. Math. J. 58, No. 4, 1241–1246 (2008; Zbl 1174.11020)] (unpublished at the time when the paper under review appeared). More accessible references for these are classical paper on primes dividing the Lucas numbers by Lagarias (see the arguments from the proof of Theorem B in [J. C. Lagarias, ”The set of primes dividing the Lucas numbers has density $$2/3$$,” Pac. J. Math. 118, 449–461 (1985; Zbl 0569.10003), errata, Pac. J. Math. 162, No. 2, 393–396 (1994; Zbl 0790.11014)] and subsequent works on the topic for more general Lucas sequences most notably by Ballot and by Moree and co-authors.

MSC:
 11B39 Fibonacci and Lucas numbers and polynomials and generalizations 11A07 Congruences; primitive roots; residue systems
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References:
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