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FFF. Fibonacci: di Fiore in Fiore

Paulo Ribenboim (2002)

Bollettino dell'Unione Matematica Italiana

In occasione della commemorazione dell’800-esimo anniversario della pubblicazione del Liber Abaci, desidero richiamare l’attenzione del lettore su alcuni dei fatti che preferisco riguardanti numeri di Fibonacci. Tali fatti includono la presenza di quadrati, di multipli di quadrati e di numeri potenti tra i numeri di Fibonacci, la rappresentazione di numeri reali e la costruzione di numeri trascendenti mediante numeri di Fibonacci, la possibilità di costruire una serie zeta ed un dominio a fattorizzazione...

Fibonacci numbers and Fermat's last theorem

Zhi-Wei Sun (1992)

Acta Arithmetica

Let Fₙ be the Fibonacci sequence defined by F₀=0, F₁=1, F n + 1 = F ₙ + F n - 1 ( n ≥ 1 ) . It is well known that F p - ( 5 / p ) ≡ 0 ( m o d p ) for any odd prime p, where (-) denotes the Legendre symbol. In 1960 D. D. Wall [13] asked whether p ² | F p - ( 5 / p ) is always impossible; up to now this is still open. In this paper the sum ∑ k ≡ r ( m o d 10 ) n k is expressed in terms of Fibonacci numbers. As applications we obtain a new formula for the Fibonacci quotient F p - ( 5 / p ) / p and a criterion for the relation p | F ( p - 1 ) / 4 (if p ≡ 1 (mod 4), where p ≠ 5 is an odd prime. We also prove that the affirmative answer to...

Fibonacci Numbers with the Lehmer Property

Florian Luca (2007)

Bulletin of the Polish Academy of Sciences. Mathematics

We show that if m > 1 is a Fibonacci number such that ϕ(m) | m-1, where ϕ is the Euler function, then m is prime

Field of moduli versus field of definition for cyclic covers of the projective line

Aristides Kontogeorgis (2009)

Journal de Théorie des Nombres de Bordeaux

We give a criterion, based on the automorphism group, for certain cyclic covers of the projective line to be defined over their field of moduli. An example of a cyclic cover of the complex projective line with field of moduli ℝ that can not be defined over ℝ is also given.

Fields of definition of ℚ -curves

Jordi Quer (2001)

Journal de théorie des nombres de Bordeaux

Let C be a ℚ -curve with no complex multiplication. In this note we characterize the number fields K such that there is a curve C ' isogenous to C having all the isogenies between its Galois conjugates defined over K , and also the curves C ' isogenous to C defined over a number field K such that the abelian variety Res K / ℚ ( C ' / K ) obtained by restriction of scalars is a product of abelian varieties of GL 2 -type.

Fields of moduli of three-point G -covers with cyclic p -Sylow, II

Andrew Obus (2013)

Journal de Théorie des Nombres de Bordeaux

We continue the examination of the stable reduction and fields of moduli of G -Galois covers of the projective line over a complete discrete valuation field of mixed characteristic ( 0 , p ) , where G has a cyclic p -Sylow subgroup P of order p n . Suppose further that the normalizer of P acts on P via an involution. Under mild assumptions, if f : Y → ℙ 1 is a three-point G -Galois cover defined over ℚ ¯ , then the n th higher ramification groups above p for the upper numbering of the (Galois closure of the) extension K / ℚ vanish,...

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