Divisibility properties of q-Fibonacci numbers Johann ...
Recommend Documents
q q q q. q q q. q q q. +. + +. + + + +. + + + +. + +.. For. 1 q â we obviously get (1) ... p. F q. + is divisible by [ ]q p if p is any prime with. 2(mod 5). p ⡠±. This.
is divisible by [ ]q p if p is any prime with. 2(mod 5). p ⡠±. This result has been proposed as problem by George Andrews and has been solved by Leonard.
Mar 13, 2017 - Many kinds of generalizations of Stirling and Bell numbers have ...... (ν2(Bn,4))0â¤nâ¤23 = {0, 0, 1, 0, 0, 7, 0, 0, 2, 0, 0, 1,0, 0,1, 0,0, 3,0, 0, 2, 0, ...
The set A has the property GCD(k) if the greatest common divisor of any k elements of A ... represented as the sum of h elements of A. On the other hand, if h > 2, then there exists a ... TAere ejtw/s a family ÃF of finite sets of positive integers
Comp. 59(1992), 717-722. 3. H. Davenport and D.J. Lewis, Homogeneous additive equations, Proc. Roy. ... R. K. Guy, Unsolved problems in number theory, Vol.
The set A has the property GCD(k) if the greatest common divisor of any k elements of A ... represented as the sum of h elements of A. On the other hand, if h > 2, then there exists a ... TAere ejtw/s a family ÃF of finite sets of positive integers
where k is any nonnegative integer and n is any positive integer greater than 1. In this paper a new result on divisibility of generalized repunits is stated.
which is clearly homogeneous of degree n - 1. Now it is well known (see, for example, [l, p. 376, problem 5]) that a hom
Consider the Pell Equation $x^{2}-dy^{2}=-1$. ... Now, Pell equation theory .... Phil. Soc., 100 (1986), 237-248. [6] FlorianLuca: A note on the divisibility of class ...
Apr 23, 2018 - For all real numbers X > 1, we let N3,l(X) denote the number of real quadratic ... In this short note, we shall study the following related problem.
Oct 10, 2017 - NT] 10 Oct 2017. DIVISIBILITY OF ... In the present. Date: October 11, 2017. ..... 7 41 -15126 -15126 120 7 43 -14958 -1662 20. 7 47 -14598 ...
Apr 21, 2016 - More precisely, for a given squarefree odd integer n > 1, we will show that there ... Finally, we give some numerical examples in Section ... Let K be an imaginary quadratic field with discriminant D. Let n be an odd divisor of.
Nov 7, 2014 - The divisibility property of Fibonacci and Lucas numbers has always ... of Theorem 1, and generalize the result to include the divisibility and ...
Theorem A Let f(x) = 1/Q(x) be the generating function of the integer sequence yn such that Q(x) is .... The last case easily follows from Lucas' Theorem. Actually ...
Mar 4, 2014 - Gaussian Fibonacci numbers. ... of some well-known divisibility properties of these sequences. .... Since no element of the Lucas sequence.
aElectronic mail: [email protected]. FIG. 1. Color online .... A. Kossyrev, A. Yin, S. G. Cloutier, D. A. Cardimona, D. Huang, P. M.. Alsing, and ...
Sep 30, 2015 - Theorem 1: For the general sequence e(n) = Ae(n â 1) + Be(n â 2) the ... to be the Lucas sequence with starting values 2 and 1. Wall [3] studied.
... fue en 1454 cuando construyó seis prensas y comenzó a compo- ner el libro
más grande de todos: la Biblia. Momentos estelares de la ciencia, Isaac Asimov,.
Hence, the word y distinguishes u and v for the language 0â repU (mN). Now assume ... Therefore, the number of states of the trim minimal automa- ton of the ...
We denote by Ï(N) the sum of divisors of N a positive integer and define h(N) = Ï(N)/N. N is said to be ... It has been known an odd perfect number must satisfy ... β = 5, 12, 24, 17, 62(McDaniel and Hagis[15]), β = 6, 8, 11, 14, 18(Cohen and.
As an example, the number of states of the trim minimal automaton accepting ...... On representation of integers in linear numeration systems, in Ergodic theory.
Jan 25, 2006 - arXiv:math/0601604v1 [math.NT] 25 Jan 2006 ... authors (see for instance [24, 25]), this result was recently proved in [3, 6]. However it is still ...
Feb 13, 2009 - NT] 13 Feb 2009. EXACT PROPERTIES OF FROBENIUS NUMBERS. AND FRACTION OF THE SYMMETRIC SEMIGROUPS. IN THE WEAK ...
Abstract In this study, a generalization of the sequence of balancing numbers called as k-balancing numbers is considered and some of their properties are ...
Divisibility properties of q-Fibonacci numbers Johann ...
is divisible by [ ]q p if p is any prime with. 2(mod 5). p ⡠±. This result has been proposed as problem by George Andrews and has been solved by Leonard.
Divisibility properties of q-Fibonacci numbers Johann Cigler Let q be a real number with 0 q 1 and let Fn n0 0,1,1, 2,3,5,8, be the sequence of Fibonacci numbers. Define q Fibonacci numbers by the recurrence Fn (q) Fn 1 (q) q n 2 Fn 2 (q) with initial values F0 (q) 0 and F1 (q) 1. The first terms are 0,1,1,1 q,1 q q 2 ,1 q q 2 q 3 q 4 ,1 q q 2 q 3 2q 4 q 5 q 6 ,. For q 1 we obviously get Fn (1) Fn . Let [n] [n]q
1 qn 1 q q n 1. 1 q
It is known that Fp 1 (q) is divisible by [ p]q if p is any prime with p 2(mod 5). This result has been proposed as problem by George Andrews and has been solved by Leonard Carlitz (Fibonacci Quarterly 8 (1970), Problem H 138, p. 76-81). For example F4 (q) [3]q and F8 (q ) [7]q 1 q 4 q 6 .
It is rather trivial that F5 n (q) is divisible by [5]q . To show this observe that q n q n (mod 5) (mod[5]q ). Therefore F5 n (q ) 0 mod[5]q implies F5 n 2 (q) F5 n 1 (q), F5 n 3 (q ) F5 n 2 (q ) 1 q F5 n 1 (q ) 1 q ,
F5 n 4 (q ) F5 n 3 (q ) q 2 F5 n 2 ( q ) 1 q q 2 F5 n 1 ( q ) and finally
