Divisibility properties of q-Fibonacci numbers Johann
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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.
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Divisibility properties of q-Fibonacci numbers Johann
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.
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
