FIBONACCI NUMBERS MODULO CUBES OF PRIMES
Letpbe an odd prime. It is well known that F p − ( p 5 ) ≡ 0 (modp), where {Fn } n⩾0is the Fibonacci sequence and (–) is the Jacobi symbol. In this paper we show that ifp≠ 5 then we may determine F p − ( p 5 ) modp3 in the following way: ∑ k = 0 ( p − 1 ) / 2 ( k 2 k ) ( − 16 ) k ≡ ( p 5 ) ( 1 + F p...
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| Published in | Taiwanese journal of mathematics Vol. 17; no. 5; pp. 1523 - 1543 |
|---|---|
| Main Author | |
| Format | Journal Article |
| Language | English |
| Published |
Mathematical Society of the Republic of China
01.10.2013
|
| Subjects | |
| Online Access | Get full text |
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| Summary: | Letpbe an odd prime. It is well known that
F
p
−
(
p
5
)
≡
0
(modp), where {Fn
}
n⩾0is the Fibonacci sequence and (–) is the Jacobi symbol. In this paper we show that ifp≠ 5 then we may determine
F
p
−
(
p
5
)
modp3
in the following way:
∑
k
=
0
(
p
−
1
)
/
2
(
k
2
k
)
(
−
16
)
k
≡
(
p
5
)
(
1
+
F
p
−
(
p
5
)
2
)
(
mod
p
3
)
.
We also use Lucas quotients to determine
∑
k
=
0
(
p
−
1
)
/
2
(
k
2
k
)
/
m
k
modulop
2for any integerm≢ 0 (modp); in particular, we obtain
∑
k
=
0
(
p
−
1
)
/
2
(
k
2
k
)
16
k
≡
(
3
p
)
(
mod
p
2
)
.
In addition, we pose three conjectures for further research.
2010Mathematics Subject Classification: Primary 11B39, 11B65; Secondary 05A10, 11A07.
Key words and phrases: Fibonacci numbers, Central binomial coefficients, Congruences, Lucas sequences. |
|---|---|
| ISSN: | 1027-5487 2224-6851 |
| DOI: | 10.11650/tjm.17.2013.2488 |