Wall-Sun-Sun prime
التعريفات والمعاني
== English ==
=== Alternative forms ===
Wall-Sun-Sun prime number
=== Etymology ===
Named after American mathematician Donald Dines Wall and Chinese mathematicians Sun Zhihong and Sun Zhiwei, who have all contributed to the study of such primes.
=== Noun ===
Wall-Sun-Sun prime (plural Wall-Sun-Sun primes)
(number theory) A (hypothetical) prime number
p
{\displaystyle p}
such that
p
2
{\displaystyle p^{2}}
divides
F
π
(
p
)
{\displaystyle F_{\pi (p)}}
, where
F
n
{\displaystyle F_{n}}
is the Fibonacci sequence and
π
(
p
)
{\displaystyle \pi (p)}
is the
p
{\displaystyle p}
th Pisano period (the period length of the Fibonacci sequence reduced modulo
p
{\displaystyle p}
).
Synonym: Fibonacci-Wieferich prime
==== Usage notes ====
Definition (slightly expanded):
Consider the Fibonacci sequence
F
n
{\displaystyle F_{n}}
. For any prime number
p
{\displaystyle p}
, reducing the sequence modulo
p
{\displaystyle p}
produces a periodic sequence. The period length of the reduced sequence is called the
p
{\displaystyle p}
th Pisano period, denoted
π
(
p
)
{\displaystyle \pi (p)}
. Since
F
0
=
0
{\displaystyle F_{0}=0}
, it follows that
p
|
F
π
(
p
)
{\displaystyle p\vert F_{\pi (p)}}
.
A Wall-Sun-Sun prime is a prime number
p
{\displaystyle p}
such that
p
2
|
F
π
(
p
)
{\displaystyle p^{2}\vert F_{\pi (p)}}
.
Alternative definitions:
Denote by
α
(
m
)
{\displaystyle \alpha (m)}
the rank of apparition modulo
m
{\displaystyle m}
(the smallest
k
{\displaystyle k}
such that
m
|
F
k
{\displaystyle m\vert F_{k}}
). For prime
p
≠
2
,
5
{\displaystyle p\neq 2,5}
, it is known that
α
(
p
)
|
p
−
(
p
5
)
{\displaystyle \alpha (p)\vert p-\left({\tfrac {p}{5}}\right)}
, where
(
p
5
)
{\displaystyle \textstyle \left({\frac {p}{5}}\right)}
is the Legendre symbol. Then:
A prime
p
{\displaystyle p}
is a Wall-Sun-Sun prime if and only if
p
2
|
F
α
(
p
)
{\displaystyle p^{2}\vert F_{\alpha (p)}}
.
A prime
p
{\displaystyle p}
is a Wall-Sun-Sun prime if and only if
p
2
|
F
p
−
(
p
5
)
{\displaystyle p^{2}\vert F_{p-\left({\frac {p}{5}}\right)}}
.
A prime
p
{\displaystyle p}
is a Wall-Sun-Sun prime if and only if
π
(
p
2
)
=
π
(
p
)
{\displaystyle \pi (p^{2})=\pi (p)}
.
A prime
p
{\displaystyle p}
is a Wall-Sun-Sun prime if and only if
L
p
≡
1
(
mod
p
2
)
{\displaystyle L_{p}\equiv 1{\pmod {p^{2}}}}
, where
L
p
{\displaystyle L_{p}}
is the
p
{\displaystyle p}
th Lucas number.
==== Translations ====
=== See also ===
Pisano period
rank of apparition
=== Further reading ===
Fibonacci prime § Wall–Sun–Sun primes on Wikipedia.Wikipedia