- Source: Wilson prime
In number theory, a Wilson prime is a prime number
p
{\displaystyle p}
such that
p
2
{\displaystyle p^{2}}
divides
(
p
−
1
)
!
+
1
{\displaystyle (p-1)!+1}
, where "
!
{\displaystyle !}
" denotes the factorial function; compare this with Wilson's theorem, which states that every prime
p
{\displaystyle p}
divides
(
p
−
1
)
!
+
1
{\displaystyle (p-1)!+1}
. Both are named for 18th-century English mathematician John Wilson; in 1770, Edward Waring credited the theorem to Wilson, although it had been stated centuries earlier by Ibn al-Haytham.
The only known Wilson primes are 5, 13, and 563 (sequence A007540 in the OEIS). Costa et al. write that "the case
p
=
5
{\displaystyle p=5}
is trivial", and credit the observation that 13 is a Wilson prime to Mathews (1892). Early work on these numbers included searches by N. G. W. H. Beeger and Emma Lehmer, but 563 was not discovered until the early 1950s, when computer searches could be applied to the problem. If any others exist, they must be greater than 2 × 1013. It has been conjectured that infinitely many Wilson primes exist, and that the number of Wilson primes in an interval
[
x
,
y
]
{\displaystyle [x,y]}
is about
log
log
x
y
{\displaystyle \log \log _{x}y}
.
Several computer searches have been done in the hope of finding new Wilson primes.
The Ibercivis distributed computing project includes a search for Wilson primes. Another search was coordinated at the Great Internet Mersenne Prime Search forum.
Generalizations
= Wilson primes of order n
=Wilson's theorem can be expressed in general as
(
n
−
1
)
!
(
p
−
n
)
!
≡
(
−
1
)
n
mod
p
{\displaystyle (n-1)!(p-n)!\equiv (-1)^{n}\ {\bmod {p}}}
for every integer
n
≥
1
{\displaystyle n\geq 1}
and prime
p
≥
n
{\displaystyle p\geq n}
. Generalized Wilson primes of order n are the primes p such that
p
2
{\displaystyle p^{2}}
divides
(
n
−
1
)
!
(
p
−
n
)
!
−
(
−
1
)
n
{\displaystyle (n-1)!(p-n)!-(-1)^{n}}
.
It was conjectured that for every natural number n, there are infinitely many Wilson primes of order n.
The smallest generalized Wilson primes of order
n
{\displaystyle n}
are:
= Near-Wilson primes
=A prime
p
{\displaystyle p}
satisfying the congruence
(
p
−
1
)
!
≡
−
1
+
B
p
(
mod
p
2
)
{\displaystyle (p-1)!\equiv -1+Bp\ (\operatorname {mod} {p^{2}})}
with small
|
B
|
{\displaystyle |B|}
can be called a near-Wilson prime. Near-Wilson primes with
B
=
0
{\displaystyle B=0}
are bona fide Wilson primes. The table on the right lists all such primes with
|
B
|
≤
100
{\displaystyle |B|\leq 100}
from 106 up to 4×1011.
= Wilson numbers
=A Wilson number is a natural number
n
{\displaystyle n}
such that
W
(
n
)
≡
0
(
mod
n
2
)
{\displaystyle W(n)\equiv 0\ (\operatorname {mod} {n^{2}})}
, where
W
(
n
)
=
±
1
+
∏
gcd
(
k
,
n
)
=
1
1
≤
k
≤
n
k
,
{\displaystyle W(n)=\pm 1+\prod _{\stackrel {1\leq k\leq n}{\gcd(k,n)=1}}{k},}
and where the
±
1
{\displaystyle \pm 1}
term is positive if and only if
n
{\displaystyle n}
has a primitive root and negative otherwise. For every natural number
n
{\displaystyle n}
,
W
(
n
)
{\displaystyle W(n)}
is divisible by
n
{\displaystyle n}
, and the quotients (called generalized Wilson quotients) are listed in OEIS: A157249. The Wilson numbers are
If a Wilson number
n
{\displaystyle n}
is prime, then
n
{\displaystyle n}
is a Wilson prime. There are 13 Wilson numbers up to 5×108.
See also
PrimeGrid
Table of congruences
Wall–Sun–Sun prime
Wieferich prime
Wolstenholme prime
References
Further reading
Crandall, Richard E.; Pomerance, Carl (2001). Prime Numbers: A Computational Perspective. Springer-Verlag. p. 29. ISBN 978-0-387-94777-8.
Pearson, Erna H. (1963). "On the Congruences (p − 1)! ≡ −1 and 2p−1 ≡ 1 (mod p2)" (PDF). Math. Comput. 17: 194–195.
External links
The Prime Glossary: Wilson prime
Weisstein, Eric W. "Wilson prime". MathWorld.
Status of the search for Wilson primes
Kata Kunci Pencarian:
- Prime Minister's Questions
- Transformers (film)
- Daftar film Indonesia tahun 2024
- Daftar presiden Amerika Serikat
- Perang Dunia I
- Transformers: Age of Extinction
- Bliss (film 2021)
- Transformers: Revenge of the Fallen
- Titik panas Anahim
- Bilangan prima
- Wilson prime
- Harold Wilson
- List of prime numbers
- Wilson's theorem
- James Callaghan
- Wilson quotient
- William Wilson
- Prime (drink)
- Harold Wilson plot allegations
- Prime Minister of the United Kingdom
No More Posts Available.
No more pages to load.
