Wilson prime number

Wilson prime numbers (after Sir John Wilson ) are prime numbers that are divisible by . It is a stronger form of Wilson's theorem . So far, only the Wilson primes 5, 13 and 563 are known. ${\displaystyle p}$ ${\displaystyle (p-1)!+1}$ ${\displaystyle p^{2}}$

definition

For notation, see faculty , divisibility and congruence

Wilson's theorem says that is divisible by if and only if is a prime number. So for every prime number : ${\displaystyle (p-1)!+1}$ ${\displaystyle p}$ ${\displaystyle p}$ ${\displaystyle p}$

${\displaystyle p\mid (p-1)!+1}$

This can be described as congruence as follows:

${\displaystyle (p-1)!\equiv -1{\pmod {p}}}$

or

${\displaystyle (p-1)!+1\equiv 0{\pmod {p}}}$

The integer result of the division

${\displaystyle {\frac {(p-1)!+1}{p}}}$

is also referred to in this context as the Wilson quotient (sequence A007619 in OEIS ). ${\displaystyle W(p)}$

A Wilson prime number is now any prime number that is also a divisor of “its” Wilson quotient (and thus fulfills Wilson's theorem almost twice ). ${\displaystyle p}$

proof

Without loss of generality ${\displaystyle n>2}$

• ${\displaystyle n}$ is ${\displaystyle prim\Rightarrow (n-1)!\equiv -1\mod n}$

${\displaystyle \forall a\in \mathbb {Z} _{n}^{*}}$has a clear solution ${\displaystyle ax\equiv 1(\mod n)}$ ${\displaystyle a^{-1}\mod n}$

${\displaystyle a^{2}\equiv 1\Leftrightarrow (a-1)(a+1)=y\cdot n\Leftrightarrow (a\equiv 1\mod n}$ or ${\displaystyle -1\mod n)}$

${\displaystyle (n-1)!\equiv (n-1)\equiv -1\mod n}$

• ${\displaystyle (n-1)!\equiv -1\mod n}$ ${\displaystyle \Rightarrow n}$ is ${\displaystyle prim}$

Adoption: ${\displaystyle n=a\cdot b,a,b>1}$

${\displaystyle a\ teilt\ (n-1)!}$ With ${\displaystyle (n-1)!\equiv -1\mod n\Rightarrow a\ teilt\ n-1}$

Contradiction: can not share and simultaneously ${\displaystyle a}$ ${\displaystyle n}$ ${\displaystyle n-1}$

example

The number is a factor of : ${\displaystyle p=13}$ ${\displaystyle (p-1)!+1}$

${\displaystyle {\frac {(13-1)!+1}{13}}={\frac {479.001.600+1}{13}}=36.846.277}$

So because of Wilson's Theorem, is prime. Since it is also a divisor of the corresponding Wilson quotient (36,846,277 13 = 2,834,329), it is even a Wilson prime number. ${\displaystyle p=13}$ ${\displaystyle :}$

The repeated division is equivalent to dividing by the square of the starting number. Analogously to Wilson's theorem, it is true that every prime number is a Wilson prime number if and only if: ${\displaystyle p}$

${\displaystyle p^{2}\mid (p-1)!+1}$

Or:

${\displaystyle (p-1)!\equiv -1{\pmod {p^{2}}}}$

or

${\displaystyle {\frac {(p-1)!+1}{p}}=W(p)\equiv 0{\pmod {p}}}$

Occurrence

So far, only the Wilson primes 5, 13 and 563 are known (sequence A007540 in OEIS ). Should further Wilson prime numbers exist, they are greater than . It is assumed that there are an infinite number of Wilson prime numbers between and . ${\displaystyle 2\cdot 10^{13}}$ ${\displaystyle \ln(\ln(y)/\ln(x))}$ ${\displaystyle x}$ ${\displaystyle y}$

Generalizations

Wilson prime numbers of order n

The generalization of Wilson's theorem states that a natural number is prime if and only if it holds for all : ${\displaystyle p\in \mathbb {N} }$ ${\displaystyle 1\leq n\leq p}$

${\displaystyle (n-1)!\cdot (p-n)!\equiv (-1)^{n}{\pmod {p}}}$

It is therefore a prime if is an integer. ${\displaystyle p}$ ${\displaystyle {\frac {(n-1)!\cdot (p-n)!-(-1)^{n}}{p}}}$

A generalized Wilson prime number of order n is a prime number for which the following applies: ${\displaystyle p}$

${\displaystyle p^{2}}$is divisor of with , ${\displaystyle (n-1)!\cdot (p-n)!-(-1)^{n}}$ ${\displaystyle n\geq 1}$ ${\displaystyle p\geq n}$

It is therefore a generalized Wilson Prime of order n if is an integer. ${\displaystyle p}$ ${\displaystyle {\frac {(n-1)!\cdot (p-n)!-(-1)^{n}}{p^{2}}}}$

This can be described as congruence as follows:

${\displaystyle (n-1)!\cdot (p-n)!\equiv (-1)^{n}{\pmod {p^{2}}}}$

or

${\displaystyle (n-1)\cdot (p-n)!-(-1)^{n}\equiv 0{\pmod {p^{2}}}}$

It is conjectured that for any natural number there are infinitely many generalized Wilson prime numbers of the order . ${\displaystyle n}$ ${\displaystyle n}$

example

Let be a prime number and . The square number is a factor of : ${\displaystyle p=17\in \mathbb {P} }$ ${\displaystyle n=7}$ ${\displaystyle p^{2}=17^{2}=289}$ ${\displaystyle (n-1)!\cdot (p-n)!-(-1)^{n}=6!\cdot (p-7)!+1}$

${\displaystyle {\frac {6!\cdot (17-7)!+1}{17^{2}}}={\frac {720\cdot 3628800+1}{289}}={\frac {2612736001}{289}}=9040609}$

So is a divisor of the corresponding generalized Wilson quotient and is therefore a generalized Wilson prime number of order . ${\displaystyle p=17}$ ${\displaystyle n=7}$

The following table shows the generalized Wilson prime numbers of the order for : ${\displaystyle n}$ ${\displaystyle 1\leq n\leq 30}$

${\displaystyle n}$ ${\displaystyle (n-1)!\cdot (p-n)!-(-1)^{n}}$ Prime , so divider of is ${\displaystyle p}$ ${\displaystyle p^{2}}$ ${\displaystyle (n-1)!\cdot (p-n)!-(-1)^{n}}$ OEIS link 1 ${\displaystyle (p-1)!+1}$ 5, 13, 563 ... (Follow A007540 in OEIS ) 2 ${\displaystyle (p-2)!-1}$ 2, 3, 11, 107, 4931 ... (Follow A079853 in OEIS ) 3 ${\displaystyle 2!\cdot (p-3)!+1}$ 7… 4th ${\displaystyle 3!\cdot (p-4)!-1}$ 10429 ... 5 ${\displaystyle 4!\cdot (p-5)!+1}$ 5, 7, 47 ... 6th ${\displaystyle 5!\cdot (p-6)!-1}$ 11 ... 7th ${\displaystyle 6!\cdot (p-7)!+1}$ 17 ... 8th ${\displaystyle 7!\cdot (p-8)!-1}$ ... 9 ${\displaystyle 8!\cdot (p-9)!+1}$ 541… 10 ${\displaystyle 9!\cdot (p-10)!-1}$ 11, 1109 ... 11 ${\displaystyle 10!\cdot (p-11)!+1}$ 17, 2713 ... 12 ${\displaystyle 11!\cdot (p-12)!-1}$ ... 13 ${\displaystyle 12!\cdot (p-13)!+1}$ 13… 14th ${\displaystyle 13!\cdot (p-14)!-1}$ ... 15th ${\displaystyle 14!\cdot (p-15)!+1}$ 349 ...

${\displaystyle n}$ ${\displaystyle (n-1)!\cdot (p-n)!-(-1)^{n}}$ Prime , so divider of is ${\displaystyle p}$ ${\displaystyle p^{2}}$ ${\displaystyle (n-1)!\cdot (p-n)!-(-1)^{n}}$ OEIS link 16 ${\displaystyle 15!\cdot (p-16)!-1}$ 31 ... 17th ${\displaystyle 16!\cdot (p-17)!+1}$ 61, 251, 479 ... (Follow A152413 in OEIS ) 18th ${\displaystyle 17!\cdot (p-18)!-1}$ 13151527 ... 19th ${\displaystyle 18!\cdot (p-19)!+1}$ 71 ... 20th ${\displaystyle 19!\cdot (p-20)!-1}$ 59, 499 ... 21st ${\displaystyle 20!\cdot (p-21)!+1}$ 217369 ... 22nd ${\displaystyle 21!\cdot (p-22)!-1}$ ... 23 ${\displaystyle 22!\cdot (p-23)!+1}$ ... 24 ${\displaystyle 23!\cdot (p-24)!-1}$ 47, 3163 ... 25th ${\displaystyle 24!\cdot (p-25)!+1}$ ... 26th ${\displaystyle 25!\cdot (p-26)!-1}$ 97579 ... 27 ${\displaystyle 26!\cdot (p-27)!+1}$ 53… 28 ${\displaystyle 27!\cdot (p-28)!-1}$ 347 ... 29 ${\displaystyle 28!\cdot (p-29)!+1}$ ... 30th ${\displaystyle 29!\cdot (p-30)!-1}$ 137, 1109, 5179 ...

The smallest generalized Wilson prime numbers of the order are (with increasing ): ${\displaystyle n}$ ${\displaystyle n=1,2,3,\ldots }$

5, 2, 7, 10429, 5, 11, 17 ... (Follow A128666 in OEIS )

The next generalized Wilson prime of the order is not already known, but it must be greater than . ${\displaystyle n=8}$ ${\displaystyle 1,4\cdot 10^{7}}$

Almost Wilson primes

A prime number representing the congruence ${\displaystyle p\in \mathbb {P} }$

${\displaystyle (p-1)!\equiv -1+Bp{\pmod {p^{2}}}}$with a small amount ${\displaystyle |B|}$

met, is called fast-Wilson Prime (English near-Wilson primes ).

Is , we get and get the Wilson prime numbers. ${\displaystyle B=0}$ ${\displaystyle (p-1)!\equiv -1{\pmod {p^{2}}}}$

The following table shows all such Fast Wilson prime numbers for with : ${\displaystyle |B|\leq 100}$ ${\displaystyle 10^{6}<p<4\cdot 10^{11}}$

${\displaystyle p}$ ${\displaystyle B}$ 1282279 +20 1306817 −30 1308491 −55 1433813 −32 1638347 −45 1640147 −88 1647931 +14 1666403 +99 1750901 +34 1851953 −50 2031053 −18 2278343 +21 2313083 +15 2695933 −73 3640753 +69 3677071 −32

${\displaystyle p}$ ${\displaystyle B}$ 3764437 −99 3958621 +75 5062469 +39 5063803 +40 6331519 +91 6706067 +45 7392257 +40 8315831 +3 8871167 −85 9278443 −75 9615329 +27 9756727 +23 10746881 −7 11465149 −62 11512541 −26 11892977 −7

${\displaystyle p}$ ${\displaystyle B}$ 12632117 −27 12893203 −53 14296621 +2 16711069 +95 16738091 +58 17879887 +63 19344553 −93 19365641 +75 20951477 +25 20972977 +58 21561013 −90 23818681 +23 27783521 −51 27812887 +21 29085907 +9 29327513 +13

${\displaystyle p}$ ${\displaystyle B}$ 30959321 +24 33187157 +60 33968041 +12 39198017 −7 45920923 −63 51802061 +4 53188379 −54 56151923 −1 57526411 −66 64197799 +13 72818227 −27 87467099 −2 91926437 −32 92191909 +94 93445061 −30 93559087 −3

${\displaystyle p}$ ${\displaystyle B}$ 94510219 −69 101710369 −70 111310567 +22 117385529 −43 176779259 +56 212911781 −92 216331463 −36 253512533 +25 282361201 +24 327357841 −62 411237857 −84 479163953 −50 757362197 −28 824846833 +60 866006431 −81 1227886151 −51

${\displaystyle p}$ ${\displaystyle B}$ 1527857939 −19 1636804231 +64 1686290297 +18 1767839071 +8 1913042311 −65 1987272877 +5 2100839597 −34 2312420701 −78 2476913683 +94 3542985241 −74 4036677373 −5 4271431471 +83 4296847931 +41 5087988391 +51 5127702389 +50 7973760941 +76

${\displaystyle p}$ ${\displaystyle B}$ 9965682053 −18 10242692519 −97 11355061259 −45 11774118061 −1 12896325149 +86 13286279999 +52 20042556601 +27 21950810731 +93 23607097193 +97 24664241321 +46 28737804211 −58 35525054743 +26 41659815553 +55 42647052491 +10 44034466379 +39 60373446719 −48

${\displaystyle p}$ ${\displaystyle B}$ 64643245189 −21 66966581777 +91 67133912011 +9 80248324571 +46 80908082573 −20 100660783343 +87 112825721339 +70 231939720421 +41 258818504023 +4 260584487287 −52 265784418461 −78 298114694431 +82

Wilson numbers

A Wilson number is a natural number for which the following applies: ${\displaystyle n}$

${\displaystyle W(n)\equiv 0{\pmod {n^{2}}}}$, With ${\displaystyle W(n)=\prod _{\stackrel {1\leq k\leq n}{ggT(k,n)=1}}{k}+e}$

Here is if and only if a primitive root has, otherwise is . ${\displaystyle e=1}$ ${\displaystyle n}$ ${\displaystyle e=-1}$

For every natural number is by divisible. The quotient is called the generalized Wilson quotient . The first generalized Wilson quotients are: ${\displaystyle n}$ ${\displaystyle W(n)}$ ${\displaystyle n}$ ${\displaystyle {\frac {W(n)}{n}}}$

2, 1, 1, 1, 5, 1, 103, 13, 249, 19, 329891, 32, 36846277, 1379, 59793, 126689, 1230752346353, 4727, 336967037143579, 436486, 2252263619, 56815333, 488695968598959812886, 154925968598959812886, 15492596859895986087, 154925968598959812886382505 Follow A157249 in OEIS )

If the generalized Wilson quotient is divisible by, you get a Wilson number. These are: ${\displaystyle n}$

1, 5, 13, 563, 5971, 558771, 1964215, 8121909, 12326713, 23025711, 26921605, 341569806, 399292158 (follow A157250 in OEIS )

If a Wilson number is prime, then Wilson is prime. There are 13 Wilson numbers for . ${\displaystyle n}$ ${\displaystyle n}$ ${\displaystyle n<5\cdot 10^{8}}$

literature

• NGWH Beeger : Quelques remarques sur les congruences r ^{p − 1} ≡ 1 (mod p ² ) et (p − 1)! ≡ −1 (mod p ² ) . In: The Messenger of Mathematics , 43, 1913–1914, pp. 72–84 (French) Textarchiv - Internet Archive
• Emma Lehmer : On congruences involving Bernoulli numbers and the quotients of Fermat and Wilson . (PDF; 747 kB) In: Annals of Mathematics , 39, April 1938, pp. 350–360 (English)
• Paulo Ribenboim : The world of prime numbers. Secrets and Records . Springer, Berlin 2006, ISBN 3-540-34283-4 (updated translation of The little book of bigger primes . Springer, New York 2004)

Web links

• Eric W. Weisstein : Wilson Prime . In: MathWorld (English).
• Chris K. Caldwell: Wilson prime . The Prime Glossary.
• Here is the latest update on… - Email from Richard McIntosh to Paul Zimmermann on March 9, 2004
• Emma Lehmer : On congruences involving Bernoulli numbers and the quotients of Fermat and Wilson. Annals of Mathematics 39 (2), April 1938, pp. 350-360 , accessed February 3, 2020 . 
• Distributed search for Wilson primes. mersenneforum.org, accessed February 3, 2020 . 
• Erna H. Pearson : On the Congruences ( p  - 1)! ≡ −1 and 2 ^{p −1}  ≡ 1 (mod  p ² ). Mathematics of Computation 17 , April 6, 1962, pp. 194-195 , accessed February 3, 2020 . 

Individual evidence

1. ↑ Eric W. Weisstein : Wilson Quotient . In: MathWorld (English).
2. ^ Karl Goldberg: A table of Wilson quotients and the third Wilson prime . In: Journal of the London Mathematical Society , April 28, 1953, pp. 252-256 (English)
3. ^ ^{A b} Edgar Costa, Robert Gerbicz, David Harvey : A search for Wilson primes. October 27, 2012, pp. 1–25 , accessed February 1, 2020 . 
4. ^ Richard Crandall , Karl Dilcher, Carl Pomerance : A search for Wieferich and Wilson primes . Mathematics of Computation 66 , January 1997, pp. 433–449 (English)
5. ↑ Chris K. Caldwell: Wilson prime . The Prime Glossary.
6. ↑ Takashi Agoh, Karl Dilcher, Ladislav Skula: Wilson Quotients for composite moduli. Mathematics of Computation 67 (222), April 1998, pp. 843-861 , accessed February 2, 2020 .