Numero di Wilson

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Un primo di Wilson, che prende il nome dal matematico inglese John Wilson, è un numero primo p tale che p² divide (p − 1)! + 1, dove il simbolo ! indica la funzione fattoriale; si confronti questo risultato con le asserzioni del teorema di Wilson, il quale afferma che ogni numero primo p divide (p − 1)! + 1.

Gli unici numeri primi di Wilson conosciuti sono 5, 13 e 563^[1]; se ne esistono altri devono essere maggiori di $2\cdot 10^{13}$ .^[2] È stato congetturato che esistano infiniti primi di Wilson, e che il loro numero in un dato intervallo [x, y] sia circa uguale a $\log \left({\frac {\log x}{\log y}}\right)$ .^[3]

Nella speranza di trovare nuovi primi di Wilson sono state svolte diverse ricerche attraverso computer.^[4]^[5]^[6] Il progetto di calcolo distribuito Ibercivis include una ricerca dei primi di Wilson.^[7] Un'altra ricerca è svolta al mersenneforum.^[8]

Generalizzazioni

Primi di Wilson di ordine n

Il teorema di Wilson può essere espresso in generale come ${\displaystyle (n-1)!(p-n)!\equiv (-1)^{n}\ {\bmod {p}}}$ per ogni intero $n\geq 1$ e primo $p\geq n$ . I primi di Wilson generalizzati di ordine $n$ sono i primi $p$ tali che $p^{2}$ divida $(n-1)!(p-n)!-(-1)^{n}$ .

È stato congetturato che per ogni numero naturale $n$ esistano infiniti primi di Wilson di ordine $n$ .

$n$	primi $p$ tali che $p^{2}$ divida $(n-1)!(p-n)!-(-1)^{n}$ (fino a 10000)
1	5, 13, 563, ...
2	2, 3, 11, 107, 4931, ...
3	7, ...
4	10429, ...
5	5, 7, 47, ...
6	11, ...
7	17, ...
8	...
9	541, ...
10	11, 1109, ...
11	17, 2713, ...
12	...
13	13, ...
14	...
15	349, ...
16	31, ...
17	61, 251, 479, ...
18	13151527, ...
19	71, ...
20	59, 499, ...
21	217369, ...
22	...
23	...
24	47, 3163, ...
25	...
26	97579, ...
27	53, ...
28	347, ...
29	...
30	137, 1109, 5179, ...

Numeri di Wilson

Un numero di Wilson è un numero naturale $n$ tale che $W(n)\equiv 0{\bmod {n}}^{2}$ dove ${\displaystyle W(n)=\prod _{\stackrel {1\leq k\leq n}{\gcd(k,n)=1}}{k}+e}$ , e dove $e=1$ se $n$ ha una radice primitiva, altrimenti $e=-1$ .^[9] Per ogni numero naturale $n$ , $W(n)$ è divisibile per $n$ . I numeri di Wilson sono

1, 5, 13, 563, 5971, 558771, 1964215, 8121909, 12326713, 23025711, 26921605, 341569806, 399292158, ...

Se un numero di Wilson $n$ è primo, allora è considerato un primo di Wilson. Ci sono 13 numeri di Wilson fino a $5\cdot 10^{8}$ .

Note

^ (EN) Sequenza A007540, su On-Line Encyclopedia of Integer Sequences, The OEIS Foundation.
^ A Search for Wilson primes Retrieved on November 2, 2012.
^ The Prime Glossary: Wilson prime
^ R. McIntosh, WILSON STATUS (Feb. 1999), in E-Mail to Paul Zimmermann, 9 marzo 2004. URL consultato il 6 giugno 2011.
^ A search for Wieferich and Wilson primes, p 443
^ (DE) P. Ribenboim e W. Keller, Die Welt der Primzahlen: Geheimnisse und Rekorde, Berlin Heidelberg New York, Springer, 2006, p. 241, ISBN 3-540-34283-4.
^ Ibercivis site, su ibercivis.net. URL consultato il 13 settembre 2018 (archiviato dall'url originale il 20 giugno 2012).
^ Distributed search for Wilson primes (at mersenneforum.org)
^ see Gauss's generalization of Wilson's theorem

Voci correlate

Collegamenti esterni

(EN) Eric W. Weisstein, Numero di Wilson, su MathWorld, Wolfram Research.
(EN) The Prime Glossary: Wilson prime, su primes.utm.edu.
(EN) Status of the search for Wilson primes, su loria.fr.

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