Proving Wilson's theorem for $n=11$

Question

Wilson's theorem establishes that if a $n$ number is prime then: \begin{align} (n-1)! &\equiv -1\ \textrm{mod}\ (n) \end{align}

I have probed the theorem for the particular case where $n = 7$ like this:

I first consider the set $\{2,..,n-2\}$, in my case, $\{2,3,4,5\}$ and then I take the pair of numbers $a$ and $a¹$ where:

\begin{align} 2.4 &\equiv 1\ \textrm{mod}\ (7) \end{align} \begin{align} 3.5 &\equiv 1\ \textrm{mod}\ (7) \end{align} Then, \begin{align} 2.3.4.5 &\equiv 1\ \textrm{mod}\ (7) \end{align} So, \begin{align} 6 &\equiv -1\ \textrm{mod}\ (7) \end{align} And finally, \begin{align} 6! &\equiv -1\ \textrm{mod}\ (7) \end{align}

How can I prove the theorem where $n=11$? How can I retrieve $a$ and $a¹$?

$2\cdot4\equiv0\pmod8$, so what about $7!=7\cdot6\cdot5\cdots$? — Jyrki Lahtonen, Oct 18 '13 at 18:24
Related : http://math.stackexchange.com/questions/164852/if-n-is-composite-then-n-divides-n-1 — lab bhattacharjee, Oct 18 '13 at 18:28
Please don't change the question after it was already answered! — Potato, Oct 18 '13 at 18:47
$2\cdot 6 = 12;, 3\cdot 4 = 12;, 5\cdot 9 = 45;, 7\cdot 8 = 56$. — Daniel Fischer, Oct 18 '13 at 18:48
It was a mistake to put $n=8$, i really meant $n=11$. Sorry. — Haritz, Oct 18 '13 at 18:48

Potato · Accepted Answer · 2013-10-18T18:49:07.060

4

Answer to Old Question

You can't prove the statement when $n=8$ because it's false. Note $7!$ is $0$ mod $8$.

Wilson's theorem is in fact an if and only if statement. The proposition $(n-1)! \equiv -1 \pmod n$ is true if and only if $n$ is prime.

Answer to Edit

To answer your new question, use the pairs $(1,10)$, $(2,6)$, $(3,4)$, $(5,9)$, and $(7,8)$.

edited Oct 18 '13 at 18:49

answered Oct 18 '13 at 18:23

Potato

40,171

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because $7! = 7 \times ... \times 4 \times ... \times 2 $ – what'sup Oct 18 '13 at 18:25
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@what'sup Indeed. – Potato Oct 18 '13 at 18:25
Note: the OP has changed the ... P – The Chaz 2.0 Oct 18 '13 at 18:47
is the pair $(1,10)$ correct? Is it necessary? – Haritz Oct 20 '13 at 10:46

score 1 · Answer 2 · edited Oct 18 '13 at 18:56

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Note that Wilson's Theorem is true if and only if $n$ is prime. You cannot prove it for composite $n$.

edited Oct 18 '13 at 18:56

Potato

40,171

answered Oct 18 '13 at 18:25

Ahaan S. Rungta

7,706

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You can certainly probe this statement for composite $n$, but you can't prove it. – Potato Oct 18 '13 at 18:30
OK, $\text{prove}\ne\text{probe}$. Thanks! :P – Ahaan S. Rungta Oct 18 '13 at 18:33

Proving Wilson's theorem for $n=11$

2 Answers2