I'm reading up on Wilson's Theorem, and see a symbol I don't know... what does an equal sign with three lines mean?

I'm looking at the example table and I still can't infer what they are trying to say about that relationship between equations.

  • $\begingroup$ This concept is called modular arithmetic. $\endgroup$
    – Tomas
    Oct 21, 2013 at 23:10
  • $\begingroup$ $a \equiv b \bmod m$, read $a$ is congruent to $b$ modulo $m$, represents equality between the remainders of $a$ and $b$ when divided by $m$. $\endgroup$ Oct 22, 2013 at 2:44

3 Answers 3


$$(n-1)! \equiv -1\pmod n$$ means that $(n-1)!$ and $-1$ differ by a multiple of $n$. Or, if you prefer, that $(n-1)!+1$ is a multiple of $n$.

In general, $$a\equiv b\pmod n$$ means that $a$ and $b$ differ by a multiple of $n$, or that $a-b$ is a multiple of $n$.

It's explained in detail in the Wikipedia article on "modular equivalence". The $\equiv$ symbol itself is pronounced "is equivalent to".

  • $\begingroup$ This is Perfect! Thanks... now I have a better route to take to study. Since I never learned this in school, what class should have exposed this to me? (given I went to school in the early 90's and am certain I was never taught it) $\endgroup$ Oct 22, 2013 at 3:11
  • $\begingroup$ I don't know what you mean by "school", and where or when you might have learned it depends on the curriculum. An undergraduate-level class in elementary number theory would have covered it. $\endgroup$
    – MJD
    Oct 22, 2013 at 3:19

Wilson's Theorem:

$$(n-1)!\ \equiv\ -1 \pmod n.$$

It means "is congruent to" (modulo n): that is, $$(n - 1)! -(-1) = (n - 1)! + 1 \equiv 0 \pmod n$$ And that simply means that $n$ divides $(n - 1)! + 1$.

  • $\begingroup$ Needs a TU ----> =+1 $\endgroup$
    – Amzoti
    Oct 22, 2013 at 1:21
  • $\begingroup$ $\Longrightarrow + \Longleftarrow$ $\endgroup$
    – Mikasa
    Oct 22, 2013 at 5:07

a ≡ b (mod n)

is equivalent to

a mod n = b :(if b less than n)

a mod n = b mod n :(else)

(I am not sure, if also b>0).


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