My textbook states: If $p$ is a prime, then $(p-1) \equiv -1\pmod p$.

But the online version is $(p-1)! \equiv -1\pmod p$.

Which one is correct?

  • 25
    $\begingroup$ Both are correct, but the first one is trivial rather than deserving to be named after Wilson. $\endgroup$ – Tobias Kildetoft Nov 1 '16 at 10:06
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    $\begingroup$ Which book is it? $\endgroup$ – lhf Nov 1 '16 at 11:14
  • $\begingroup$ My book has $(p-2)! \equiv 1 \pmod p$. $\endgroup$ – I. J. Kennedy Nov 1 '16 at 20:31
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    $\begingroup$ Everything written in textbooks is also modulo possible typos. $\endgroup$ – Count Iblis Nov 1 '16 at 21:36

The second one is Wilson's theorem.

Though the first one is not absurd, since

$$p\equiv 0\pmod p$$

you always have

$$p-1\equiv -1\pmod p$$

whether $p$ is prime or not.

  • 12
    $\begingroup$ The first one isn't absurd, but calling it "Somebody's Theorem" is. $\endgroup$ – David Richerby Nov 1 '16 at 11:56
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    $\begingroup$ @DavidRicherby, by the powers vested in me by the State of Insanity, I'm officially calling $(p − 1) ≡ −1 (\mod p)$ to be "Somebody's Theorem". $\endgroup$ – Ben Hocking Nov 1 '16 at 21:32
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    $\begingroup$ @BenHocking That's absurd! $\endgroup$ – David Richerby Nov 1 '16 at 22:02

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