Exercise 2.1.16 (Pairs of associated integer in examples of Wilson's theorem)

Question

Illustrate the proof of Theorem 2.11 for $p=11$ and $p=13$ by actually determining the pairs of associated integers.

Richard Ganaye · Accepted Answer

\begin{proof} 
If $p = 11$, since $1 \equiv 2 \cdot 6 \equiv 3 \cdot 4 \equiv 5 \cdot 9 \equiv 7 \cdot 8 \pmod {11}$,
$$ (p-1)!  \equiv 1\cdot (2 \cdot 6) \cdot (3 \cdot 4) \cdot (5 \cdot 9) \cdot (7 \cdot 8) \cdot 10 \equiv -1 \pmod {11}.$$

Similarly, if $p = 13$,
$$(p-1)!  \equiv 1 \cdot (2\cdot 7) \cdot (3 \cdot 9) \cdot (4 \cdot 10) \cdot (5 \cdot 8) \cdot (6 \cdot 11) \cdot 12 \equiv -1 \pmod {13}.$$
\end{proof}

Exercise 2.1.16 (Pairs of associated integer in examples of Wilson's theorem)

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Exercise 2.1.16 (Pairs of associated integer in examples of Wilson's theorem)

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