Wilson's Theorem

��w m $\in$ $\mathbb {N}$ , ��N�M m ��誺�� a, �� Proposition 3.2.5 ��i�H��@�өM m ��誺�� b �ϱo ab $\equiv$ 1(mod m), �ڭ̤]��M�o�˪� b �ä��ߤ@, ��b modulo m ��U��|�O�ߤ@��. �]�N�O��u��b��H m ��U�M b �P�l��Ƥ~�|�ŦX. �o�ئb modulo m ��U��k�Ϥ��s�b�ߤ@�ʥ� modulo m ��U�� reduced residue system �̮e��F.

�� . �]�� S �O�@�� reduced residue system modulo m, �C�@�� S �� s_i �ҩM m ��, �G�Q�� Proposition 3.2.5 ��s�b b $\in$ $\mathbb {Z}$ �ϱo r_ib $\equiv$ 1(mod m). �ѩ� b �M m �]�O��誺, �G�� S �O�@�� reduced residue system modulo m ��w�q��s�b r_j $\in$ S �M b �b modulo m ��U�O�P��, �]�N�O�� b $\equiv$ r_j(mod m). �]�� Lemma 3.1.3 ��, r_ir_j $\equiv$ r_ib $\equiv$ 1(mod m). �ұo�s�b��.

��ߤ@��, �ڭ̥��] r_j, r_k $\in$ S �Һ�� r_ir_j $\equiv$ 1(mod m) �H�� r_ir_k $\equiv$ 1(mod m). �]��o r_ir_j $\equiv$ r_ir_k(mod m). ��ѩ� gcd(m, r_i) = 1, �Q�� Corollary 3.2.4 �o r_j $\equiv$ r_k(mod m). �� S �O reduced residue system modulo m �� S ��۲��b modulo m ��U��O��P��, �G�� r_j $\equiv$ r_k(mod m) �� r_j = r_k. �o�Ұߤ@��. $\qedsymbol$

�Ҧp S = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} �O�@�� reduced residue system modulo 11, �b modulo 11 ��U�ڭ̦�

�� . ��Y a $\equiv$ ±1(mod p), �h a² $\equiv$ (±1)²(mod p). �o�� a² $\equiv$ 1(mod p).

�Ϥ�, �Y a² $\equiv$ 1(mod p), �� p| a² - 1, �]�N�O�� p|(a - 1)(a + 1), �G�] p �O��, �Q�� Lemma 1.4.2 �o p| a - 1 �� p| a + 1. �]�N�O�� a $\equiv$ 1(mod p) �� a $\equiv$ - 1(mod p). $\qedsymbol$

Theorem 3.4.3 (Wilson's Theorem) ��w�@�� p. �] {r₁,..., r_{p - 1}} ��@ reduced residue system modulo p. �h

r₁^...r_{p - 1} $\displaystyle \equiv$ - 1(mod p).

�S�O�a, �ڭ̦�

(p - 1)! $\displaystyle \equiv$ - 1(mod p).

�� . �Y p = 2, �h modulo 2 ��U�� reduced residue system �� {r₁} �@�Ӥ��, �䤤 r₁ $\equiv$ 1(mod 2). ��b modulo 2 ��U�ڭ̦� 1 $\equiv$ - 1(mod 2), �G�o�� r₁ $\equiv$ - 1(mod 2).

�{�Ҽ{ p > 2 ��, �O S = {r₁,..., r_{p - 1}} �ѩ� gcd(p, 1) = gcd(p, - 1) = 1 �B 1 $\not\equiv$ - 1(mod p) (�_�h p| 2), �G��O�s�b r_i, r_j $\in$ S �䤤 r_i $\ne$ r_j �� r_i $\equiv$ 1(mod p) �B r_j $\equiv$ - 1(mod p). �]��, �ڭ̥i��] r₁ $\equiv$ 1(mod p) �B r₂ $\equiv$ - 1(mod p). �{�Ҽ{ r_i $\in$ S, �䤤 3 $\le$ i $\le$ p - 1. �� Lemma 3.4.1 ��s�b�ߤ@�� r_j $\in$ S �ϱo r_ir_j $\equiv$ 1(mod p). �]�� r_i $\not\equiv$ ±1(mod p), �G�� r_j $\not\equiv$ ±1(mod p), �]�N�O�� 3 $\le$ j $\le$ p - 1. �S�Y r_i = r_j, �|�ɭP r_i² $\equiv$ 1(mod p), �o�P Lemma 3.4.2 �ۥ٬�, �G�� i $\ne$ j. �]�N�O��b T = {r₃,..., r_{p - 1}} ��@�� r_i ��i��ߤ@��t�@�� r_j $\in$ T �ϱo r_ir_j $\equiv$ 1(mod p). �]��ڭ̥i�H�� T ��o p - 3 �Ӥ��t�� (�`�N p �O�_��), �ϱo�C�@�襤��ۭ��ᰣ�H p �|�l 1. �]�N�O�� r₃^...r_{p - 1} $\equiv$ 1(mod p). �]��ڭ̱o��

r₁r₂r₃^...r_{p - 1} $\displaystyle \equiv$ r₁r₂ $\displaystyle \equiv$ - 1(mod p).

�̫�ѩ� {1, 2,..., p - 1} �O�@�� modulo p �� reduced residue system, �G��

1×2×^...×(p - 1) = (p - 1)! $\displaystyle \equiv$ - 1(mod p).

$\qedsymbol$

�Y p �O�@��ƥB a �O�M p ��誺��, �ڭ̥i�H�Q�� Wilson's Theorem ��b modulo p ��U, a ��k�Ϥ��. �ѩ�� a $\equiv$ ±1(mod p) �� a² $\equiv$ 1(mod p), �]�N�O�� a ��b modulo p ��U�O�ۤv��k�Ϥ��, �ҥH�ڭ̶ȰQ�� a $\not\equiv$ ±1(mod p) ��p.

�� . �ѩ� 2 $\le$ i $\le$ p - 2, �ڭ̪� b �O�@�Ӿ��. ��

ab $\displaystyle \equiv$ i $\displaystyle {\frac{(p-2)!}{i}}$ $\displaystyle \equiv$ (p - 2)!(mod p)

�S�ѩ� (p - 1)! = (p - 1)^.(p - 2)! �B p - 1 $\equiv$ - 1(mod p), �G�o��

ab $\displaystyle \equiv$ (p - 2)! $\displaystyle \equiv$ - ((p - 1)!) $\displaystyle \equiv$ 1(mod p).

$\qedsymbol$

�ڭ̤��n�j�դ@�U��M Lemma 3.4.1 �b�@�몺 m $\in$ $\mathbb {N}$ ��, �� Lemma 3.4.2 �ݭ��b��Ʈɤ~��, �ҥH Wilson's Theorem �b modulo �@�몺 m �ä��@�w��. �]�N�O��Y {r₁,..., r_(m)} �O�@�� reduced residue system modulo m, �ä��@�w�i�H�o r₁^...r_(m) $\equiv$ - 1(mod m). �Ҧp�b modulo 15 ��U�ڭ̤��٦� 4 �M -4 �� 4² $\equiv$ (- 4)² $\equiv$ 1(mod 15), �ҥH�Q�� Theorem 3.4.3 ��ҩ��k (�Ϊ��p��) �ڭ̥i�o, �Y {r₁,..., r₈} �O�@�� reduced residue system modulo 15, �h r₁^...r₈ $\equiv$ 1(mod 15). ��M�Q�� Theorem 3.4.3 ��k�ڭ̥i�H�N Wilson's Theorem ��s��@�� m ��, ��L��ɹ�@�� modulo m �� reduced residue system {r₁,..., r_(m)} �� r_i² $\equiv$ 1(mod m) �� r_i �|��ܦh�ر��, �Q�װ_�Ӹ��, �b�o�̧ڭ̴N��h��Q�F.