v (3)Congruences (congruence) 數 , 數 , 數

(1)

數

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數論基數數 ,

數 . 基 , 數論

. 數論 ( ) ,

Silverman A Friendly Introduction to Number Theory (Prentice Hall, Third Edition

2006). .

v

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Congruences

(congruence) 數 , 數

, 數 . congruence

congruence .

3.1.

Congruence relation equivalent relation. equivalent relation 基 .

. , ;

;

, .

equivalence relation. 數 equivalence relation .

Deﬁnition 3.1.1. S a∼ b a b ,

equivalence relation:

(equiv1): a∈ S, a∼ a (reﬂexivity).

(equiv2): a∼ b, b∼ a (symmetry).

(equiv3): a∼ b b∼ c, a∼ c (transitivity).

“=” equivalent relation.

(equiv2) : a∼ b b∼ a. (equiv3)

a∼ a. (equiv1) ? (equiv1) S

a a∼ a. (equiv2) (equiv3), S a S

b a∼ b, a a∼ a .

. equivalence relation

( ).

35

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equivalence relation ? 前 (equiv1) . (equiv2) (equiv3)

; b A B , A a b a∼ b

B c b b∼ c. (equiv2) (equiv3) a∼ c.

A B . A B . 言

equivalent relation .

.

Deﬁnition 3.1.2. 數 m, a, b∈ Z m 數 , a, b

m (a is congruent to b modulo m), a≡ b (mod m) .

a b m (a is incongruent to b modulo m), a̸≡ b (mod m) .

m . a b ,

a b .

a, b m , a b m 數 ,

m a− b.

Lemma 3.1.3. 數 m, a, b∈ Z, a≡ b (mod m) m|a − b.

Proof. a≡ b (mod m) h₁, h₂∈ Z a = mh₁+ r b = mh₂+ r 0≤ r < m. a− b = m(h1− h2) m|a − b.

a, b m 數 r1 r2, h1, h₂∈ Z a = mh1+ r₁

b = mh2+ r2, 0≤ r1, r2< m, a− b = m(h1− h2) + (r1− r2). m|a − b m|r1− r2. 0≤ r1, r₂< m, −m < r1− r2< m, m|r1− r2 r₁= r₂.

Lemma 3.1.3 congruent relation equivalent relation.

Proposition 3.1.4. 數 m, 數 m equivalent

relation. .

(1) a∈ Z a≡ a (mod m).

(2) a≡ b (mod m) b≡ a (mod m).

(3) a≡ b (mod m) b≡ c (mod m), a≡ c (mod m).

Proof. (1) a∈ Z, a− a = 0, m|a − a. Lemma 3.1.3 a≡ a (mod m).

(2) a≡ b (mod m) Lemma 3.1.3 m|a − b, m|b − a b≡ a (mod m).

(3) a≡ b (mod m) b≡ a (mod m), m|a−b m|b−c. m|(a−b)+(b−c),

m|a − c. a≡ c (mod m).

論 ,

“a b modulo m ” : a b m 數 .

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數 , m∈ N, modulo m

? 數 m 數 0, 1, . . . , m− 1, m .

,

, .

Deﬁnition 3.1.5. 數 m, S m , modulo m

, S complete residue system modulo m.

S complete residue system modulo m, 數 modulo m

equivalent relation, S , S

, Z modulo m m , S 數 m ,

S . 言 , S Z modulo m

. {0,1,...,m − 1} complete residue system modulo m.

complete residue system modulo m.

Question 3.1. m∈ N.

(1) S⊆ Z S 數 m. a∈ Z s∈ S a≡ s

(mod m). S complete residue system modulo m.

(2) S⊆ Z S complete residue system modulo m a∈ Z

s∈ S a≡ s (mod m).

equivalent relation , .

. modulo m ,

m 數 .

Lemma 3.1.6. 數 m, a≡ b (mod m), gcd(a, m) = gcd(b, m).

Proof. a≡ b (mod m), a b m 數 , r.

Lemma 1.3.1 gcd(a, m) = gcd(r, m) = gcd(b, m).

a m , modulo m a m .

S complete residue system modulo m, S m ,

m . modulo m

m ? S ={0,1,...,m − 1} complete residue system modulo m !

S m 數 Euler ϕ-function ϕ(m) , 數 modulo m

ϕ(m) m . ϕ(m)

, .

Deﬁnition 3.1.7. 數 m, S ϕ(m) , m

modulo m , S reduced residue system modulo m.

m 數 p ,{1,..., p − 1} reduced residue system modulo p.

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Question 3.2. m∈ N.

(1) S⊆ Z S 數 ϕ(m). gcd(a, m) = 1 數 a,

s∈ S a≡ s (mod m). S reduced residue system modulo m.

(2) S⊆ Z S reduced residue system modulo m S

m gcd(a, m) = 1 數 a, s∈ S a≡ s

(mod m).

3.2.

, 數 (

).

m∈ N, modulo m (

), ?

, .

? modulo 5

, 5 數 2 數 3 . 數 2 3

2 3 , 2 + 3 = 5 2×3 = 6

0 1 . ? 2 3

7 −12 , 7 + (−12) = −5 7× (−12) = −84,

5 0 , 5 1 , 前 .

, .

Lemma 3.2.1. m∈ N, a, b∈ Z a≡ b (mod m), c∈ Z a + c≡ b + c (mod m) and ac ≡ bc (mod m).

Proof. a≡ b (mod m) m|a − b. m|(a + c) − (b + c), a + c≡ b + c

(mod m). m|(a − b)c m|ac − bc, ac≡ bc (mod m)..

Lemma 3.2.1 數數數 . 數

數數 . 數 ( )

數 .

Proposition 3.2.2. m∈ N, a, b, c, d∈ Z a≡ b (mod m) c≡ d (mod m), a + c≡ b + d (mod m) and ac ≡ bd (mod m).

Proof. a≡ b (mod m), Lemma 3.2.1 a + c ≡ b + c (mod m). c≡ d (mod m) b + c≡ b + d (mod m), equivalent relation ( Proposition 3.1.4(3)) a + c≡ b + d (mod m).

, a≡ b (mod m) c≡ d (mod m) ac≡ bc (mod m) bc≡ bd (mod m),

ac≡ bd (mod m).

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, 1752 388 5 數,

5 數 . 1752≡ 2 (mod 5) 388≡ 3 (mod 5)

1752× 388 ≡ 6 ≡ 1 (mod 5).

Proposition 3.1.4 ( congruence relation equivalent relation) m∈ N

“≡” . Lemma 3.2.1 c =−1, a≡ b (mod m)

−a ≡ −b (mod m). Proposition 3.2.2 ≡ “ ”

( ) 數 , , .

5742 11 數 , 5742 = 5×10³+ 7×10²+ 4×10+2. 10≡ −1 (mod 11) 5742≡ 5×(−1)³+ 7×(−1)²+ 4×(−1)+2 ≡ −5+7−4+2 ≡ 0 (mod 11).

5742 11 , 11 數 . 9

數 10≡ 1 (mod 9) . 10≡ 3 (mod 7) 7

數 ( ).

: , modulo 數 ,

≡ . a = 3, a≡ 3 (mod 5) a≡ 3 (mod 7),

a²≡ 3²≡ 4 (mod 5) a²≡ 4 (mod 7). ( )

congruence . a̸= 0 ab = ac, b = c; congruence

. a = 2, b = 2, c = 5 modulo 6 a̸≡ 0 (mod 6) ab≡ ac (mod 6), b̸≡ c (mod 6). congruence

數 . , .

Proposition 3.2.3. m∈ N a, b, c∈ Z. d = gcd(m, a) ab≡ ac (mod m) b≡ c (mod m/d).

Proof. d = gcd(m, a), m = m^′d a = a^′d, Corollary 1.1.8 gcd(m^′, a^′) = 1.

ab≡ ac (mod m), m|ab−ac. Lemma 1.1.5(2) (m/d)|(a/d)(b−c), m^′|a^′(b−c). gcd(m^′, a^′) = 1 Proposition 1.2.6(1) m^′|b−c, b≡ c (mod m/d).

, b≡ c (mod m/d), m^′|b − c. Lemma 1.1.5(1) dm^′|d(b − c), m|d(b − c). db≡ dc (mod m). Lemma 3.2.1 a^′db≡ a^′dc (mod m),

ab≡ ac (mod m).

前 , m = 6 a = 2, gcd(m, a) = 2. ab≡ ac (mod 6) b≡ c (mod 3). , b = 2, c = 5, 2≡ 5 (mod 3).

a modulo m congruence ? Proposition 3.2.3

gcd(m, a) = 1, m a . .

Corollary 3.2.4. m∈ N a, b, c∈ Z. m a , ab≡ ac (mod m) b≡ c (mod m).

數 , a̸= 0 ab = ac a b = c, “ ”

, 數 a̸= 0 b̸= 0 ab̸= 0 . congruence

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, 2̸≡ 0 (mod 6) 3̸≡ 0 (mod 6) 2× 3 ≡ 0 (mod 6).

congruence . 數 , a̸= 0,

數 a⁻¹ a· a⁻¹= 1, ab = bc, a⁻¹, b = c.

“ ” a. 數 0 a, ( a⁻¹)

, 數 . 數 ±1

數, 論 congruence .

Proposition 3.2.5. m∈ N, a∈ Z, b∈ Z ab≡ 1 (mod m)

a m .

Proof. b∈ Z ab≡ 1 (mod m), m|ab − 1. d = gcd(m, a), d|m d|ab.

m|ab − 1 d|m d|ab − 1, d|ab d|1. a m .

, a m , gcd(m, a) = 1, Corollary 1.2.4 r, s∈ Z

mr + as = 1. b = s, m|ab − 1, ab≡ 1 (mod m).

, a m 數 b ab≡ 1 (mod m),

b modulo m . c∈ Z ac≡ 1 (mod m),

ab≡ 1 ≡ ac (mod m) gcd(m, a) = 1, Corollary 3.2.4 b≡ c (mod m).

, b a modulo m .

3.3. Euler’s Theorem

, . m∈ N, a ∈ Z gcd(a, m) =

1 , b∈ Z ab≡ 1 (mod m) . Proposition 3.2.5

mx + ay = 1 數 b, m a

數 . Euler’s Theorem m, a b .

m∈ N, a, b∈ Z ab≡ 1 (mod m), Proposition 3.2.5 a b m

. 言 , m 數 , reduced residue system

modulo m.

Lemma 3.3.1. m∈ N, a∈ Z gcd(m, a) = 1. {r1, . . . , r_ϕ_(m)} reduced residue system modulo m, {ar1, . . . , ar_ϕ_(m)} reduced residue system modulo m.

Proof. ,{r1, . . . , r_ϕ_(m)} reduced residue system modulo m gcd(m, r_i) = 1 i̸= j, ri̸≡ rj (mod m). {ar1, . . . , ar_ϕ_(m)} reduced residue system modulo m, gcd(m, ar_i) = 1 i̸= j ar_i̸≡ arj (mod m).

gcd(m, ar_i)̸= 1, 數 p p|m p|ari. p 數, Lemma 1.4.2 p|a p|ri. 言 , p m, a 數 m, ri 數. gcd(m, a) = 1 gcd(m, ri) = 1 , gcd(m, ari) = 1.

, i̸= j ari≡ arj (mod m), gcd(m, a) = 1, Corollary 3.2.4 r_i≡ rj (mod m). r_i̸≡ rj (mod m) , ar_i̸≡ arj (mod m).

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前 , m∈ N, m , m 數 ϕ(m)

. reduced residue system modulo m.

S ={a1, . . . , a_ϕ_(m)} T ={b1, . . . , b_ϕ_(m)} reduced residue system modulo m,

a_i∈ S, m , T a_i

. 言 , bj∈ T ai ≡ bj (mod m). bj ,

S T modulo m . ,

ai≡ bi (mod m). a1···a_ϕ(m)≡ b1···b_ϕ(m) (mod m).

Euler’s Theorem.

Theorem 3.3.2 (Euler’s Theorem). m∈ N, a∈ Z gcd(m, a) = 1, a^ϕ^(m)≡ 1 (mod m).

Proof. S ={r1, . . . .r_ϕ_(m)} reduced residue system modulo m.

gcd(m, r₁···r_ϕ(m)) = 1. gcd(m, r₁···r_ϕ(m))̸= 1, 數 p p|m p|r1···r_ϕ(m). Corollary 1.4.3 ri∈ S p|ri, gcd(m, ri)̸= 1. S reduced residue system modulo m r_i∈ S , gcd(m, r₁···r_ϕ(m)) = 1.

gcd(m, a) = 1, Lemma 3.3.1 {ar1, . . . , ar_ϕ_(m)} reduced residue system modulo m,

r1···r_ϕ(m)≡ (ar1)···(ar_ϕ(m))≡ a^ϕ^(m)(r₁···r_ϕ(m)) (mod m).

gcd(m, r1···r_ϕ(m)) = 1, Corollary 3.2.4 a^ϕ^(m)≡ 1 (mod m). m∈ N a∈ Z gcd(m, a) = 1, b = a^ϕ^(m)⁻¹, Euler’s Theorem

ab≡ a^ϕ^(m)≡ 1 (mod m). a modulo m .

Corollary 3.3.3. m∈ N, a∈ Z gcd(m, a) = 1, b = a^ϕ^(m)⁻¹, ab≡ ba ≡ 1 (mod m).

, m 數 p , Euler’s Theorem Fermat’s Little Theorem.

.

Theorem 3.3.4 (Fermat’s Little Theorem). 數 p, a∈ Z p- a, a^p⁻¹≡ 1 (mod p).

, b = a^p⁻², ab≡ ba ≡ 1 (mod p).

Proof. p 數, p- a gcd(p, a) = 1. ϕ(p) = p − 1,

Theorem 3.3.2 a^p−1≡ 1 (mod p).

p|a Ferma’s Little Theorem , a≡ 0 (mod p), a^p⁻¹≡ 0 (mod p).

數 a .

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Corollary 3.3.5. 數 p, 數 a a^p≡ a (mod p).

Proof. p 數 a∈ Z, p|a p- a . p|a

, a≡ 0 (mod p), a^p≡ 0 ≡ a (mod p). p- a , Theorem 3.3.4 a^p−1≡ 1

(mod p), a a^p≡ a (mod p).

3.4. Wilson’s Theorem

p 數 , p- a, Fermat’s Little Theorem a^p−2 modulo p

a . a modulo p , Wilson’s Theorem

modulo p a .

m∈ N, m 數 a, Proposition 3.2.5 m

數 b ab≡ 1 (mod m), b , modulo m

. m b 數 .

modulo m modulo m reduced residue system

.

Lemma 3.4.1. m∈ N, S ={r1, . . . , r_ϕ_(m)} reduced residue system modulo

m. r_i∈ S r_j∈ S r_ir_j≡ 1 (mod m).

Proof. S reduced residue system modulo m, S s_i m

, Proposition 3.2.5 b∈ Z r_ib≡ 1 (mod m). b m ,

S reduced residue system modulo m rj∈ S b modulo m , b≡ rj (mod m). Lemma 3.1.3 , r_ir_j≡ rib≡ 1 (mod m).

.

, rj, r_k∈ S rirj≡ 1 (mod m) rirk≡ 1 (mod m).

rirj≡ rir_k (mod m). gcd(m, ri) = 1, Corollary 3.2.4 rj≡ rk (mod m). S

reduced residue system modulo m S modulo m ,

rj≡ rk (mod m) rj= rk. .

S ={1,2,3,4,5,6,7,8,9,10} reduced residue system modulo 11, modulo 11

1× 1 ≡ 2 × 6 ≡ 3 × 4 ≡ 5 × 9 ≡ 7 × 8 ≡ 10 × 10 ≡ 1 (mod 11).

, S 1 10 , modulo

數 .

Lemma 3.4.2. 數 p. a∈ Z a²≡ 1 (mod p) a≡ ±1 (mod p).

Proof. a≡ ±1 (mod p), a²≡ (±1)² (mod p). a²≡ 1 (mod p).

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, a²≡ 1 (mod p), p|a²− 1, p|(a − 1)(a + 1), p 數, Lemma 1.4.2 p|a − 1 p|a + 1. a≡ 1 (mod p) a≡ −1 (mod p).

Lemma 3.4.2 modulo 數 , modulo 15

1 14 , 4 4²≡ 1 (mod 15), 4̸≡ ±1 mod 15.

Lemma 3.4.2, 數 , Wilson’s Theorem.

Theorem 3.4.3 (Wilson’s Theorem). 數 p. {r1, . . . , r_p−1} reduced residue system modulo p.

r₁···rp−1≡ −1 (mod p).

,

(p− 1)! ≡ −1 (mod p).

Proof. p = 2, modulo 2 reduced residue system {r1} , r₁≡ 1 (mod 2). modulo 2 1≡ −1 (mod 2), r1≡ −1 (mod 2).

p > 2 , S ={r1, . . . , r_p₋₁} gcd(p, 1) = gcd(p,−1) = 1 1̸≡ −1 (mod p) ( p|2), ri, rj ∈ S ri ̸= rj ri ≡ 1 (mod p) rj ≡ −1

(mod p). , r₁≡ 1 (mod p) r₂≡ −1 (mod p). r_i∈ S,

3≤ i ≤ p−1. Lemma 3.4.1 rj∈ S rirj≡ 1 (mod p). ri̸≡ ±1 (mod p), r_j̸≡ ±1 (mod p), 3≤ j ≤ p−1. r_i= r_j, r²_i ≡ 1 (mod p),

Lemma 3.4.2 , i̸= j. T ={r3, . . . , r_p₋₁} ri

rj∈ T rirj≡ 1 (mod p). T p− 3

( p 數), p 1. r₃···rp−1≡ 1

(mod p).

r1r2r3···rp−1≡ r1r2≡ −1 (mod p).

{1,2,..., p − 1} modulo p reduced residue system, 1× 2 × ··· × (p − 1) = (p − 1)! ≡ −1 (mod p).

p 數 a p 數, Wilson’s Theorem modulo

p , a . a≡ ±1 (mod p) a²≡ 1 (mod p), a

modulo p , 論 a̸≡ ±1 (mod p) .

Corollary 3.4.4. 數 p a∈ Z p- a. a≡ i (mod p), 2≤ i ≤ p−2.

b =(p− 2)!

i ab≡ 1 (mod p).

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Proof. 2≤ i ≤ p − 2, b 數.

ab≡ i(p− 2)!

i ≡ (p − 2)! (mod p) (p− 1)! = (p − 1) · (p − 2)! p− 1 ≡ −1 (mod p),

ab≡ (p − 2)! ≡ −((p − 1)!) ≡ 1 (mod p).

Lemma 3.4.1 m∈ N , Lemma 3.4.2

數 , Wilson’s Theorem modulo m .

{r1, . . . , r_ϕ_(m)} reduced residue system modulo m, r1···r_ϕ(m)≡ −1

(mod m). modulo 15 4 −4 4²≡ (−4)²≡ 1 (mod 15),

Theorem 3.4.3 ( ) , {r1, . . . , r₈} reduced

residue system modulo 15, r1···r8≡ 1 (mod 15). Theorem 3.4.3

Wilson’s Theorem m , modulo m reduced

residue system{r1, . . . , r_ϕ_(m)} r²_i ≡ 1 (mod m) ri , 論 , .