Congruence Equations

基礎數論

數

數論 基 數 數 ,

數 . 基 , 數論

. 數論 ( ) ,

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

2006). .

v

Chapter 4

Congruence Equations

modulo m “≡” “=” ,

. congruence equation. , 論 數

congruence equation. , congruence equation , 論

congruence equation.

4.1. Congruence Equation

數 f (x) ( f (x) = cnxⁿ+··· + c1x + c0, ci∈ Z), f (x)

數 數, x 數 a , f (a) 數. m∈ N,

數 a f (a)≡ 0 (mod m) ( m| f (a)). 數 congruence

equation.

f (x) = cnxⁿ+··· + c1x + c0, ci ∈ Z. m∈ N, a ∈ Z f (x)≡ 0 (mod m) , f (a)≡ 0 (mod m). b≡ a (mod m), Proposition 3.2.2 ,

i∈ N bⁱ≡ aⁱ (mod m). Proposition cibⁱ≡ ciaⁱ (mod m), f (b)≡ f (a) (mod m). , x = a f (x)≡ 0 (mod m) 數 ,

b∈ Z b≡ a (mod m), x = b f (x)≡ 0 (mod m) . x = a f (x)≡ 0

(mod m) 數 , x≡ a (mod m) f (x)≡ 0 (mod m) .

modulo m a 數 f (x)≡ 0 (mod m) .

modulo m , 數

. f (x)≡ 0 (mod m) , modulo m ,

f (x)≡ 0 (mod m) 數 , modulo m

f (x)≡ 0 (mod m), 數 .

, modulo m complete residue system S, S

f (x) , f (x)≡ 0 (mod m),

. m . 論,

45

congruence equation . , congruence

equation modulo m 數 m.

, 前 congruence equation . modulo m

a∈ Z ax≡ 1 (mod m) ( ax−1 ≡ 0 (mod m))

congruence equation. Proposition 3.2.5 a m , congruence equation

. Proposition 3.2.3, a m congruence equation

modulo m .

Lemma 3.4.2 論 p 數 x²≡ 1 (mod p) . Lemma 3.4.2

p 數 , x≡ 1 (mod p) x≡ −1 (mod p). m

數 , x≡ ±1 (mod m) x²≡ 1 (mod m) congruence equation

, congruence equation . x²≡ 1 (mod 15) x≡ ±1

(mod 15) x≡ ±4 (mod 15) 4 . n n

, .

n 數 n 數

, 數 . 數 1

數 , . ,

.

Lemma 4.1.1. f (x) n (n≥ 1) 數 a∈ Z. n− 1

數 h(x) r∈ Z

f (x) = (x− a)h(x) + r.

Proof. f (x) 數 n 數 . f (x) 1 , f (x) = c₁x + c₀, h(x) = c₁ r = ac₁+ c₀, (x− a)h(x) + r = f (x).

數 , 數 n < k 數 g(x), n− 1 數

h0(x) r0∈ Z g(x) = (x− a)h0(x) + r0. f (x) 數 n = k ,

f (x) = c_kx^k+ c_k−1x^k−1+···+c1x + c₀, c_i∈ Z c_k̸= 0. g(x) = f (x)−(x−a)ckx^k−1, g(x) = (ck−1+ c_ka)x^k−1+···c1x + c0 數 k 數 .

數 k− 1 數 h0(x) r0∈ Z g(x) = (x− a)h0(x) + r0. f (x) = (x− a)ckx^k−1+ (x− a)h0(x) + r₀. h(x) = c_kx^k−1+ h₀(x) r = r₀,

h(x) 數 k− 1 數 r∈ Z f (x) = (x− a)h(x) + r. 

Lemma 4.1.1, p 數 modulo p n

congruence equation n . congruence equation 數

.

Definition 4.1.2. f (x) = cnxⁿ+··· + c1x + c0 數 , m∈ N.

(1) m- cn, f (x) modulo m 數 (degree) n .

4.1. Congruence Equation 47

(2) m- cr m|ci, for r < i≤ n, f (x) modulo m 數 r .

數 g(x) modulo m 數 n, g(x)≡ 0 (mod m)

n congruence equation.

f (x) modulo m 數 n 數 ,

f (x) 數 n . 數 n 數 g(x) (

f (x) m ) 數 a, f (a)≡ g(a) (mod m).

f (x)≡ 0 (mod m) g(x)≡ 0 (mod m) . congruence equation

, 論 n congruence equation f (x)≡ 0 (mod m) , ,

f (x) 數 n.

Theorem 4.1.3 (Lagrange). 數 p 數 f (x). modulo p

f (x)≡ 0 (mod p) 數 n , f (x)≡ 0 (mod p) modulo p

n .

Proof. , f (x) = cnxⁿ+··· + c1x + c0, p- cn. n . f (x) = c1x + c0 數 , x≡ a (mod p) f (x)≡ 0 (mod p)

. x≡ b (mod p) , c₁a + c₀≡ c1b + c₀ (mod p).

gcd(p, c1) = 1, Lemma 3.2.4 a≡ b (mod p). n = 1 .

n < k n congruence equation n . n = k

. x≡ a (mod p) f (x)≡ 0 (mod p) , Lemma 4.1.1 數

k− 1 數 h(x) r∈ Z f (x) = (x− a)h(x) + r. x≡ a (mod p) f (x)≡ 0 (mod p) , f (a)≡ 0 (mod p), a f (a) = r≡ 0 (mod p).

x≡ b (mod p) , f (b) = (b−a)h(b)+r (b−a)h(b) ≡ 0 (mod p).

言 , b̸≡ a (mod p), p- (b−a), Lemma 1.4.2 , p|h(b), x≡ b (mod p) h(x)≡ 0 (mod p) . k congruence equation f (x)≡ 0 (mod p) x≡ a (mod p) h(x)≡ 0 (mod p) . h(x)≡ 0 (mod p) 數 k

congruence equation, k− 1 , f (x)≡ 0 (mod p)

k . 

, congruence equation f (x)≡ 0 (mod m)

, x≡ a (mod m) .

modulo 數 . x²≡ 1 (mod 8), 數 ,

x≡ 1 (mod 2) .

數 modulo 數. x²≡ 1 (mod 8)

modulo 8 x≡ 1,3,5,7 (mod 8), 4 , modulo 2 .

4.2.

congruence equation ,

.

f (x) = anxⁿ+··· + a1x + a0, ai∈ Z, m∈ N 數. 論 f (x)≡ 0 (mod m) congruence equation.

: d a_n, . . . , a₁, a₀ m 數.

ai m an= a^′_nd, . . . , a1= a^′₁d, a0= a^′₀d m = m^′d, a^′_i∈ Z m^′ ∈ N.

g(x) = a^′_nxⁿ+···a^′₁x + a^′₀, f (x)≡ 0 (mod m) g(x)≡ 0 (mod m^′) congruence equation .

Proposition 4.2.1. m∈ N f (x) = anxⁿ+··· + a1x + a0, ai ∈ Z. d a_n, . . . , a₁, a₀ m 數 a_n= a^′_nd, . . . , a₁= a^′₁d, a₀ = a^′₀d m = m^′d.

g(x) = a^′_nxⁿ+··· + a^′₁x + a^′₀.

x≡ c (mod m^′) g(x)≡ 0 (mod m^′) , t∈ Z, x ≡ c + m^′t (mod m) f (x)≡ 0 (mod m) . , g(x)≡ 0 (mod m^′) , f (x)≡ 0 (mod m) .

Proof. x≡ c (mod m^′) g(x)≡ 0 (mod m^′) , m^′|a^′ncⁿ+··· + a^′₁c + a^′₀. m^′d|a^′_ndcⁿ+··· + a^′₁dc + a^′₀d, m|ancⁿ+···a1c + a0. x≡ c (mod m) f (x)≡ 0 (mod m) .

t∈ Z c^′= c + m^′t. c≡ c^′ (mod m^′), x≡ c^′ (mod m^′) g(x)≡ 0 (mod m^′) . 論 c^′= c + m^′t , x≡ c+m^′t (mod m)

f (x)≡ 0 (mod m) . t∈ Z, x ≡ c + m^′t (mod m) f (x)≡ 0

(mod m) .

, x≡ c (mod m) f (x)≡ 0 (mod m) , m|ancⁿ+··· + a1c + a₀, m^′|a^′_ncⁿ+··· + a^′₁c + a^′₀. x≡ c (mod m^′) g(x)≡ 0 (mod m^′) .

g(x)≡ 0 (mod m^′) , f (x)≡ 0 (mod m) . 

Proposition 4.2.1 , x≡ c (mod m^′) g(x)≡ 0 (mod m^′) , t∈ Z, x ≡ c + m^′t (mod m) f (x)≡ 0 (mod m) .

modulo m , . t≡ t^′ (mod d), d|t − t^′,

dm^′|m^′(t−t^′). c + m^′t≡ c + m^′t^′ (mod m). x≡ c + m^′t (mod m)

0≤ t ≤ d −1, . , modulo m^′ g(x)≡ 0 (mod m^′) ,

f (x)≡ 0 (mod m) modulo m d . f (x)≡ 0 (mod m)

g(x)≡ 0 (mod m^′) , .

Corollary 4.2.2. m∈ N f (x) = anxⁿ+··· + a1x + a0, ai ∈ Z. d a_n, . . . , a₁, a₀ m 數 a_n= a^′_nd, . . . , a₁= a^′₁d, a₀= a^′₀d m = m^′d. g(x) =

4.2. 49

a^′_nxⁿ+··· + a^′₁x + a^′₀. g(x)≡ 0 (mod m^′) modulo m^′ k , congruence equation f (x)≡ 0 (mod m) modulo m kd .

Proposition 4.2.1 modulo m congruence equation modulo

m^′ congruence equation. modulo m^′ 數 ,

. an, . . . , a1, a0 m , modulo

. , .

Lemma 4.2.3. m∈ N 數 f (x). m^′|m f (x)≡ 0 (mod m^′) , f (x)≡ 0 (mod m) .

Proof. f (x)≡ 0 (mod m) x≡ c (mod m) , m| f (c). m^′|m, m^′| f (c), x≡ c (mod m^′) f (x)≡ 0 (mod m^′) . f (x)≡ 0 (mod m^′)

, f (x)≡ 0 (mod m) . 

Lemma 4.2.3 Proposition 4.2.1 Proposition 4.2.1 數

數 modulo m^′ , modulo m ,

Lemma 4.2.3 , modulo m^′

modulo m . modulo m^′ modulo m

, . m 數 congruence

equations, . .

m 數 , m = pⁿ₁¹··· pⁿr^r, p_i 數. i = 1, . . . , r, f (x)≡ 0 (mod pⁿ_iⁱ) ,

.

Proposition 4.2.4. m = pⁿ₁¹··· pⁿr^r, pi 數 f (x) 數 . i∈ {1,...,r}, f (x)≡ 0 (mod pⁿ_iⁱ) , f (x)≡ 0 (mod m) .

, x≡ c (mod m) f (x)≡ 0 (mod m) i∈ {1,...,r}, x ≡ c (mod pⁿ_iⁱ) f (x)≡ 0 (mod pⁿ_iⁱ) .

Proof. , pⁿ_iⁱ|m, Lemma 4.2.3 , f (x)≡ 0 (mod pⁿ_iⁱ) , f (x)≡ 0 (mod m) .

x≡ c (mod m) f (x)≡ 0 (mod m) , m| f (c),

i∈ {1,...,r} pⁿ_iⁱ|m, pⁿ_iⁱ| f (c). i∈ {1,...,r}, x ≡ c (mod pⁿ_iⁱ) f (x)≡ 0 (mod pⁿ_iⁱ) .

, i∈ {1,...,r}, x ≡ c (mod pⁿ_iⁱ) f (x)≡ 0 (mod pⁿ_iⁱ) . pⁿ_iⁱ| f (c).

pⁿ_iⁱ , Proposition 1.2.6(2) pⁿ₁¹··· pⁿr^r| f (c), m| f (c).

x≡ c (mod m) f (x)≡ 0 (mod m) . 

Proposition 4.2.4 , pi f (x)≡ 0 (mod pⁿ_iⁱ) , f (x)≡ 0

(mod m) . p_i, f (x)≡ 0 (mod pⁿ_iⁱ) , f (x)≡ 0

(mod m) ? . p_i ,

數 modulo pⁿ_iⁱ ,

Proposition 4.2.4 f (x)≡ 0 (mod m) . .

4.3. Congruence Equations

congruence equation, congruence equation.

數 .

m∈ N modulo m congruence equation ax≡ b (mod m) congruence equation, a, b∈ Z m- a.

congruence equation .

Proposition 4.3.1. m∈ N. congruence equation ax≡ b (mod m), m- a. d = gcd(m, a). d|b ax≡ b (mod m) .

Proof. d = gcd(m, a), d| m, congruence equation ax≡ b (mod d).

d| a, modulo d ax≡ 0x (mod d). d- b, b̸≡0 (mod d), congruence equation ax≡ b (mod d) ( 0x≡ b (mod d)) . Lemma 4.2.3 ax≡ b (mod m) .

, d|b, d = gcd(d, b) = gcd(gcd(m, a), b). a = a^′d, b = b^′d, m = m^′d.

Proposition 4.2.1 ax≡ b (mod m) a^′x≡ b^′ (mod m^′) .

gcd(a, m) = d gcd(a^′, m^′) = 1, Proposition 3.2.5 e∈ Z a^′e≡ 1 (mod m^′).

a^′x≡ b^′ (mod m^′) e

x≡ a^′ex≡ b^′e (mod m^′).

x≡ b^′e (mod m^′) a^′x≡ b^′ (mod m^′) , Proposition 4.2.1

x≡ b^′e (mod m) ax≡ b (mod m) . 

Proposition 4.3.1 , a^′x≡ b^′ (mod m^′) modulo m^′

. , gcd(a^′, m^′) = 1, a^′x≡ b^′ (mod m^′) modulo m^′ . Lemma 4.3.2. m∈ N. congruence equation ax≡ b (mod m).

gcd(a, m) = 1, ax≡ b (mod m) modulo m .

Proof. x≡ c (mod m) x≡ c^′ (mod m) ax≡ b (mod m) , ac≡ b≡ ac^′ (mod m) m| a(c − c^′). gcd(m, a) = 1, m| c − c^′ (Proposition 1.2.6),

c≡ c^′ (mod m). 

Lemma 4.3.2 congruence equation ax≡ b (mod m) ,

modulo m 數.

4.3. Congruence Equations 51

Proposition 4.3.3. m∈ N. congruence equation ax≡ b (mod m).

d = gcd(m, a) d| b, ax≡ b (mod m) modulo m d . , x≡ c

(mod m/d) (a/d)x≡ (b/d) (mod m/d) , ax≡ b (mod m) modulo m

x = c + tm

d, t = 0, 1, . . . , d− 1.

Proof. d|b, d = gcd(d, b) = gcd(gcd(m, a), b). a = a^′d, b = b^′d, m = m^′d.

gcd(a^′, m^′) = 1, Lemma 4.3.2 a^′x≡ b^′ (mod m^′) modulo m^′ . x≡ c (mod m^′) , Proposition 4.2.1 ax≡ b (mod m) x = c + tm^′

t∈ Z. Corollary 4.2.2 modulo m ax≡ b (mod m) d ,

x = c + t(m/d), t = 0, 1, . . . , d− 1. 

, Proposition 4.3.1 Proposition 4.3.3 .

Theorem 4.3.4. m∈ N, a,b ∈ Z congruence equation ax≡ b (mod m).

d = gcd(m, a).

(1) d- b, ax≡ b (mod m) .

(2) d- b, ax≡ b (mod m), modulo m d . x≡ c (mod m)

,

x≡ c +m

dt, t = 0, 1, . . . , d− 1

ax≡ b (mod m) modulo m .

, a m , b∈ Z, ax ≡ b (mod m) , modulo m

.

Example 4.3.5. 16x≡ 8 (mod 52). gcd(52, 16) = 4 4|8, congruence

equation , modulo 28 4 .

4x≡ 2 (mod 13). 4× 10 ≡ 1 (mod 13), x≡ 2 × 10 ≡ 7 (mod 13) 4x≡ 2 (mod 13) . x≡ 7 (mod 52) 16x≡ 8 (mod 52)

( 16× 7 = 112 = 52 × 2 + 8).

, 52/4 = 13 Theorem 4.3.4 modulo 52 x≡ 7,20,33,46 (mod 52) 16x≡ 8 (mod 52) .

, Theorem 4.3.4 ax≡ b (mod m) ,

modulo m . d = gcd(a, m)| b.

ax + my = b 數 x = r, y = s. ar≡ b (mod m) x≡ r

(mod m) ax≡ b (mod m) .

4.4. Chinese Remainder Theorem

m = pⁿ₁¹··· pⁿ_r^r pi 數 f (x) 數 . Proposition 4.2.4 i∈ {1,...,r}, f (x) ≡ 0 (mod pⁿ_iⁱ) , f (x)≡ 0 (mod m)

. ? (Chinese Remainder Theorem)

f (x)≡ 0 (mod pⁿ_iⁱ) , .

Theorem 4.4.1 (Chinese Remainder Theorem). m1, . . . , mr∈ N mi

( i̸= j , gcd(m_i, m_j) = 1). c₁, . . . , c_r∈ Z 數 c

c≡ ci (mod mi),∀i ∈ {1,...,r}.

Proof. , M = m1···mr i∈ {1,...,r}, Mi= M/mi.

Mj mi : (1) i̸= j, mi|Mj. (2) gcd(Mi, mi) = 1.

(1) M_j , (2) ( m_i ),

gcd(M1, m₁) = 1. M1, m₁ , 數 p p|M1 p|m1. M1= m2···mr, Corollary 1.4.3 i∈ {2,...,r} p|mi. i̸= 1, gcd(m₁, m_i) = 1, p|m1 p|mi m₁, m_i , gcd(M₁, m₁) = 1.

t₁, . . . ,t_r∈ Z i∈ {1,...,r}, t = c₁M₁t₁+··· + crM_rt_r

t≡ ci (mod m_i). t1, . . . ,t_r∈ Z i∈ {1,...,r}, (1) ( mi|Mj for i̸= j) t≡ ciMiti (mod mi). ti ∈ Z ciMiti ≡ ci

(mod m_i) . (2) ( gcd(M_i, m_i) = 1) Proposition 3.2.5 e_i ∈ Z Miei ≡ 1 (mod mi), ti= ei, t≡ ciMiei≡ ci (mod mi). i∈ {1,...,r}, e_i M_ie_i ≡ 1 (mod mi), c = c₁M₁e₁+··· + crM_re_r,

c≡ ci (mod m_i),∀i ∈ {1,...,r}. 

! mi , c1, . . . , cr 數 c

c≡ ci (mod m_i) i∈ {1,...,r} . m₁= 4, m₂= 6 c₁= 1,

c2= 2, 數 c c≡ 1 (mod 4) c≡ 2 (mod 6). c≡ 1

(mod 4) c 4k + 1 , 數. c≡ 2 (mod 6), c 6k + 2 ,

數. 數 數 數.











x≡ c1 (mod m₁) x≡ c2 (mod m₂)

... ... x≡ cr (mod m_r)

4.4. Chinese Remainder Theorem 53

. r

. Theorem 4.4.1 , 數 t₁, . . .t_r , r

r ti , . .

Example 4.4.2. m1= 3, m₂= 4, m₃= 5 c1= 2, c₂= 1, c₃= 3 數 c c≡ ci (mod mi),∀i ∈ {1,2,3}. c





c≡ 2 (mod 3) c≡ 1 (mod 4) c≡ 3 (mod 5)

Theorem 4.4.1 M₁= 20, M₂= 15 M₃= 12.

e1∈ Z M1e1≡ 1 (mod m1), 20e1≡ 1 (mod 3), 2e1≡ 1 (mod 3).

e1= 2. e2, e3 15e2 ≡ 1 (mod 4) ( 3e2≡ 1 (mod 4)) 12e₃≡ 1 (mod 5) ( 2e₃≡ 1 (mod 5)). e₂= 3 e₃= 3 . c = 2× 20 × 2 + 1 × 15 × 3 + 3 × 12 × 3 = 233 233≡ 2 (mod 3), 233 ≡ 1 (mod 4) 233≡ 3 (mod 5).

前 , m∈ N, m = pⁿ₁¹··· pⁿr^r, p_i 數. f (x)

數 , f (x)≡ 0 (mod m), pi f (x)≡ 0 (mod pⁿ_iⁱ).

pi f (x)≡ 0 (mod pⁿ_iⁱ) , Proposition 4.2.4 f (x)≡ 0 (mod m) . p_i f (x)≡ 0 (mod pⁿ_iⁱ), Proposition 4.2.4 ,











f (x)≡ 0 (mod pⁿ₁¹) f (x)≡ 0 (mod pⁿ₂²)

... ... f (x)≡ 0 (mod pⁿ_r^r)

f (x)≡ 0 (mod m) . ,

, .

Corollary 4.4.3. m = pⁿ₁¹··· pⁿr^r, p_i 數 f (x) 數 . i∈ {1,...,r}, f (x) ≡ 0 (mod pⁿ_iⁱ) f (x)≡ 0 (mod m) .

Proof. Proposition 4.2.4 , f (x)≡ 0 (mod m) , i∈ {1,...,r}, f (x) ≡ 0 (mod pⁿ_iⁱ) .

i∈ {1,...,r}, f (x) ≡ 0 (mod pⁿ_iⁱ) x≡ ci (mod pⁿ_iⁱ) .

pⁿ_iⁱ Theorem 4.4.1 , c∈ Z i∈ {1,...,r}

c≡ ci (mod pⁿ_iⁱ). i∈ {1,...,r}, x ≡ c (mod pⁿ_iⁱ) f (x)≡ 0 (mod pⁿ_iⁱ) . Proposition 4.2.4 x≡ c (mod m) f (x)≡ 0 (mod m) . 

. 數 ,

, .

Example 4.4.4. x² ≡ 1 (mod 15). 前 x²≡ 1 (mod 3) x²≡ 1 (mod 5) . 3 5 數, Lemma 3.4.2 x≡ ±1 (mod 3)

x≡ ±1 (mod 5) x²≡ 1 (mod 3) x²≡ 1 (mod 5) . congruence equation:

(1)

{ x≡ 1 (mod 3) x≡ 1 (mod 5) , (2)

{ x≡ −1 (mod 3) x≡ −1 (mod 5) , (3)

{ x≡ −1 (mod 3) x≡ 1 (mod 5) , (4)

{ x≡ 1 (mod 3) x≡ −1 (mod 5) .

(1) (2) 數 1 −1 (1) (2). 11

(3), 4 (4). Proposition 4.2.4 x≡ 1,−1,11,4 (mod 15) x²≡ 1

(mod 15) . x²≡ 1 (mod 15) modulo 15 4 ,

4 . modulo 15 4 .

x²≡ 1 (mod 15) modulo 15 4

4 , . Theorem

4.4.1 , . ,

? , ,

.

Theorem 4.4.5. m1, . . . , m_r∈ N mi . M = m1···mr, c1, . . . , cr∈ Z congruence equation











x≡ c1 (mod m1) x≡ c2 (mod m₂)

... ... x≡ cr (mod mr)

modulo M . c∈ Z congruence equation,

c^′∈ Z c^′≡ c (mod M) congruence equation.

Proof. Theorem 4.4.1 , modulo m₁···mr .

c, c^′∈ Z congruence equation. i∈ {1,...,r}

c≡ ci (mod m_i) c^′≡ ci (mod m_i). i∈ {1,...,r} mi|c − c^′. mi , Proposition 1.2.10(2), m1···mr|c − c^′, c≡ c^′ (mod M).

modulo M .

, c congruence equation c^′ ∈ Z c^′≡ c (mod M), i∈ {1,...,r}, mi|M, c^′≡ c ≡ ci (mod mi). c^′ congruence

equation. 

Example 4.4.2 , x = 233





x≡ 2 (mod 3) x≡ 1 (mod 4) x≡ 3 (mod 5)

4.4. Chinese Remainder Theorem 55

congruence equation, Theorem 4.4.5 數 c c≡ 233 ≡ 53

(mod 60) congruence equation. c≡ 53 (mod 60)

數 congruence equation.

Theorem 4.4.5 Theorem 4.4.1 . Theorem 4.4.1

, Theorem 4.4.5 modulo m₁···mr ,

. , 論 Theorem 4.4.5

Chinese remainder theorem.

.