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- ゆみか いまいだ
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1 3 m = [n, n1, n 2,..., n r, 2n] p q = [n, n 1, n 2,..., n r ] p 2 mq 2 = ±1
2
3 T T : r = : r = : r i
4 ii Pell Pell Pell Pell Pell Pell Pell I Conrad II A B C D Pell E Pell
5 1 (2 2 ) continued fractions Hardy-Wright continued fractions expansion continued fractions expansion 1
6 Pell Pell 5 2 Pell 2 Pell 2017 Saradha[15] 2014 Dummit[16] 2016 Lahn-Spiegel[17] Dummit Pell x 2 my 2 = ±4 2 Pell x 2 my 2 = ±1 Pell 3 6 Pell x 2 my 2 = ±d Conrad Conrad
7 3 Dirichlet-Dedekind Hardy-Wright An Introduction to the Theory of Numbers Sierpinski Elementary Theory of Numbers 4 Dirichlet-Dedekind Pell Dirichlet-Dedekind Hardy-Wright Hardy-Wright Gauss ( H(...) ) Sierpinski 10 5 Lamé E Fibonacci
8 a b a b a b a b gcd(a, b,...) a, b,... [x] Gauss (0 x [x] < 1) M M ( ) (M ) N Z Q 3... ( ) (a, b, c,...) {... } (curly bracket) [a, b, c,...]
9 5 Pell Python Python Python Python arisawa@aichi-u.ac.jp R
10 Chapter ( ) = = = = = ( ) a b c a b b c a = a d, b = b d a = nb + c (a nb )d = c c d d b c ( ) 1 E 6
11 ( ) : x 0 = n 0 x 1 + x 2... (1.1) x l 1 = n l 1 x l + x l+1 x l = n l x l x 1 1 x 0 x 2, x 3,... x 1 > x 2 > x 3 > > x l+1 > 0 0 x l+2 = 0 n 1,..., n l 1 n 0 x 1,..., x l 1 x l+1 1 x l+1 x 0 = 11349, x 1 = 6201, l = 4, x 5 = / = = = = = ( ) Hardy-Wright 11349/6201 = [1, 1, 4, 1, 8] (1.1) x 0 /x 1 = [n 0, n 1,..., n l ] n 0, n 1,..., n l Hardy-Wright: the partial quotients, or simply the quotients, of the continued fraction
12 8 1.2 [n 1, n 2,..., n l ] (1.1) xk x k+1 = n k + x k+2 x k+1 (k = 1, 2,..., l 1) x l x l+1 = n l x k /x k+1 = [n k,..., n l ] 1 [n k,..., n l ] = n k + (k = 1, 2,..., l 1) [n k+1,..., n l ] [n l ] = n l (1.2) 1. [1, 1, 4, 1, 8] [8] = 8, [1, 8] = = 9 8, [4, 1, 8] = = 44 9, [1, 4, 1, 8] = = 53 44, [1, 1, 4, 1, 8] = = [1, 1, 4, 1, 8] 11349/ / / ( 1, 1, 4, 1, 8) , 1, 4, 1, 8 97/53
13 [1, 1, 4, 1, 8] 11349/6201 = 97/53 p q d pd qd x n 97/ x n : x n x 1 1 x 2 1 x 3 4 x 4 1 x : x 6 = 1 x 1 x 2 x 1 x 5 x 5, x 4,..., x x 5,..., x 1 1 (1.2) [n k,..., n l ] = n k + 1/[n k+1,..., n l ] (1.1) x 5, x 4,..., x 1 x 6 = 1 x 5 = n 5 = 8 x 4 = = 9 x 3 = = 44, x 2 = = 53, x 1 = = 97 x k n k, n k+1,..., n l
14 10 x k = H(n k, n k+1,..., n l ) 4 H 2. H() = 1, H(n l ) = n l H(n k,..., n l ) = n k H(n k+1,..., n l ) + H(n k+2,..., n l ) (k = l 1,..., 1) (1.3) H(a) = a, H(a, b) = ab + 1, H(a, b, c) = abc + a + c, H(a, b, c, d) = abcd + ab + cd + ad + 1 x 1 = H(n 1, n 2,..., n l ), x 2 = H(n 2,..., n l ) H : [n 1, n 2,..., n l ] = H(n 1, n 2,..., n l ) H(n 2,..., n l ) H H(n 1, n 2,..., n l 1, n l ) = H(n l, n l 1,..., n 2, n 1 ) (1.4) H(n 1, n 2,..., n l 1, n l ) = n l H(n 1, n 2,..., n l 1 ) + H(n 1, n 2,..., n l 2 ) (1.5) C H(n 1,..., n l ) H(n 1,..., n l 1 ) H(n 2,..., n l ) H(n 2,..., n l 1 ) = ( 1)l p l, q l, p l 1, q l 1 ( ) ( ) pl p l 1 H(n1,..., n l ) H(n 1,..., n l 1 ) = H(n 2,..., n l ) H(n 2,..., n l 1 ) q l q l 1 p l q l 1 p l 1 q l = ( 1) l gcd(p l, q l ) = 1 gcd(p l 1, q l 1 ) = 4 Gauss [n k, n k+1,..., n l ] Dirichlet-Dedekind Gauss Hardy-Wright Gauss Hardy-Wright Gauss H(n k, n k+1,..., n l ) Gauss
15 p l /q l = [n 1,..., n l ], p l 1 /q l 1 = [n 1,..., n l 1 ] p l, q l, p l 1, q l 1 H [n 1,..., n l ] [n 1,..., n l 1 ]? p l /q l = [n 1,..., n l ], p l 1 /q l 1 = [n 1,..., n l 1 ] p l /q l p l 1 /q l 1 H 1. p l /q l p l 1 /q l 1 p l /q l = [n 1, n 2,..., n l 1, n l ], p l 1 /q l 1 = [n 1, n 2,..., n l 1 ] ( ) p l q l 1 q l p l 1 = ( 1) l 1 1 [n 1, n 2,..., n l ] p/q p/q Hardy-Wright 3. [1, 1, 4, 1, 8] = 1 + 1/[1, 4, 1, 8] = 97/53, [1, 1, 4, 1] = 11/ = 1 px qy = ±1 97x 53y = ±1 (x, y) = (6, 11) p > q
16 12 p/q [n 1, n 2,..., n l 1, n l ] y/x = [n 1, n 2,..., n l 1 ] p < q (p, q) p > q > 0 p/q [n 1, n 2,..., n l ] p k /q k = [n 1, n 2,..., n k ] p l /q l = p/q p 1 /q 1, p 2 /q 2,..., p l /q l p/q? 1 p k q k p k 1 q k 1 = ( 1)k q k 1 q k (1.6) k p k /q k p k 1 /q k 1 : p 1 p 2 p 3 p 4 q 1 q 2 q 3 q 4 p k = H(n 1, n 2,..., n k ) = n kh(n 1, n 2,..., n k 1 ) + H(n 1, n 2,..., n k 2 ) q k H(n 2,..., n k ) n k H(n 2,..., n k 1 ) + H(n 2,..., n k 2 ) = n kp k 1 + p k 2 n k q k 1 + q k 2 (1.7) n k > 0 p k /q k p k 1 /q k 1 p k 2 /q k 2 p 1 q 1 < p 3 q 3 < p 5 q 5 <, p 2 q 2 > p 4 q 4 > p 6 q 6 > p/q 4. p/q = [1, 1, 4, 1, 8] p 1, p 2, p 3, p 4, p 5 = 1 q 1 q 2 q 3 q 4 q 5 1, 2 1, 9 5, 11 6, :
17 (1.7) [1] = 1 1, [1, 1] = , [1, 1, 4] = = 9 5 [1, 1, 4, 1] = = , [1, 1, 4, 1, 8] = = [n 1, n 2,..., n l + 1 n l+1 ] = [n 1, n 2,..., [n l, n l+1 ]] = [n 1, n 2,..., n l, n l+1 ] 5 n l+1 = 1 [n 1, n 2,..., n l + 1] = [n 1, n 2,..., n l, 1] 2 [1, 2, 3] = [1, 2, 2, 1] 1.3 ω (ω > 1) ω = [n 1, n 2, n 3,...] n k ( ) (ω 1 n 1 )ω 2 = 1, (ω 2 n 2 )ω 3 = 1, (ω 3 n 3 )ω 4 = 1, ω 1 = ω n k n k ω k < n k+1 ω k 1 ω k 0 < ω k n k < 1 ω k > 1 n k 1 ω k ω = [n 1, n 2, n 3,..., n k, ω k+1 ] n 1, n 2, n 3,..., n k ω k ω (ω > 1) [n 1, n 2, n 3,...] 5 C 6 p.131
18 14 p k /q k = [n 1, n 2, n 3,..., n k ] ω p k q k < ω : 1 q k q k+1 lim ω p k = 0 k q k : ω = [n 1, n 2, n 3,..., n k, ω k+1 ] ω = H(n 1, n 2, n 3,..., n k, ω k+1 ) H(n 2, n 3,..., n k, ω k+1 ) ( ) ( ) ( ) pk p k 1 H(n1, n 2, n 3,..., n k ) H(n 1, n 2, n 3,..., n k 1 ) = H(n 2, n 2, n 3,..., n k ) H(n 2, n 3,..., n k 1 ) q k q k 1 ω = p kω k+1 + p k 1 q k ω k+1 + q k 1 ω k+1 > 0 ω p k /q k p k 1 /q k 1 (1.6) ω p k 1 q k 1 < 1 q k q k 1 k k + 1 q k (p k ) k k q k q k+1
19 Chapter θ θ 0 = θ 1 θ k = θ k 1 n k 1 (k = 1, 2, 3,...) (2.1) θ k n k n k n k θ k < n k + 1 n k = [θ k ] θ k = 0 n 0, n 1, n 2,... θ 1 θ Hardy-Wright θ = [n 0, n 1, n 2,...] 2 n 1, n 2,... n k > 0 (k > 0) n 0 n 0 0 Hardy-Wright [2] 2 15
20 16 [n 0, n 1,..., n k, n k+1, n k+2,..., n k+r ] 3 Gauss 1 [x, ], 4 (a) θ (b) θ (c) θ 2 5 (a),(b) (c) (2.1) n k = [θ k ], (θ k n k )θ k+1 = 1 (k = 0, 1, 2,...) (2.2) θ k θ k+1 θ 0 (2.2) θ k, n k (k 1) θ k > 1, n k > 0, 0 < θ k n k < 1, : k 0 0 < θ k n k < 1 n k = [θ k ] (2.2) (θ k n k )θ k+1 = 1 θ k+1 > 1 k 1 2 (2.2) 2 θ k m + bk θ k = (2.3) a k 2 a k, b k (2.2) m + bk m + bk+1 ( n k )( ) = 1 (2.4) a k a k+1 3 Hardy-Wright Lagrange
21 b k+1 = n k a k b k b k+1 b k+1 < m n k b k+1 (2.4) m b 2 k+1 a k a k+1 = 1 a k+1 a k+1 = (m b 2 k+1 )/a k k ( 1) : 0 < b k < m + bk m bk m, > 1, 0 < < 1, a k (m b 2 a k a k) (2.5) k 0 < b < m b m b 2 7 b = 1, 2 m b m b (2.4) a k, b k, n k (k = 1, 2,...) : ( 2)( ) = ( 1)( ) = ( 1)( ) = ( 1)( ) = ( 4)( ) = = [2, 1, 1, 1, 4] n k = [( m + b k )/a k ] n k = [(n 0 + b k )/a k ]
22 18 : n k ( m + b k )/a k < n k + 1 n k a k m + b k < (n k + 1)a k n 0 = [ m] n 0 m < n n 0 + b k m + b k < n 0 + b k + 1 n k a k < n 0 + b k + 1 n k a k n 0 +b k n 0 +b k < (n k +1)a k n k (n 0 +b k )/a k < n k +1 n k = [(n 0 + b k )/a k ] n 0 = [ m], a 0 = 1, b 0 = 0 (2.6) b k = n k 1 a k 1 b k 1, a k = m b2 k a k 1, n k = [ n 0 + b k a k ] (k = 1, 2,...) (2.7) m > b k > 0 (k = 1, 2,...) a k 1 (m b 2 k ) (k = 2, 2,...) ξ ξ [ξ] ξ [ξ] : (min): b Z m + b ( ) min m b ( ) a n a [ ] m + b n =, b = na b a m b 0 < ( ) < 1 a b
23 : (inv): b Z m b ( ) inv m + b ( a a ) (2.8) aa = m b 2 1 m b m + b ( )( a a ) = 1 m b m + b 0 < ( ) < 1 = ( a a ) > 1 b m = ( ) min 7 2 ( ) inv ( ) ( ) min 7 1 ( ) inv ( ) ( ) min 7 1 ( ) inv ( ) ( ) min 7 2 ( ) inv ( ) ( ) min 7 2 ( ) inv ( ) m + b : S: S(m) (b Z, a N) S(m) a m > b > 0 and a (m b 2 )
24 20 m (b, a) S(m) S(m) S(m) S S(5) : S S + : S = {ξ S; ξ < 1} S + = {ξ S; ξ > 1} 1. m = 5 S S + S S S S + 1 : f: f 7 1 f f S S k f k ξ 0 = m [ m] f k (k = 0, 1, 2,...) (S ) f k (ξ 0 ) = f k (ξ 0 ) k k m m ξ 0 ξ 0 = f l (ξ 0 ) l 6 6 p.205
25 S + S ξ S ξ 1 S + ξ S + ξ 1 S : ξ S ξ = ( m b)/a m b m + b ( )( a a ) = 1 aa = m b 2 S a ξ 1 S + : b > < 1, > 1, ( )( ) = ξ S +, ξ = ξ [ξ] ξ S ξ = ( m + b)/a, ξ = ( m b )/a 0 < ξ < 1 (a) 0 < b < m (b) a (m b 2 ) (c) ( m + b )/a > 1 : n = [ξ] ξ > 1 n 1 b = na b (b) m b 2 = m (na b) 2 = m b 2 na(na 2b) a (m b 2 ) a (m b 2 ) S + a (m b 2 ) (c) m + b a = m + na b a = m b + (n 1)a > 0 n 1 S + m > b > b (a) 0 < ξ < 1 (c) m b m + b 0 < < 1 < a a m > b > 0
26 22 : ξ = ( 5 + 1)/4 ξ < 1 ξ S ξ = ξ [ξ] ξ S + ξ = ( 5 1)/4 ( 5 + 2)/1, ( 5 + 1)/1, ( 5 1)/1 S + ( 5 2)/1 3. ( m b)/a S ( m + b)/a > 1 m + b m b m + b = + n and > 1 and m > b (2.9) a a a n b (a) b > 0 (b) ( m b )/a < 1 : n n n = 1 0 < ( m b)/a m b m + (a b) m + b 1 < + 1 = = a a a b = a b ( m + b)/a > 1 m > a b = b m > b = na b b m < b + a (b) (a) (2.9) (b) ( m b )/a < 1 < ( m + b )/a : (2.9) n 1 (m, b, a) = (12, 3, 1) n = 4, 5, 6 (2.9) : : ( m + b)/a S b + a > m ( m + b)/a 1 ( m + b)/a S and ( m + b)/a + 1 S
27 b + a > m ( m b)/a < 1 4. ( m b)/a S ( m + b)/a ( m + b)/a > 1 ( m + b)/a ( ( m b)/a < 1 ) : ( m b)/a < 1 < ( m + b)/a b > 0 0 < ( m b)/a m > b m b m + b m + b 1 = ( )( a a ) = ( a m + b 1 < 0 < a m b )( a ) m b a < 1 m < b + a : ξ : ξ = ( m + b)/a S ξ = ( m b)/a b : T T + : T = {ξ S ; ξ > 1} T + = {ξ S + ; ξ < 1} 3 T S + ( 2) T + S ( 3) 4 T T + ξ T ξ = ( m b)/a b > 0 ( 5 1)/4 T 5. ( m b)/a T ( m + b)/a T + 0 < b < m a (m b 2 ) 0 < ( m b)/a < 1, ( m + b)/a > 1
28 24 : 2 3 T T + 6. inv (a) T T + (b) T + min T : (a) T inv T + aa = m b 2 m b m + b m b m + b ( )( a a ) = ( a )( ) = 1 a T ( m b)/a T ( m b)/a < 1, ( m + b)/a > 1 ( m + b)/a > 1, ( m b)/a < 1 ( m + b)/a T + T + inv T (b) ξ T = f(ξ) T f T T : f(ξ) ξ 6 f T T 1. ξ T ξ = f l (ξ) l 1 : f T T T f k (ξ) = f k (ξ) (k < k ) k k 7 f f k (ξ) = f k (ξ) f k 1 (ξ) = f k 1 (ξ)
29 ξ = f 0 (ξ) = f k k (ξ) = f k k (ξ) l = k k ξ T 1 ξ = f l (ξ) l f ξ θ 0 = ξ n k = [θ k ], ξ k = θ k n k, θ k+1 = 1/ξ k (k = 0, 1, 2,...) 0 < ξ < 1 n 0 = [θ 0 ] = [ξ] = 0 ξ 0 = θ 0 θ 1 = 1/ξ 0 θ l = 1/ξ l 1, θ l+1 = 1/ξ l ξ l = ξ 0 θ l+1 = θ 1 n l+1 = n l ξ = [0, n 1, n 2,..., n l ] θ 0 = m n 0 = [ m], ξ = θ n 0 ξ T m = [n0, n 1, n 2,..., n l ] m
30 b a m ( m + b)/a ( m + b)/a (b, a) (b, a) b a T T f (b, a) (b < 0) ( b, (m b 2 )/a) (b, a) (b > 0) ( b, a) b 0 < b = na b < m f T T + g T = T T + g T ξ T g ξ g T T g 1 g (m=5) T 2 : 5 2 ( ) inv ( ) min 5 2 ( ) ( ) inv ( ) min 5 1 ( ) (m=6) 1 T : 6 2 ( ) inv ( ) min 6 2 ( ) inv ( ) min 6 2 ( ) S ( 7 1)/6 ( 7 + 1)/6 < ( 8 1)/7
31 m = 5 2.1: : m = m = 6 2.2: : m = m = 7 2.3: : m = m = 8 2.4: : m = 8
32 ( 5 + 1)/3 > 1, > 0 3 (5 1 2 ) ( ) = ( 3( 5 + 1) ) ) = ( ) ( ) ( )/9 T + (45) ) ( ) min 45 6) ( ) inv ( ) min min 45 3 ( ) inv ( ) min 45 5 ( inv ( ) min 45 3 ( ) inv ( ) ( 45 6 ) inv ( ) ) min 45 5 ( ) ) inv ( ( ) min ( ) = ( 2( 2 + 2) ) = ( ) inv ( 4 8 ) min ( 3 8 ) inv ( ) min 8 2 ( ) T (8) ( ) min ( ) = ( 5( 2 + 2) ) = ( ) inv ( ) min ( 8 50 ) inv ( ) min 50 6 ( ) T (50)
33 ( ) min ( ) inv ( min 3 2 ( ) T (3) 4 1 ) min 2.5 ( θ (2.2) θ 0 = θ ) inv n k = [θ k ], (θ k n k )θ k+1 = 1 (k = 0, 1, 2,...) ( ) 1 η k = θ k [θ k ] 0 < η k < 1 η 0, η 1, η 2,... T θ = (b ± m)/a a m b m m b 2 a (m b 2 ) a (m b 2 ) (m b 2 )/a = d/c d c a = ac, b = bc, m = mc 2 (b ± m)/a = (bc ± mc 2 )/(ac) = (b ± m )/a (m b 2 )/a = (c 2 (m b 2 ))/(ac) = d a (m b 2 ) d/c 1. θ = (a, b, m) = (6, 5, 11) b 2 m = = /6 = 7/ = η = η 0 0 < η < 1 a aa = m b 2 b b a ( ) n = [θ ] 1, b = b na b < b
34 30 2.1: TYPE η m b 2 θ = η 1 η = θ [θ ] η TYPE b + m (A) m > b 2 b + m b + m a a a T b + m (B) m < b 2 b m b m a a a (D) b m (C) m > b 2 a b m (D) m < b 2 b + m b + m a a a (A) or (B) TYPE (A) (b + m)/a S (S ) T θ S + ( 1) η T ( 2 T ) TYPE (B) η > 0 m < b 2 b > 0 b > m 0 < η < 1 b > m η TYPE(D) TYPE (C) m > b 2 b m < 0 η > 0 TYPE (D) η > 0 b > 0 m < b 2 b > m η m b 2 2 m > b 2 η TYPE (A) T m < b 2 TYPE (B) TYPE (D) TYPE (B) TYPE (D) b 0 ± m b 1 m b 2 ± m a 0 a 1 a 2 b k (k = 0, 1, 2...) : b k > m b 0 > b 1 > b 2 >
35 η = = η = = TYPE(B) ( ) inv ( ( 7 45 ) inv ( T ) min ( 7 45 ) TYPE(D) 1 2 ( ) TYPE(A) 6 2 ) min
36 Chapter T : M : M M M T (m) T (m) S S(m) ( m + b)/a m a b b m > b > 0 a a (m b 2 ) [ m] S(m) = 2 d(m b 2 ) b=1 d(n) n n = 6 1, 2, 3, 6 d(6) = 4 2 b 32
37 3.1. T S S(m) = S (m) S + (m), S (m) S + (m) =, S (m) = S + (m) S (m) = [ m] b=1 d(m b 2 ) (3.1) ξ S ξ < T T T ξ S ξ < 1 ξ > 1 m b 2 = ac m b m + b m b m + b ( )( ) = ( )( ) = 1 a c c a ( m b)/a > 1 ( m + b)/c < 1 ( m b)/c < 1 ( m b)/c T b 0 = [ m] ( m b)/a < 1 b 0 b < a b = 1, 2,..., b 0 a a (m b 2 ) b 0 b < a and b 0 b < c c = (m b 2 )/a 1. m = 7: b 0 = 2, b = 1, 2 b = 1 m b 2 = 6 = ac = (1 6), 2 3, 3 2, (6 1) b = 2 m b 2 = 3 = ac = 1 3, 3 1 T (7) = 4 : (...)
38 34 2. m = 8: b 0 = 2, b = 1, 2 b = 1 m b 2 = 7 = ac = (1 7), (7 1) b = 2 m b 2 = 4 = ac = 1 4, 2 2, 4 1 T (8) = 3 : (...) 3. m = 13: b 0 = 3, b = 1, 2, 3 b = 1 m b 2 = 12 = ac = (1 12), (2 6), 3 4,... b = 2 m b 2 = 9 = ac = (1 9), 3 3,... b = 3 m b 2 = 4 = ac = 1 4, 2 2,... : a c T (13) = = 6 : (...) 4. m = 19: b 0 = 4, b = 1, 2, 3, 4 b = 1 m b 2 = 18 = ac = (1 18), (2 9), (3 6),... b = 2 m b 2 = 15 = ac = (1 15), 3 5,... b = 3 m b 2 = 10 = ac = (1 10), 2 5,... b = 4 m b 2 = 3 = ac = 1 3,... : a c T (19) = = 6 : (...) T S 3.1 S (m) T (m) m
39 3.1. T : S (m) T (m) m S (m) T (m) T (m) S (m) m 2 m T (m) / m m m S (m) T (m) 3.2 "ratio" 100 T (m) / S (m) m 0 3.2: S (m) T (m) m S (m) T (m) ratio(%)
40 T m l(m) T (m) T (m) θ k (2.1) θ 1, θ 2,... T (m) l(m) T (m) m 3.3 T (m) l(m) Y N n = [ m] m θ 1, θ 2,... T (m) 3.3: T m T (m) = l(m)? T (m) l(m) n 2 Y Y Y N Y Y Y Y Y Beceanu[12] l(m)/ m m m m log(m)? m log(m) l(m) ( m ) 1 l(m) 1 m log log(m)
41 3.2. T 37 Saradha[15] Beceanu l(m) (3.1) 2 l(m) S (m) T (m) 3.4 m Beceanu l(m)/ m m m T (m) l(m) 3.4 T (m) = l(m) m 3.4: T m T (m) = l(m)? T (m) l(m) n 2 Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Beceanu d(n) 1 n
42 Y Y Y Y Y Y Y Y Y Y Y Y Y N Y Y Y Y Y Y Y Y Balková-Hru sková[14] m l(m) l(m) 2m l(m) 2n ( m 1000 ) m = 1726 n = [ m] 3.4 m l(m) 2m l(m) 2m l(m) < m
43 3.2. T n d(n) 2[ n] : a 1, a 2,..., a l n n l [ n] a k a k b k = n b k b k n n d(n) = 2l 2[ n] d(n) = 2l 1 2[ n] 1 2. S (m) < m : S (m) = [ m] b=1 [ m] d(m b 2 ) 2 [ m] b=1 b=1 [ m b 2 ] 2 [ m] b=1 m b2 < π 4 m m b 2 m 1/4 1. m m : l(m) T (m) S (m) < m Hardy-Wright ( ) 3 3. lim n log d(n) log log n log n = log 2 d(n) ˆd(n) ˆd(n) = max(d(1), d(2),..., d(n)) ˆd(n) n 3 Hardy-Wright p.262 Ramanujan[8] p.44
44 40 3a. log lim ˆd(n) log log n = log 2 (3.2) n log n : Hardy-Wright 3 (1 + ϵ) log 2 log n log d(n) log log n d(n) ˆd(n) ( n ) n = P (P ) (1 ϵ) log 2 log n log d(n) > log log n d(n) ˆd(n) ( ˆd(n) d(n) ) (3.2) α n = (log 2)/(log log n) ˆd(n) lim n n = 1 αn α n n α n 0 ˆd(n) lim m S (m) = [ m] b=1 S (m) m ˆd(m) = lim m d(m b 2 ) < [ m] ˆd(m) < m ˆd(m) (3.3) lim m S (m) ˆd(m) m ˆd(m) m = lim S (m) αm m mm α m l(m) mm α m 1 4 (3.3) ˆd(n) d(n) lim ˆd(n) lim 5 Hickerson[9]
45 [n 0, n 1, n 2,...] : : ξ = x + y m (x, y Q) ξ x y m ξ 6 2. n 0 = [ m] 7 (a) m [n 0, n 1, n 2,..., n r 1, n r, 2n 0 ] (b) n k n 0 (k = 1, 2,..., r) (c) [n 0, n 1, n 2,..., n r 1, n r, 2n 0 ) = [n 0, n r, n r 1,..., n 2, n 1, 2n 0 ] (d) m = [n 0, n 1, n 2,..., n r 1, n r, 2n 0 ] m n 1, n 2,..., n r 1, n r m : (a) : θ 0 = m θ k (2.1) θ k = (θ k 1 [θ k 1 ]) 1 (k = 1, 2,...) n k = [θ k ], η k = θ k n k (k = 0, 1, 2,...) η 1 = m n 0 T (m) T (m) ( 2 : 1) η 1 = η l+1 l (> 0) l 6 2 Hardy-Wright (conjugate) 2 Hardy-Wright 2? 7 Sierpinski[4] (b) n k < 2n 0 Balková-Hru sková[14] (b) n k n 0 observation (d) Sierpinski[4] Kraïtchik[7] (c) Galois Theorem [13, 14]
46 42 η k = ( m b k )/a k 1 a 1, a 2,..., a l 1 1 a k = 1 (1 k < l) b k+1 = n k b k n k b k+1 n 0 b k+1 n 0 η k+1 = ( m n k+1 )/a k = ( m n 0 )/1 = η 1 l η l+1 = η 1 a 1, a 2,... 1 a l = 1, b 1 = n 0 b l = n 0 n l = 2n 0 ( m + b)/a n = ( m b )/a ( m + b)/a n = ( m b )/a ( m b)/a = ( m + b )/a n m b m + b ( )( a a ) = 1 m + b m b ( )( a a ) = 1 inv m + b min m b inv a n a inv m b a min n m + b a m + b a inv m b a min n min n θ l θ 1 θ 1 = (θ l n l ) 1 min m bl n l 1 min n l 1 a l 1 m + bl a l 1 inv inv m + bl a l m bl a l min nl min nl m b1 a l m + b1 a l inv m + b1 a 1 m b1 inv a l+k = a k, b l+k = b k, n l+k = n k a 1 min n1 (3.4) min n1 (3.5) θ = m η 0 = η l = ( m n 0 )/1 a l = 1, b 1 = n 0 (3.5) b l = b 1 = n 0, n l = 2n 0
47 (b) : m + bk m bk+1 n k = a k a k b k + b k+1 = n k a k 2.2 : 5 b k (k = 1, 2,..., r) 0 < b k < m 0 < b k n 0 b k + b k+1 2n 0 a k = 1 k = l n k = n l = 2n 0 a k 2 n k a k 2n 0 n k n 0 (c) : (3.4) ( m b 1 )/a l (3.5) ( m b l )/a l (3.4) n 1, n 2,... (3.5) n l 1, n l 2,... (d) : : 5 1
48 Chapter 4 Hardy-Wright [n 0, n 1,...] [n 0, n 1,..., n k, n k+1, n k+2,..., n k+r ] [n] Gauss [n, ] m m = [n, n1, n 2,..., n r, 2n] (4.1) n 1, n 2,..., n r m 4.1 : r = 0 r = 0 m = [n, 2n] m = [n, θ 1 ], θ 1 = [2n, θ 1 ] : ( m n)θ 1 = 1 (4.2) (θ 1 2n)θ 1 = 1 (4.3) 44
49 4.2. : r = 1 45 θ 1 ξ (4.2) (4.3) (m n 2 )ξ 2 2nξ 1 = 0 ξ 2 2nξ 1 = 0 2 m = n (n = 1, 2, 3,...) (4.4) m = 2, 5, 10, 17, : r = 1 r = 1 m = [n, c, 2n] m m = [n, θ 1 ], θ 1 = [c, 2n, θ 1 ] θ 1 = [c, θ 2 ], θ 2 = [2n, θ 1 ] : ( m n)θ 1 = 1 (4.5) (θ 1 c)θ 2 = 1 (θ 2 2n)θ 1 = 1 θ 1 ξ θ 2 = 2n + 1 ξ = 2nξ + 1, ξ = c + 1 ξ c(2nξ + 1) + ξ = c + = ξ θ 2 2nξ + 1 2nξ + 1 (4.5) 2 2nξ 2 2ncξ c = 0 (4.6) aξ 2 2nξ 1 = 0, a = m n 2 (4.7) ac = 2n (4.8) m = n 2 + a a = 1 m = [n, 2n] a 2 c c n m = [n, c, 2n] m c (2n) a = (2n)/c
50 46 m = n 2 + a c (a, n) c a (2n) (a, n) = (2k, ck) m = k(kc 2 + 2) c c = 2d (4.8) n = ad (a, n) = (k, kd) a 2 k 2 m = k(kd 2 + 1) m m = n 2 + a (a 2) a 2n c = (2n)/a [n, c, 2n] c c 3 k a, n, m c k 1 2 c k a n m m m = 3, 6, 8, 11, 12, 15, : r 2 (4.1) m = [n, ξ], ξ = [n1, n 2,..., n r, 2n] (4.9)
51 4.3. : r 2 47 m = n + 1 ξ (4.10) (m n 2 )ξ 2 2nξ 1 = 0 (4.11) a = m n 2 aξ 2 2nξ 1 = 0 (4.12) a n m = n 2 + a m n (4.9) ξ ξ = [n 1, n 2,..., n r, 2n, ξ] ξ = H(n 1, n 2,..., n r, 2n.ξ) H(n 2,..., n r, 2n, ξ) = H(n 1, n 2,..., n r, 2n)ξ + H(n 1, n 2,..., n r ) H(n 2, n 2,..., n r, 2n)ξ + H(n 2, n 2,..., n r ) ( C) ξ Aξ 2 Bξ C = 0 A = H(n 2, n 3,..., n r, 2n) B = H(n 1, n 2,..., n r, 2n) H(n 2, n 2,..., n r ) C = H(n 1, n 2,..., n r ) r 2 2n A = 2nH(n 2, n 3,..., n r ) + H(n 2, n 3,..., n r 1 ) B = 2nH(n 1, n 2,..., n r ) + H(n 1, n 2,..., n r 1 ) H(n 2, n 3,..., n r ) (4.13) C = H(n 1, n 2,..., n r ) H(n 1, n 2,..., n r 1 ) = H(n 2, n 3,..., n r )
52 48 : H H(n 2, n 3,..., n r ) = H(n r,..., n 3, n 2 ) n 1, n 2,..., n r 1, n r ( 3.3 : 2) H(n r,..., n 3, n 2 ) = H(n 1, n 2, n 3,..., n r 1 ) ξ Aξ 2 2nCξ C = 0 A = 2nH(n 2, n 3,..., n r ) + H(n 2, n 3,..., n r 1 ) C = H(n 1, n 2,..., n r ) (4.12) C acξ 2 2nCξ C = 0 ac = A ah(n 1, n 2,..., n r ) = 2nH(n 2, n 3,..., n r ) + H(n 2, n 3,..., n r 1 ) (4.14) a, n (4.14) p k = H(n 1, n 2,..., n k ), q k = H(n 2, n 3,..., n k ) (1 k r) (4.15) (4.14) ap r = 2nq r + q r 1 (4.16) x, y xp r yq r = q r 1 (4.17) 1. n 1, n 2,..., n r > 0 p k, q k > 0 : H p r H(n 1 ) > 0, H(n 1, n 2 ) > 0 H(n 1, n 2,..., n r ) = n r H(n 1, n 2,..., n r 1 ) + H(n 1, n 2,..., n r 2 )
53 4.3. : r 2 49 q r 2. n 1, n 2,..., n r 1 p r > p r 1, q r > q r 1 : p r p r 1 = H(n 1,..., n r ) H(n 1,..., n r 1 ) = H(n 1,..., n r 1 )(n r 1) + H(n 1,..., n r 2 ) > 0 q r q r 1 3. : p r > q r > q r 1 : n 1,..., n r p r = H(n 1,..., n r ) = n 1 H(n 2,..., n r ) + H(n 3,..., n r ) > H(n 2,..., n r ) = q r q r = H(n 2,..., n r ) = H(n 2,..., n r 1 )n r + H(n 2,..., n r 2 ) > H(n 2,..., n r 1 ) = q r 1 4. x, y xp r yq r = q r 1 : (a) x 0 x y (b) x q r y < p r (c) y 0 0 < x y (d) 0 y p r 0 < x q r : (a) x(p r q r ) + (x y)q r = q r 1 x 0 p r > q r (x y)q r q r 1 q r > q r 1 x y (b) (x q r )p r + (p r y)q r = q r 1
54 50 x q r (p r y)q r q r 1 q r > q r 1 p r > y (c) x > 0 (x y)p r + y(p r q r ) = q r 1 p r > q r (x y)p r q r 1 p r > q r 1 x y (d) x > 0 (x q r )p r + (p r y)q r = q r 1 p r y (x q r )p r q r 1 p r > q r 1 x q r (4.15) p k, q k 1 p k q k 1 q k p k 1 = ( 1) k (2 k r) (4.18) p k q k 2 q k p k 2 = ( 1) k+1 n k (3 k r) (4.19) n 1, n 2,..., n r : p k q k 1 q k p k 1 p r 1 = q r (4.20) (4.18) p r q r 1 q 2 r = ( 1) r (4.21) p r, q r, q r 1 1 Appendix C Appendix C Hadry-Wright p n = H(a 0, a 1,..., a n) p n = H(a 1,..., a n)
55 4.3. : r : 1 p r q r q r 1 x 0, y 0 (4.17) ( ) x 0, y 0 x 0 p r y 0 q r = q r 1 (4.22) (x x 0 )p r (y y 0 )q r = 0 (4.17) x = x 0 + kq r, y = y 0 + kp r (k Z) (4.21) q r 1 p r qr 1 2 q r 1 qr 2 = ( 1) r q r 1 x 0, y 0 (4.22) x 0 = ( 1) r qr 1, 2 y 0 = ( 1) r q r 1 q r (4.23) x = kq r + ( 1) r qr 1, 2 y = kp r + ( 1) r q r 1 q r (k Z) (4.24) a, n (4.16) x, y (a) x > 0, y > 0 (b) y (c) n max y/2
56 52 n max = max(n 1, n 2,..., n r ) a = x, n = y/2 q r 1 q r 4.1 p r y k q r 1 q r y k : p r k y p r k k y + y y 2n max k x, y x 1, y 1 p r, q r, q r 1 > 0 (4.17) x 1 a = kq r + x 1, 2n = kp r + y 1, (k 0) (4.25) y 1 p r k 1. n 1, n 2,..., n r (n 1, n 2,..., n r ) = (n r,..., n 2, n 1 ) p r, q r, q r 1 (4.15) (4.16) (a, n) q r 1 q r 0 (mod 2) (4.26) n a n = [ m], m = n 2 +a (n + 1) 2 > m 2n + 1 > a y x 4 (4.26) 4.1
57 4.3. : r : 2 p r q r q r 1 k / a, n ( ) (S1) p r, q r, q r 1 n max (S2) 0 < x < q r, 0 < y < p r x, y x ( 1) r qr 1 2 (mod q r ), y ( 1) r q r 1 q r (mod p r ) (S3) y < 2n max x q r y p r (S4) y x q r y p r (S5) y (S7) (S6) (S7) a = x, n = y/2 Balková-Hru sková[14] m a 3 (mod 4) 1. r a 3 (mod 4) 2 : r (4.24) x = kq r + qr 1, 2 y = kp r + q r 1 q r (k Z) 2 Pell 5.3 : 2
58 54 q r 1 q r 4.2 p r k q r 1 = 2k + 1 x = kq r + qr 1 2 = kq r + (2k + 1) (mod 4) a = x 3 (mod 4) q r 1 p r 4.2 k q r x a = x 3 (mod 4) q r 1 p r r 4.2 q r (4.21) p r q r 1 = qr q r = 2k r 2 r = 0 a = 1 1 (4.17) : xp r yq r = q r 1 x = kq r + ( 1) r qr 1, 2 y = kp r + ( 1) r q r 1 q r (k Z) x, y y ( 1) r q r 1 q r (mod p r ) (4.27) y > 0 y k k x 4 x 0 < x q r x x 0 < x < q r x ( 1) r qr 1 2 (mod q r ) (4.28) q r q r 1 x = 0 (4.27,4.28) x, y
59 4.3. : r 2 55 (4.28) x = kq r + ( 1) r qr 1 2 k (4.27) y = k p r + ( 1) r q r 1 q r k k = k k = [( 1) r q2 r 1 ], k = [( 1) r q r 1q r ] q r p r k = k [( 1) r q2 r 1 q r ] = [( 1) r q r 1q r p r ] [ q2 r 1 ] = [ q r 1q r ] (4.29) q r p r : ω n = [ω] n ω < n + 1 q p q p n = [ p ] n q nq < p < nq + q p = nq + r n r (0 < r < q) p n = [ p q ] p = n q + r r (0 < r < q) 0 = (n + n )q + (r + r ) 0 < r + r < 2q n + n = 1 [ p q ] + [ p ] = 1 q [ 5 3 ] = 1 [ 5 ] = 1 1 = 2 3 (4.29) n 1, n 2,..., n r (r 2) n 1, n 2,..., n r = 2, 3, 5 (p 3, q 3, q 2 ) = (37, 16, 3) [qr 1/q 2 r ] = [9/16] = 0 [(q r 1 q r )/p r ] = [48/37] = 1 (4.29) n 1, n 2,..., n r (4.29) (4.29)
60 n k (k = 1,..., r) p k q k p 0 = 1, p 1 = n 1, q 0 = 0, q 1 = 1, q 2 = n 2 p k = n k p k 1 + p k 2, q k = n k q k 1 + q k 2 (k = 2,..., r) q 0 = 0 p k q k (? ) k r 1 r n k * n 1 n 2 n 3... n r 1 n r p k 1 n 1??...?? q k 0 1 n 2?...?? * k n k * p k q k p r = 224, q r = 97, q r 1 = 42 q r q r 1 : x (mod 97), y (mod 224) n = 91, a = 79 m = = 8360
61 k n k * p k q k p r = 949, q r = 411, q r 1 = 178 q r q r 1 : x (mod 411), y (mod 949) y x, y q r, p r x = 448, y = 1034 n = 517, a = 448 m = = k n k * p k q k p r = 4, q r = 3, q r 1 = 2 q r q r 1 : x (mod 3), y (mod 4) n = y/2 = 1 n k 2 x, y q r, p r x = 5, y = 6 n = 3, a = 5 m = = 14
62 k n k * p k q k p r = 15, q r = 11, q r 1 = 8 q r q r 1 : r = 5 x (mod 11), y (mod 15) n = y/2 = 1 n k 2 x, y q r, p r x = 13, y = 17 y = 17 x, y q r, p r x = 24, y = 32 n = 16, a = 24 m = = m = [n, 1, 1, 1,..., 2n] n 1 = n 2 = = n r = 1 r = r = r xp r yq r = q r 1 y a = x, n = y/2, m = n 2 + a m p k = H(n 1, n 2,..., n k ) (0 k r), q k = H(n 2,..., n k ) (1 k r)
63 H() = 1, H(1) = 1 H(n 1, n 2,..., n k ) = n k H(n 1, n 2,..., n k 1 ) + H(n 1, n 2,..., n k 2 ) p 0 = 1, p 1 = 1 p k = p k 1 + p k 2 (k = 2, 3, 4,..., r) H(n 2,..., n k ) = n k H(n 2,..., n k 1 ) + H(n 2,..., n k 2 ) q 1 = 1, q 2 = 1 q k = q k 1 + q k 2 (k = 3, 4,..., r) p k q k Fibonacci Fibonacci F k (k = 1, 2, 3...) Fibonacci k 1 3 p k = F k+1, q k = F k m q r 1 q r = F r 1 F r 0 (mod 2) F r 1 F r F 3k+0, F 3k+1, F 3k+2 0, 1, 1 (mod 2) (k = 0, 1, 2, 3,...) r 2 (mod 3) m 4 p r m m = n 2 + a Fibonacci x = a, y = 2n x, y (4.24) x = kq r + ( 1) r qr 1, 2 y = kp r + ( 1) r q r 1 q r (k Z) 3 Hardy-Wright[3] Sierpinski[4] 4 [7] observation
64 60 x = kf r + ( 1) r Fr 1, 2 y = kf r+1 + ( 1) r F r 1 F r (k Z) (4.30) y x, y 5. F r+1 F r 1 Fr 2 = ( 1) r (4.31) F r+1 F r 2 F r F r 1 = ( 1) r+1 (4.32) : (4.18) : p k q k 1 q k p k 1 = ( 1) k (2 k r) p k q k 2 q k p k 2 = ( 1) k+1 n k (3 k r) n 1, n 2,..., n r q k = F k, p k 1 = F k k = r n r = 1 (4.30) k = ( 1) r+1 F r 2 x = ( 1) r ( F r 2 F r + Fr 1) 2 = ( 1) r { ( 1) r 1 } = 1 y = ( 1) r ( F r 2 F r+1 + F r 1 F r ) = ( 1) r { ( 1) r+1 } = 1 x, y y k = ( 1) r+1 F r 2 + k x = k F r + 1, y = k F r F r+1 y k F r k a = x, n = y/2 1. k = 1 r = 2 r = 3 m = 7 r = 4 m = 13 r = 5 r = 6 m = 58 r = 1
65 m = [n, c, c, c,..., 2n] n 1 = n 2 = = n r = c r = r = r xp r yq r = q r 1 y a = x, n = y/2, m = n 2 + a m p k = H(n 1, n 2,..., n k ) (0 k r), q k = H(n 2,..., n k ) (1 k r) H() = 1, H(c) = c H(n 1, n 2,..., n k ) = n k H(n 1, n 2,..., n k 1 ) + H(n 1, n 2,..., n k 2 ) p 0 = 1, p 1 = c p k = cp k 1 + p k 2 (k = 2, 3, 4,..., r) q k q 1 = 1, q 2 = c q k = cq k 1 + q k 2 (k = 3, 4,..., r) q k = p k 1 : q 1 = 1, q 2 = c, q 3 = c 2 +1, q 4 = c 3 +2c, q 5 = c 4 +3c 2 +1, q 6 = c 5 +4c 3 +3c q r 1 q r c q 1 q 2 1 (mod 2) q k q k 1 + q k 2 (mod 2) (k = 3, 4,...) q 1, q 2, q 3, q 4, q 5, q 6,... 1, 1, 0, 1, 1, 0,... (mod 2)
66 62 r 2 (mod 3) c q 1 1 q 2 0 (mod 2) q k q k 2 (mod 2) (k = 3, 4,...) q k 1 (mod 2) (k = 1, 3, 5,...) q k 0 (mod 2) (k = 2, 4, 6,...) r y x, y x = a, y = 2n x, y (4.24) x = kq r + ( 1) r qr 1, 2 y = kp r + ( 1) r q r 1 q r (k Z) k = ( 1) r+1 q r 2 x = ( 1) r ( q r 2 q r + qr 1) 2 y = ( 1) r ( q r 2 p r + q r 1 q r ) 6. q r q r 2 q 2 r 1 = ( 1) r+1 p r q r 2 q r q r 1 = ( 1) r+1 c : (4.18) : p k q k 1 q k p k 1 = ( 1) k (2 k r) p k q k 2 q k p k 2 = ( 1) k+1 n k (3 k r) n 1, n 2,..., n r k = r 1 p r 1 = q r, p r 2 = q r k = r p r 2 = q r 1, n k = c 2 x = 1, y = c c x, y a = 1, n = y/2 4.1 r = 0 c k = ( 1) r+1 q r 2 + k
67 x = k q r + 1, y = k p r + c k 1 c y k c p r y p r y c r m c r = 2 r = 3 r = 4 r = 5 r = 6 1 None 7 13 None None None None None ( ) : (x, y) (1, c) : xp r yq r = q r 1 q r 1 = H(n 2, n 3,..., n r 1 ) p r = n 1 q r + H(n 3, n 4,..., n r ) H(n 2, n 3,..., n r 1 ) = H(n 3, n 4,..., n r ) (x, y) = (1, n 1 ) n 1, n 2,..., n r c (x, y) = (1, c)
68 Chapter 5 Pell 5.1 Pell Pell m x 2 my 2 = ±1 (5.1) x, y 1 (x, y) = (1, 0) (5.1) +1 Pell 2 ±1 Pell ( 2 2a 4) 3 [2] Pell 1 ± 2 p.317 Pell Euler Fermat 3 2 Evan Dummit[16]! 64
69 5.1. Pell 65 (5.1) (x + y m)(x y m) = ±1 (5.2) 1 x, y (x + y m)(x y m) = = (x + y m) 2 (x y m) 2 = (x 2 + my 2 + 2xy m)(x 2 + my 2 2xy m) = (x 2 + my 2 ) 2 (2xy) 2 m (5.3) +1 (5.2) x, y (x + y m) n (x y m) n = ±1 n = 2 n > 1 n Pell x 2 my 2 = 1 (5.4) m 3 (mod 4) x 2 0, 1 (mod 4) y 2 0, 1 (mod 4) x 2 + y 2 0, 1, 2 (mod 4) (5.4) x 2 + y 2 3 (mod 4) m 0 (mod 4) (5.4) m m = n (n = 1, 2, 3,...) x = n, y = 1 m (5.4) B (m < 100) 2 (5.4) m = 41 2 (x, y) = (32, 5)
70 66 Pell 5.2 : Q( m) : {x + y m; x, y Q} Q( m) m Q( m) : : ξ = x + y m Q( m) ξ x y m ξ ξ 4 Q( m) : ξ 1 ξ 2 = ξ 1 ξ 2 : N(ξ): ξ = x + y m Q( m) ξ N(ξ) = ξξ = x 2 my 2 N(ξ) ξ : Z(ω): ω Z(ω) {x + yω; x, y Z} Pell (5.1) Z( m) 5.5 Pell (5.6) : : 2 2 ξ 2 + bξ + c = 0 (b, c Z) Hardy-Wright
71 : ω ω 2 Z(ω) ω ξ ξ = b ± D, D = b 2 4c 2 ξ = b ± m? 2 1. (a) ω = ω 2 2ω 1 = 0 2 (b) ω = ω 2 ω 1 = 0 2 ω = 1 + m 4ω 2 4ω (m 1) = 0 ω 2 m 1 (mod 4) 1. ω Z(ω) : Z(ω) ξ 1, ξ 2 Z(ω) ξ 1 = x 1 +y 1 ω, ξ 2 = x 2 +y 2 ω ξ = ξ 1 ξ 2 = x 1 x 2 + y 1 y 2 ω 2 + (x 1 y 2 + y 1 x 2 )ω ω 2 Z(ω) 2. ω Z(ω) : ρ Z(ω) ρ 2 Z(ρ) ω 2 = uω+v ρ = x+yω (ρ x) 2 = y 2 ω 2 = y 2 (uω + v) = uy(yω) + vy 2 = uy(ρ x) + vy 2 ρ 2 = 2xρ x 2 + uy(ρ x) + vy 2 Z(ρ)
72 68 Pell : Z ( m): ω m/4 for m 0 (mod 4) 1 + m ω = for m 1 (mod 4) (5.5) 2 m for m 2, 3 (mod 4) Z ( m) = Z(ω) 5 (a) ω (b) Z ( m) (c) ξ Z ( m) ξ Z ( m) : (a) (b) (c) m 1 (mod 4) m 1 (mod 4) ω = ω + 1 : Z( m) Z ( m) : : ϵ Z ( m) ϵϵ = ±1 Z ( m) ε Z( m) εε = ±1 Z( m) ±1 : ( ) 1 : U( m) U ( m): U( m) U ( m) Z( m) Z ( m) Z( m) Z ( m) U( m) U ( m) 3. U( m) U ( m) 1 : (5.5) ω ξ Z(ω) ξξ = e = ±1 ξ 1 = eξ Z(ω) 5 Hardy-Wright m 0 (mod 4) m
73 Z( m) ε = x + y m (x, y > 0) ε 1 = x y m x+y m x+y m > 1 x, y > 0 x + y m < 1 x, y Z ( m) Z( m) Pell (5.1) x 2 my 2 = ±4 (5.6) Pell x, y θ = (x + y m)/2 θθ = ±1 4. Pell x, y (x + y m)/2 Z ( m) : m 0 (mod 4) : m = 4m (5.6) x 2 4m y 2 = ±4 x x = 2x x 2 m y 2 = ±1 (x + y m)/2 = x + y m Z( m ) = Z ( m) x + y m m 1 (mod 4) : = x + y 1 + m x = 2 2 2x + y, y = y x 2 y 2 (mod 4) x, y 2x = x y x y x + y m Z ( m) 2 m 2, 3 (mod 4) : x 2 my 2 (mod 4) x, y x = 2x, y = 2y x + y m = x + y m Z( m) = Z ( m) 2 (5.1) Z( m) m 0 (mod 4) Z( m) N(ξ) = ±1 x 2 my 2 = ±1 Pell (5.1) Z( m) Pell (5.6) Z ( m)
74 70 Pell Z( 8) 1/(3 + 8) = 3 8 Z( 8) 1 (x, y) = (3, 1) Pell x 2 8y 2 = Z( 5) 1/(2 + 5) = 5 2 Z( 5) 1 (x, y) = (2, 1) Pell x 2 5y 2 = 1 4. ϵ = (1 + 5)/2 ϵϵ = 1 Z ( 5) ϵ 2 = ϵ 2 = (3 + 5)/2 ϵ 2 ϵ 2 = 1 Z ( 5) + ϵ 3 = ϵ 3 = ϵ 3 ϵ 3 = 1 Z ( 5) Z( 5) Z( 5) 5. θ = (1 + 10)/3 N(θ) = 1 3θ 2 2θ 3 = 0 : 2 2 : ξ aξ 2 + bξ + c = 0 a, b, c gcd(a, b, c) = 1 2 ξ 2 a > 0 6 ξ 2 D = b 2 4ac ξ 7 D b D 0 (mod 4) D b D 1 (mod 4) 5. X 2 ξ X (a),(b) (a) ξ ξ = ξ n (n Z) (b) ξ ξ = 1/ξ ξ 2 aξ 2 + bξ + c = 0 a ξ 2 + b ξ + c = 0 (1) gcd(a, b, c) = gcd(a, b, c ) 6 p.197. a > 0 Hardy-Wright a > 0 (p.205) Hardy-Wright 7 p.196
75 (2) b 2 4ac = b 2 4a c (3) b b (mod 2) : (b) (a) aξ 2 + bξ + c = a(ξ + n) 2 + b(ξ + n) + c = aξ 2 + (2an + b)ξ + (an 2 + bn + c) a = a, b = 2an + b, c = an 2 + bn + c b b (mod 2) gcd(a, b, c ) = gcd(a, 2an + b, an 2 + bn + c) = gcd(a, b, an 2 + bn + c) = gcd(a, b, c) = 1 ξ D ξ D D = (2an + b) 2 4a(an 2 + bn + c) = b 2 4ac = D : (3) b : (a) (b) (1) (2) (3) m D = 4m m 4m m = 4n + 1 (n N) ξ = ( m + 1)/2 ξ 2 ξ 2 ξ n = 0 ξ m ξ 2 2 a ξ 2 + b ξ + c = 0 gcd(a, b, c ) = 1 m b ξ ( m + b)/a b (= b ) a (= 2a )
76 72 Pell : : ω ω = (rω+s)/(tω+u) ω r, s, t, u ru st = ±1 e = ru st 8 r s ru st = e t u = e ( ) r s (rω + s)/(tω + u) t u ω = (rω + s)/(tω + ( u) ) ω = ( ( rω ) r s r s s)/( tω u) t u t u ω = (rω + s)/(tω + u), ω = (r ω + s )/(t ω + u ) ω = ((r r + s t)ω + r s + s u)/((t r + u t)ω + t s + u u) : ( ) ( ) ( ) r r + s t r s + s u r s r s = t r + u t t s + u u t u t u r r + s t r s + s u r s r s = t r + u t t s + u u t u t u = e e ω = (1ω ( + 0)/(0ω ) + 1) ω = ( 1ω + 0)/(0ω 1) 1 0 ± p.173.
77 ω = (rω + s)/(tω + u) ω = ( uω +s)/(tω ( r) ) ω ( = (uω s)/( tω ) +r) u s u s t r t r 2 ( ) 1 ( ) r s u s = e t u t r e : ω ω = 2, ω = (r, s, t, u) = (1, 1, 0, 1) (r, s, t, u) = (0, 1, 1, 1) e ( 2 + 1)/1 = 1/( 2 1) : : ( ) r s (rω + s)/(tω + u) ω t u ( ) 1 n 6. η = θ n 0 1 θ = 1/η ( ) 1 n 0 1 η θ θ ( ) η θ ( )( ) n θ 9 p.176
78 74 Pell θ ( n ) θ θ θ ( ) 1 ( ) ( ) ( ) 0 1 n 1 1 n 0 1 = = 1 n θ = [n, θ ] θ θ = [n 1, n 2, n 3,...] θ = [n 2, n 3,...] θ = [n 1, θ ] θ θ 6. ω = [n 1, n 2,..., n l, ω] [n 1, n 2,..., n l, ω] ω M M = ( 1) l : l = 1 ω = [n 1, ω] ( = n 1 + ) 1/ω = (n 1 ω + 1)/ω [n 1, ω] n l 2 C 6 r, s, t, u r/t = [n 1, n 2,..., n l ], s/u = [n 1, n 2,..., n l 1 ] r/t, s/u ( ) [n 1, n 2,..., n l, ω] = s + rω u + tω ( ) r s r s C 5 t u t u = ( 1) l 7. ω = [2, 3, 5, ω] C 6
79 [2] = 2 1, [2, 3] = = , [2, 3, 5] = = [2, 3, 5, ω] = ω ω ( ) 37 7 ω = (37ω + 7)/(16ω + 3) = 1 l = : : θ 2 θ = (rθ+s)/(tθ+u) ru st = ±1 (r, s, t, u) θ θ 2 (r, s, t, u) = (1, 0, 0, 1) 7. 2 θ θ = rθ + s, ru st = e = ±1 (1) tθ + u p 2 Dq 2 = 4e D θ : (1) tθ 2 + (u r)θ s = 0 (2) θ 2 ts 0 q = gcd(t, u r, s) a = t/q, b = (u r)/q, c = s/q aθ 2 + bθ + c = 0 (ac 0, gcd(a, b, c) = 1) (3) (3) θ D = b 2 4ac p = u + r 2u = p + bq, 2r = p bq ( ) ( ) r s (p bq)/2 cq = (4) t u aq (p + bq)/2 ru st = (p 2 b 2 q 2 )/4 + acq 2 = (p 2 Dq 2 )/4 (5)
80 76 Pell p 2 Dq 2 = 4e : D = 4m p p = 2p Z( m) p 2 mq 2 = e 8. θ = 5θ + 3 3θ + 2 p = = 7 3θ2 3θ 3 = 0 q = 3 θ 2 θ 1 = 0 D = 5 p 2 5q 2 = = 4 8. θ = (rθ + s)/(tθ + u), e = ur st = ±1 ρ = tθ +u Z ( D) D θ 10 : 7 (4) ρ = tθ + u = aqθ + p + bq 2 7 (3) aθ = b + D 2 ρ = q( b + D) 2 7 (5) + p + bq 2 ρρ = p2 Dq 2 4 = p + q D 2 = e ρ Z ( D) ( 4) : D = 4m p p = 2p Z( m) p 2 mq 2 = e 9. 4θ + 3 e = 1 ρ = 3θ + 2 3θ + 2 3θ 2 2θ 3 = 0 D = 40 θ = = Z( 10) ρ = 3θ + 2 = p.212 1
81 ( D ) Z ( D) D 2 Z ( D) 9. θ 2, a, b, c aθ 2 + bθ + c = 0 (a 0) (1) θ D = b 2 4ac (2) p, q p 2 Dq 2 = 4e (e = ±1, q 0) (3) ( ) ( ) r s (p bq)/2 cq = t u aq (p + bq)/2 r, s, t, u θ θ = rθ + s tθ + u θ (4) (5) : (4) (3) ru st = (p 2 b 2 q 2 )/4 + acq 2 = (p 2 Dq 2 )/4 = e (6) (4) (1) θ(tθ + u) (rθ + s) = tθ 2 + (u r)θ s = q(aθ 2 + bθ + c) = 0 (7) a 0, q 0 t 0 θ tθ + u 0 θ = rθ + s tθ + u r u (2) D b 2 (mod 4) (3) p 2 (bq) 2 (mod 4) p bq r, u θ 1 1 (8)
82 78 Pell : (1) b (1) aθ 2 + 2bθ + c = 0 (a 0) (1 ) (2) D = 4m, m = b 2 ac (2 ) (3) p 2 mq 2 = e (e = ±1, q 0) (3 ) (4) ( ) ( ) r s p bq cq = t u aq p + bq (5) (4 ) 10. ε = (p, q) = (3, 1) θ = 8 θ 2 8 = 0 (a, b, c) ( = (1, ) 0, 8) ( ) r s 3 8 p bq = 3, p + bq = 3, aq = 1, cq = 8 = t u 1 3 θ = 8 2 θ θ 2 + 4θ 4 = 0 θ + 5 = θ(θ + 5) = θ 2 + 5θ = 4θ θ = θ + 4 θ = (θ + 4)/(θ + 5) θ θ θ θ = : (3+ 8) 2 = , ( ) 8 = θ = ( )/( ) θ = 8
83 (rθ + s)/(tθ + u) (r θ + s )/(t θ + u ) θ 2 ( ) ( ) ( ) r s r s rr + st rs + su = t u t u tr + ut ts + uu (tr + ut )θ + (ts + uu ) tθ + s t θ + s : (tr + ut )θ + (ts + uu ) = (tθ + s)(t θ + s ) 2 (tθ + s)(t θ + s ) = tt θ 2 + (ut + tu )θ + uu t θ 2 = (r u )θ + s tt θ 2 + (ut + tu )θ + uu = t((r u )θ + s ) + (ut + tu )θ + uu = (tr + ut )θ + ts + uu : 2 : θ θ θ > 1, 0 > θ > 1 : : 12 θ = [n 1, n 2,..., n l, θ] : 2 2 T θ 2 ξ = 1 ξ 2 θ [θ] : θ = θ, ξ = ξ θ > 1 0 < θ < 1 ξ > 1 0 < ξ < 1 11 p.112 (p.200) 12 p.205
84 80 Pell n = [θ] 1 ξ = 1/(θ n) > 1 0 < θ < 1 n < θ + n < n n > 1 θ + n > 1 n ξ = θ + n 1 n > ξ > 1 n < ξ < θ 2 η = 1/θ θ = [n 1, n 2,..., n l 1, n l, θ] = [n 1, n 2,..., n l 1, n l ] (1) η = [n l, n l 1,..., n 2, n 1, η] = [n l, n l 1,..., n 2, n 1 ] (2) : n 1, n 2,..., n l θ 1 = θ, n k = [θ k ], 1 θ k+1 = θ k n k (k = 1, 2,..., l 1, l) (3) θ l+1 = θ 1 θ k = θ k θ k > 1, 0 < θk < 1 (k = 1, 2,...) ( 11) η k = 1/θk η k > 1, 0 < ηk < 1 (k = 1, 2,...) ηk = η k 1 η l+1 = η 1 = η, = η k+1 n k (k = l, l 1,..., 2, 1) η k η k > 1 (k = 1, 2,...) 0 < η k+1 n k < 1 [η k+1 ] = n k η = [n l, n l 1,..., n 2, n 1, η] ω = m, n 0 = [ω] (a) ω [n 0, n 1, n 2,..., n r 1, n r, 2n 0 ] (b) [n 0, n 1, n 2,..., n r 1, n r, 2n 0 ) = [n 0, n r, n r 1,..., n 2, n 1, 2n 0 ] (c) n k n 0 (k = 1, 2,..., r) : ω = m ω > 1, ω + ω = 0 n 0 = [ω]
85 n 0 1 θ 1 = 1 ω n 0 θ 1 > 1 1 θ 1 = n 0 ω = n 0 + ω > 1 1 < θ 1 < 0 θ 1 η 1 = 1/θ 1 12 ω = [n 0, θ] (1) θ 1 = [n 1, n 2,..., n l 1, n l, θ 1 ] (2) η 1 = [n l, n l 1,..., n 2, n 1, η 1 ] (3) 1 θ 1 = ω n 0, η 1 = ω + n 0 ω + ω = 0 1 θ 1 η 1 = 2n 0 η 1 = 2n θ 1 = [2n 0, θ 1 ] (3) 2n 0 = n l θ 1 = [n l 1,..., n 2, n 1, η 1 ] = [n l 1,..., n 2, n 1, [2n 0, θ 1 ]] = [n l 1,..., n 2, n 1, 2n 0, θ 1 ] (2) (n l 1,..., n 2, n 1 ) = (n 1, n 2,..., n l 1 ) (2) (4) (r = l 1 ) (a) (b) m + bk (c) : θ k = a k m + bk m bk+1 n k = (k = 1, 2,..., r) a k a k b k + b k+1 = n k a k b k (k = 1, 2,..., r) 0 < b k < m 0 < b k n n 0 = [ m] b k + b k+1 2n 0 a k 2 n k a k 2n 0 n k n 0 a k = 1 n k b k+1 = n 0 m n 0 (θ 1 = 1/( m n 0 ) ) (4)
86 82 Pell 5.3 Pell Pell θ = m θ k = 1 θ k 1 [θ k 1 ] (k = 1, 2,...) θ k θ 0 = θ m m = [n 0, n 1, n 2,..., n l ] n k = [θ k ] k = l + 1 k = 1 θ l+1 = θ 1 θ 1 = [n 1, n 2,..., n l, θ 1 ] θ 1 θ 1 = (rθ 1 + s)/(tθ 1 + u), ru st = ( 1) l r, s, t, u ( 6) η = m n θ 1 = 1/η θ 1 η η = (uη + t)/(sη + r) 8 sη + r N(sη + r) = ru st N(r + sη) = (r + sη)(r + sη) = r 2 + sr(η + η) + s 2 ηη = r 2 2nsr s 2 (m n 2 ) = (r sn) 2 ms 2 = ( 1) l (5.7) x = ns r, y = s Pell 13 1 m Pell Z( m) ±1 Z( m) 2 Pell Pell 13 p.215
87 5.3. Pell 83 Pell 1. Pell x 2 8y 2 = ±1 m = 8, 8 = [2, 1, 4], l = 2, n 0 = 2 C 6 [1] = 1/1, [1, 4] = 5/4, θ 1 = [1, 4] = [1, 4, θ 1 ] = 1 + 5θ θ ( ) ( ) 1 r s 5 1 = ( 8) (5.7) t u 4 1 x = r ns = = 3, y = s = 1 : (5.7) θ 1 θ 1 = (5θ 1 +1)/(4θ 1 +1) 4θ1 2 4θ 1 1 = 0 4θ θ = (2 ± 8)/4 4θ 1 +1 = 3± 8 Pell Pell m 2. n = [ m] m = [n, n 1, n 2,..., n r, 2n] ( 1) p/q = [n, n 1, n 2,..., n r ] p 2 mq 2 = ( 1) r+1 14 : θ 1 = 1/( m n), θ 1 = [n 1, n 2,..., n r, 2n, θ 1 ] 14 [16, 17] Dummit[16] Lahn-Spiegel[17] Dummit
88 84 Pell θ 1 = (rθ 1 + s)/(tθ 1 + u), ru st = ( 1) r+1 ( 6) r = H(n 1, n 2,..., n r, 2n), s = H(n 1, n 2,..., n r ) ( C 6) 1 C 1 (n 1, n 2,..., n r 1, n r ) = (n r, n r 1,..., n 2, n 1 ) r = H(n 1, n 2,..., n r, 2n) = H(2n, n r,..., n 2, n 1 ) = H(2n, n 1, n 2,..., n r ) r s = H(2n, n 1, n 2,..., n r ) = [2n, n 1, n 2,..., n r ] = n + [n, n 1, n 2,..., n r ] H(n 1, n 2,..., n r ) = n + p q s = q, r = p + nq = p + ns (5.7) 2. 7 = [2, 1, 1, 1, 4] p/q = [2, 1, 1, 1] [1, 1] = 1 + 1/1 = 2/1, [1, 1, 1] = 1 + 1/2 = 3/2, [2, 1, 1, 1] = 2 + 2/3 = 8/3 p/q = 8/3 p 2 7q 2 = 1 Pell l = r + 1 = 4 ( 1) l = Pell x, y m x/y m 2 (n 1, n 2,..., n r ) 2n p/q = [2, 1] p/q = [2, 1, 4, 1] p/q = [2, 1, 4, 1, 4, 1] m = 8 Pell
89 5.4. Pell 85 3 m 3 (mod 4) (5.4) 2 m (r + 1) m 3 (mod 4) m 0 (mod 4) 4.3 : 1 r ( ) m = n 2 + a a a 3 (mod 4) n 2 0, 1 (mod 4) n 2 0 (mod 4) m 3 (mod 4) n 2 1 (mod 4) m 0 (mod 4) 4 m ( ) 2 Pell 2a 5.4 Pell m Pell (5.1) 15 : : ε 1 < ε Z( m) Pell (x, y) ε = x + y m Z( m) ε = x + y m ε ε 1 x, y 1 < ε Z( m) Z( m) 3+ 8 Z( m) 1 < x+y 8 < ε 0 Z( m) Z( m) ε > 1 ε n 0 n 15
90 86 Pell : 1 < ε 0 ε > 1 ε n 0 ε < ε n+1 0 n 1 ε/ε n 0 < ε 0 ε/ε n 0 Z( m) ε 0 1 < ε 0 ε = ε n 0 m = [n0, θ], θ = [n 1, n 2,...] n 1, n 2,... l n l+k = n k (k = 1, 2,...) m D = 4m θ 4m θ = [n 1, n 2,..., n l, θ], θ = rθ + s, ru ts = ±1 tθ + u tθ + u Z( m) ( 8) k θ = [n 1, n 2,..., n l, n l+1, n l+2,..., n kl, θ] ( ) ( rk s k r s = t u t k u k t k θ + u k t k θ + u k = (tθ + u) k ( 10) θ = θ = [1, 2, 1, θ] θ = 4θ + 3 ( ) 3 3θ ( ) θ + 18 θ = [1, 2, 1, 1, 2, 1, θ] = 18θ + 13 ( ) ( ) 2 ( ) = ε = 3θ + 2 = εε = 1 Z( 10) ε = 18θ + 13 = ε ε = 1 Z( 10) ε = ε 2 2. ω = 10 ω = [3, 6], θ = [6, θ] = 6θ + 1 1θ + 0 ε = 1θ + 0 = ω + 3 ) k
91 5.4. Pell 87 ε 2 19θ + 60 = 6ω + 19 θ = [6, 6, θ] = 6θ + 19 θ ε k (k 1) 14. m θ = rθ + s tθ + u tθ + u Z( m) : Z( m) ε 0 θ ε 0 ( 9) θ = r 0θ + s 0 t 0 θ + u 0 tθ + u Z( m) ( 8) ε ε 0 ε k 0 = ε (k 1) ( 13) ε ( ) ( ) k r s r0 s 0 = k t u t 0 u 0 ε k = 1 ε : Pell : Pell Pell Pell Pell Z( m) (x, y) = (3, 1) Pell x 2 8y 2 = ±1 (3 + 8) k Pell x 2 8y 2 = ±1 2 2a. n = [ m] m = [n, n 1, n 2,...] l n l = 2n n, n 1, n 2,..., n l 1 2n ( 1) p k /q k = [n, n 1, n 2,..., n kl 1 ] (k = 1, 2,...) p 2 k mqk 2 = ( 1) kl
92 88 Pell (p 1, q 1 ) Pell ε 0 (p k, q k ) ε k 0 Pell : Z( m) ε 0 θ 1 = 1/( m n) θ 1 = [n 1, n 2,..., n r, 2n, θ 1 ] ( 14) p 1 /q 1 = [n, n 1, n 2,..., n r ] Pell 2 Pell p k /q k = [n, n 1, n 2,..., n kl 1 ] (k = 1, 2,...) ε k 0 Pell 4. Pell p 2 7q 2 = ±1 7 = [2, 1, 1, 1, 4] p/q = [2, 1, 1, 1] p/q = 8/3 (p, q) = (8, 3) Pell Z( 7) 5. (p, q) = (15, 4) p 2 14q 2 = 1 p/q [3, 1, 3] p/q = [3, 1, 2, 1] 14 = [3, 1, 2, 1, 6] (p, q) = (15, 4) 6. (p, q) = (17, 12) p 2 2q 2 = 1 p/q = [1, 2, 2, 2] 2 = [1, 2, 2, 2, 2] [1, 2, 2, 2, 2] = [1, 2] [1, 2] p/q = [1] (p, q) = (17, 12) p/q = [1] (p, q) = (1, 1) Z( 2) = (1 + 2) Pell Pell 2 : T1 x 2 my 2 = ±1 T2 x 2 my 2 = ±4
93 5.5. Pell 89 x 2 my 2 = ±4 m 1 (mod 5) x 2 my 2 (mod 4) (a) m 0 (mod 4) x = 2x, m = 4m x 2 m y 2 = ±1; (b) m 2, 3 (mod 4) x y 0 (mod 2) x = 2x, y = 2y x 2 my 2 = ±1; m 1 (mod 4) T1 m 1 (mod 4) T2 Pell m θ M M θ θ 1. ω = 1 + m 2 m : 4ω 2 4ω (m 1) = 0 m 1 (mod 4) (m 1) = 4m ω 2 ω m = 0 D = 1 + 4m = m 2. 1 m = 5, 13, 17, 21, 29, 33, 37 ω : CASE m = 5: ω = [1] CASE m = 13: ω = [2, 3] CASE m = 17: ω = [2, 1, 1, 3] CASE m = 21: ω = [2, 1, 3] CASE m = 29: ω = [3, 5] CASE m = 33: ω = [3, 2, 1, 2, 5] CASE m = 37: ω = [3, 1, 1, 5] m = ( 2)( ) = ( 1)( ) = ( 1)( ) = ( 3)( ) = 1 2 4
94 90 Pell m ω =, n 0 = [ω] 2 (a) ω [n 0, n 1, n 2,..., n r 1, n r, 2n 0 1] (b) [n 0, n 1, n 2,..., n r 1, n r, 2n 0 1) = [n 0, n r, n r 1,..., n 2, n 1, 2n 0 1] (c) n k n 0 (k = 1, 2,..., r) m + 1 : ω = ω > 1, ω + ω = 1 n 0 = [ω] 2 1 n 0 1 θ = θ > 1 ω n 0 1 θ = n 0 ω = n 0 (1 ω) = n ω > 1 1 < θ < 0 θ 2 η = 1/θ 12 ω = [n 0, θ] (1) θ = [n 1, n 2,..., n l 1, n l, θ] (2) η = [n l, n l 1,..., n 2, n 1, η] (3) 1 θ = ω n 0, η = ω + n 0 ω + ω = 1 1 θ η = 1 2n 0 η = 2n θ = [2n 0 1, θ] (3) 2n 0 1 = n l θ = [n l 1,..., n 2, n 1, η] = [n l 1,..., n 2, n 1, [2n 0 1, θ]] = [n l 1,..., n 2, n 1, 2n 0 1, θ] (2) (n l 1,..., n 2, n 1 ) = (n 1, n 2,..., n l 1 ) n l = 2n 0 1 (2) (4) (r = l 1 ) (a) (b) m + bk (c) : θ k = a k m + bk m bk+1 n k = (k = 1, 2,..., r) a k a k (4)
95 5.5. Pell 91 b k + b k+1 = n k a k b k (k = 1, 2,..., r) 0 < b k < m 0 < b k n n = [ m] b k + b k+1 2n m 1 (mod 4) a k = 1 a k ( 5 ) a k 2 n k a k 2n n k n 3. 2 m = 5, 13, 17, 21, 29, 33, 37 Pell x 2 my 2 = ±4 : CASE m = 5: ω = [1, θ], θ = [1, θ] θ θ = (θ+1)/θ θ = ( 5 + 1)/2 θ θθ = 1 (x, y) = (1, 1) CASE m = 13: ω = [2, θ], θ = [3, θ] θ θ = (3θ + 1)/θ θ = ( )/2 θ θθ = 1 (x, y) = (3, 1) CASE m = 17: ω = [2, θ], θ = [1, 1.3, θ] θ θ = (7θ + 2)/(4θ+1) θ = ( 17+3)/4 4θ+1 = (4θ + 1)(4θ + 1) = (4 + 17)(4 17) = 1 Z( 17) = 3 + 2ω Z ( 17) Pell T2 x, y m m (mod 8) ω Pell eq. 5 5 ω = [1] = [1, 1] = ω = [2, 3] = ω = [2, 1, 1, 3] = ω = [2, 1, 3] = ω = [3, 5] = ω = [3, 2, 1, 2, 5] = ω = [3, 1, 1, 5] = 1 Z( m)?
96 92 Pell 15. m 1 (mod 8) Z ( m) Z( m) : ϵ = (x + y m)/2 Z ( m) x 2 my 2 = ±4 m 1 (mod 4) x 2 y 2 (mod 4) x, y x 2 y 2 ± 4 (mod 8) x, y x, y ϵ Z( m) 16. m 5 (mod 8) Z ( m) ϵ ϵ 3 Z( m) : ϵ = (x + y m)/2 x 2 my 2 = ±4 m 1 (mod 4) x 2 y 2 (mod 4) x y (mod 2) ϵ 3 x, y ϵ 3 = x(x2 + 3my 2 ) + y(3x 2 + my 2 ) m 8 x(x 2 + 3my 2 ) x(1 + 3m) 16x 0 (mod 8) y(3x 2 + my 2 ) y(3 + m) 8y 0 (mod 8) ϵ 3 Z( m) x, y ϵ Z( m) ϵ 3 Z( m) 5.3 Pell Pell Pell ω = ( m + 1)/2 ω = [n, n 1, n 2,..., n l ] ω = [n, θ], θ = [n 1, n 2,..., n l, θ] θ θ = (rθ + s)/(tθ + u) η = 1/θ η = (uη + t)/(sη + r) m + 1 m (2n 1) η = n = 2 2
97 5.5. Pell 93 η + η = (2n 1), ηη = ((2n 1) 2 m)/4 N(sη + r) = (r 2 + sr(η + η) + s 2 ηη) = r 2 (2n 1)sr + s 2 (2n 1)2 m 4 = (r 2n 1 s) 2 m 2 4 s2 (2r (2n 1)s) 2 ms 2 = ±4 1. m = 21 ω = [2, θ], θ = [1, 3, θ] = (4θ + 1)/(3θ + 1) (n, r, s) = (2, 4, 1) (8 3 1) = +4 Pell 2. m = 33 ω = [3, θ], θ = [2, 1, 2, 5, θ] = (43θ + 8)/(16θ + 3) (n, r, s) = (3, 43, 8) ( ) = +4 (43 5 4) = m m 1 (mod 4) ω m + 1 ω = = [n, n 1, n 2,..., n r, 2n 1] 2 p q = [n, n 1, n 2,..., n r ] p/q (2p q) 2 mq 2 = ( 1) r+1 4
98 94 Pell : ω = [n, θ], θ = [n 1, n 2,..., n r, 2n 1, θ] = rθ + s tθ + u, ω = n + η η = uη + t m (2n 1) sη + r, η = ω n = 2 C 6 C 1 η = 1 θ r = H(n 1, n 2,..., n r, 2n 1) = H(2n 1, n r,..., n 2, n 1 ) = H(2n 1, n 1, n 2,..., n r ) s = H(n 1, n 2,..., n r ) C 4 r s = H(2n 1, n 1, n 2,..., n r ) = [2n 1, n 1, n 2,..., n r ] H(n 1, n 2,..., n r ) = n 1 + [n, n 1, n 2,..., n r ] = n 1 + p q s = q, r = p + (n 1)q m (2n 1) sη + r = q + p + (n 1)q = (2p q) + q m 2 2 sη + r (2p q) 2 mq 2 = ±4 ( 6) 3. m = 5 ω = [1], p/q = [1] (p, q) = (1, 1) (2 1) = 4 4. m = 6 m 1 (mod 4) ? ω = 2 4 ω = [1, 1, 2, 1] Pell x 2 6y 2 = ±4? x = 2p, y = 2q p 2 6q 2 = ±1 (p, q) = (5, 2)
99 5.5. Pell m = 13 ω = [2, 3], p/q = [2] = 2/1 (4 1) = 4 6. m = 17 ω = [2, 1, 1, 3], p/q = [2, 1, 1] = 5/2 (10 2) = 4 or = 1 7. m = 21 ω = [2, 1, 3], p/q = [2, 1] = 3/1 (6 1) = m = 29 ω = [3, 5], p/q = [3] = 3/1 (6 1) = 4 9. m = 33 ω = [3, 2, 1, 2, 5], p/q = [3, 2, 1, 2] = 27/8 (54 8) = +4 or = m = 37 ω = [3, 1, 1, 5], p/q = [3, 1, 1] = 7/2 (14 2) = 4 or = 1 m 1 (mod 4) Z ( m)? YES: Z( m) 13 Z( m) Z ( m) 16 Z( m) ε ε = ϵ k 0 (k N) ϵ 0 Z ( m) m + 1 m 1 (mod 4) 2 Z ( m)? YES: Z( m) m (mod 4) Z ( m) Z( m) Z ( m)
100 96 Pell 11. m = 13 ω = [2, 3] 1 p/q = [2] = 2/1 x = 2p q = 3, y = q = 1 ϵ 0 = x + y 13 = Z ( 13) ϵ 2 0 = Z ( 13) 2 Pell = 4 ω 2 p/q = [2, 3] = 7/3, x = 2p q = 11, y = q = m = 21 ω = [2, 1, 3] 1 p/q = [2, 1] = 3/1 x = 2p q = 5, y = q = 1 ϵ 0 = x + y 21 = Z ( 21) ϵ 2 0 = Z ( 21) 2 Pell = 4 ω 2 p/q = [2, 1, 3, 1] = 14/5, x = 2p q = 23, y = q = 5
101 Chapter 6 Pell 6.1 Pell m, d N x 2 my 2 = k (= ±d) (x, y Z) (6.1) m m = 5, d = 20 (x, y) = (0, 2), (5, 1), (5, 3), (10, 4), (15, 7) k (x, y, k) = (0, 2, 20), (5, 1, 20), (5, 3, 20), (10, 4, 20), (15, 7, 20) k k 1 d = 1 Pell d = 4 Pell Pell 2?? 1 k +d d 2 generalized Pell s equation Pell [5] 97
102 98 Pell (6.1) x, y ( 6.1) x y x y ( I Conrad II ) (6.1) (x my)(x + my) = k (6.2) (6.1) ξξ = k Z( m) ξ = x + my 3 θ Z( m) ε 0 ε 0 ε 0 = ±1 ξ = ±θε n 0 (n Z) ξξ = θθ(ε 0 ε 0 ) n = k(ε 0 ε 0 ) n ε 0 ε 0 = +1 ξ n ξξ = k ε 0 ε 0 = 1 ξ n ξξ = +k ξξ = k (6.1) 6.1 x 2 5y 2 = ±20 ( ) x + 5y = 0 x 5y = 0 ( ) x y x, y 0 (x, y) = (0, 2), (5, 1), (5, 3), (10, 4), (15, 7) 3 Z( m) 5.2
103 6.1. Pell : (0, 2) 2 5 Z( 5) ε 0 = ε 0 = (10, 4) 2 5ε 1 0 = (10, 4) (5, 1) (5 + 5)ε 0 = (5 + 5)ε 1 0 = (5, 3) (5+3 5)ε 0 = (5+3 5)ε 1 0 = 5 5 (5+3 5)ε 2 0 = (6.1) k = d k = +d ( ) k = +d k = d (6.1) x + my > 0 x, y x + my > 0 Z( m) ε 0 ε n 0 (n Z)
104 100 Pell : : ξ ξ U( m) ξ/ξ U( m) (U( m) Z( m) ) U ( m) (U ( m) Z ( m) 4 ) : Z ( m) 5 (6.1) θθ = ±d θ U( m) ξ(= ±θε n 0 ) ξξ = ±d U ( m) Z( m) ε 0 Z ( m) ϵ x + 5y > 0 9 ( ) 3 : (0, 2) (10, 4) (10, 4) (5, 1) (15, 7) ( 5, 3) (5, 3) (5, 1) ( 15, 7) U( 5) x 2 5y 2 = ±20 Z( 5) ε 0 = ± 5ε n 0, ±(5 + 5)ε n 0, ±( )ε n 0 (n Z) U ( 5) Z ( 5) ϵ 0 ϵ 0 = (1 + 5)/2, ε 0 = ϵ ϵ 0 = 5 + 5, (5 + 5)ϵ 0 = ±2 5ϵ n 0 (n Z) 4 Z ( m) p.272
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6 2 T γ T B (6.4) (6.1) [( d nm + 3 ] 2 nt B )a 3 + nt B da 3 = 0 (6.9) na 3 = T B V 3/2 = T B V γ 1 = const. or T B a 2 = const. (6.10) H 2 = 8π kc2
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LINEAR ALGEBRA I Hiroshi SUZUKI Department of Mathematics International Christian University
LINEAR ALGEBRA I Hiroshi SUZUKI Department of Mathematics International Christian University 2002 2 2 2 2 22 2 3 3 3 3 3 4 4 5 5 6 6 7 7 8 8 9 Cramer 9 0 0 E-mail:hsuzuki@icuacjp 0 3x + y + 2z 4 x + y
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![(2016 2Q H) [ ] R 2 2 P = (a, b), Q = (c, d) Q P QP = ( ) a c b d (a c, b d) P = (a, b) O P ( ) a p = b P = (a, b) p = ( ) a b R 2 {( ) } R 2 x = x, y](/thumbs/98/136512264.jpg "(2016 2Q H) [ ] R 2 2 P = (a, b), Q = (c, d) Q P QP = ( ) a c b d (a c, b d) P = (a, b) O P ( ) a p = b P = (a, b) p = ( ) a b R 2 {( ) } R 2 x = x, y")
(2016 2Q H) [ ] R 2 2 P = (a, b), Q = (c, d) Q P QP = a c b d (a c, b d) P = (a, b) O P a p = b P = (a, b) p = a b R 2 { } R 2 x = x, y R y 2 a p =, c q = b d p + a + c q = b + d q p P q a p = c R c b
液晶の物理１：連続体理論（弾性，粘性）
The Physics of Liquid Crystals P. G. de Gennes and J. Prost (Oxford University Press, 1993) Liquid crystals are beautiful and mysterious; I am fond of them for both reasons. My hope is that some readers
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GO 2016 8 26 1 a σ(a) σ(a) = 2a, 6,28,496,8128 6 = 2 (2 2 1), 28 = 2 2 (2 3 1), 496 = 2 4 (2 5 1), 8128 = 2 6 (2 7 1) 2 1 Q = 2 e+1 1 a = 2 e Q (perfect numbers ) Q = 2 e+1 1 Q 2 e+1 1 e + 1 Q = 2 e+1
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(1.2) T D = 0 T = D = 30 kn 1.2 (1.4) 2F W = 0 F = W/2 = 300 kn/2 = 150 kn 1.3 (1.9) R = W 1 + W 2 = = 1100 N. (1.9) W 2 b W 1 a = 0
1 1 1.1 1.) T D = T = D = kn 1. 1.4) F W = F = W/ = kn/ = 15 kn 1. 1.9) R = W 1 + W = 6 + 5 = 11 N. 1.9) W b W 1 a = a = W /W 1 )b = 5/6) = 5 cm 1.4 AB AC P 1, P x, y x, y y x 1.4.) P sin 6 + P 1 sin 45
18 2 F 12 r 2 r 1 (3) Coulomb km Coulomb M = kg F G = ( ) ( ) ( ) 2 = [N]. Coulomb
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r 1 r 2 r 1 r 2 2 Coulomb Gauss Coulomb 2.1 Coulomb 1 2 r 1 r 2 1 2 F 12 2 1 F 21 F 12 = F 21 = 1 4πε 0 1 2 r 1 r 2 2 r 1 r 2 r 1 r 2 (2.1) Coulomb ε 0 = 107 4πc 2 =8.854 187 817 10 12 C 2 N 1 m 2 (2.2)
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9 203 6 7 WWW http://www.math.meiji.ac.jp/~mk/lectue/tahensuu-203/ 2 8 8 7. 7 7. y fx, z gy z gfx dz dx dz dy dy dx. g f a g bf a b fa 7., chain ule Ω, D R n, R m a Ω, f : Ω R m, g : D R l, fω D, b fa,
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9 .,,, 3 () C k k C k C + C + C + + C 8 + C 9 + C k C + C + C + C 3 + C 4 + C 5 + + 45 + + + 5 + + 9 + 4 + 4 + 5 4 C k k k ( + ) 4 C k k ( k) 3 n( ) n n n ( ) n ( ) n 3 ( ) 3 3 3 n 4 ( ) 4 4 4 ( ) n n
n 2 + π2 6 x [10 n x] x = lim n 10 n n 10 k x 1.1. a 1, a 2,, a n, (a n ) n=1 {a n } n=1 1.2 ( ). {a n } n=1 Q ε > 0 N N m, n N a m
![n 2 + π2 6 x [10 n x] x = lim n 10 n n 10 k x 1.1. a 1, a 2,, a n, (a n ) n=1 {a n } n=1 1.2 ( ). {a n } n=1 Q ε > 0 N N m, n N a m](/thumbs/94/118912662.jpg "n 2 + π2 6 x [10 n x] x = lim n 10 n n 10 k x 1.1. a 1, a 2,, a n, (a n ) n=1 {a n } n=1 1.2 ( ). {a n } n=1 Q ε > 0 N N m, n N a m")
1 1 1 + 1 4 + + 1 n 2 + π2 6 x [10 n x] x = lim n 10 n n 10 k x 1.1. a 1, a 2,, a n, (a n ) n=1 {a n } n=1 1.2 ( ). {a n } n=1 Q ε > 0 N N m, n N a m a n < ε 1 1. ε = 10 1 N m, n N a m a n < ε = 10 1 N
Go a σ(a). σ(a) = 2a, 6,28,496, = 2 (2 2 1), 28 = 2 2 (2 3 1), 496 = 2 4 (2 5 1), 8128 = 2 6 (2 7 1). 2 1 Q = 2 e+1 1 a = 2
Go 2016 8 26 28 8 29 1 a σ(a) σ(a) = 2a, 6,28,496,8128 6 = 2 (2 2 1), 28 = 2 2 (2 3 1), 496 = 2 4 (2 5 1), 8128 = 2 6 (2 7 1) 2 1 Q = 2 e+1 1 a = 2 e Q (perfect numbers ) Q = 2 e+1 1 Q 2 e+1 1 e + 1 Q
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I, II 1, A = A 4 : 6 = max{ A, } A A 10 10%
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y π π O π x 9 s94.5 y dy dx. y = x + 3 y = x logx + 9 s9.6 z z x, z y. z = xy + y 3 z = sinx y 9 s x dx π x cos xdx 9 s93.8 a, fx = e x ax,. a =
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Microsoft Word - 11問題表紙（選択）.docx
A B A.70g/cm 3 B.74g/cm 3 B C 70at% %A C B at% 80at% %B 350 C γ δ y=00 x-y ρ l S ρ C p k C p ρ C p T ρ l t l S S ξ S t = ( k T ) ξ ( ) S = ( k T) ( ) t y ξ S ξ / t S v T T / t = v T / y 00 x v S dy dx