Super perfect numbers and Mersenne perefect numbers /2/22 1 m, , 31 8 P = , P =
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1 Super perfect numbers and Mersenne perefect numbers /2/22 1 m, , 31 8 P = , P =
2 1 m, 1: m = 28 m = 28 m = 10 height A A D A D m = A m = 8 m = A m = A A D A D A D A F D 2
3 2: m = 24, 22 m = F m = 24 m = A A m = 6 m = F m = F A m = A m = 4 m = A m = D A A A A D D A A m = 19 m = F m = 18 m = A A A A F 3
4 3: m = 17, 16 m = A m = 1 m = 16 m = D A D D m = D 2 2 C D C A C m = C m = C A C A C A C 4
5 2 A = σ(a) + m, σ(a) = 2a + m a A a m,a. m = 28 4: m = * a = 7p, p 7, A = 4q, q 2 B. A = σ(a) + m, σ(a) = 2a + m, 5
6 A = σ(a) + m = 8(p + 1) 28 = 4(2p 5), Q = 2p 5 σ(a) = 7σ(Q) = 14p 28 = 14(p 2). 7σ(Q) = 14p 28 = 14(p 2), σ(q) = 2(p 2) = 2p 4 = 2p 5+1 = Q+1. σ(q) = Q + 1 Q. (p, Q = 2p 5),. B a = 7p (p, 2p 5). 1 a = 7h (7 7 ) h. Proof. a = 7h, A = σ(a)+m = 8σ(h) 28 = 4(2σ(h) 7), σ(a) = 2a+m = 14h 28. Q = 2σ(h) 7, 4. σ(a) = σ(4q) = 7σ(Q). σ(a) = 2a + m = 14h 28 = 7σ(Q), 14h 28 = 7σ(Q), 2h 4 = σ(q). Q = 2σ(h) 7 σ(q) = 2h 4 1 coσ(q) = Q σ(q) = 2σ(h) 7 2h + 4 = 2coσ(h) 3 1. coσ(q) = coσ(h) = 1. Q, h,2h 4 = σ(q) = Q + 1. Q = 2h 5 (h, Q = 2h 5). End a = 2 e C, A. a = p : ( ),A = 2 e q, (q : ). a = p,a = 2 e q A, pa.. m. m A = σ(a) + m, σ(a) = 2a + m, A = σ(a) + m = p m = 2 e q, σ(a) = 2a + m = 2p + m p m = 2 e q., p = 2 e q m 1. N = 2 e+1 1 2p + m = σ(a) = σ(2 e q) = N(q + 1) = Nq + N = 2 2 e q q + N. 2p + m = 2 2 e q m 2, 2p + m = 2 2 e q m 2 = 2 2 e q q + N. 6
7 ,q = 2 e m. m 2 = q + N = q + 2 e+1 1., m 2 e+1 +1+m q, q = 2 e+1 +1+m. 2 e q m 1,p = 2 e q m 1. a = p A = σ(a) + m = p m = 2 e q σ(a) = σ(2 e q) = N(q + 1) = Nq + N = 2 2 e q q + N = 2(p m) q + 2 e+1 1 = 2(p m) q + q 1 m 1 = 2p + m. σ(a) = 2p + m = 2a + m., 2 q = 2 e m p = 2 e q m 1 a = p. A = σ(a) + m = p m σ(a) = 2a + m. A = p m p = A 1 m, σ(a) = 2p + m = 2A 2 m. µ 0 = 2 + m,σ(a) = 2A µ 0., σ(a) = 2A µ 0 A 0 A 0 1 m = A 0 1 (µ 0 2) = A 0 +1 µ 0 2 p, a = p A = σ(a) + m, σ(a) = 2a + m. 7
8 5: m = 28, 26; q = 2 e m, p = 2 e q m 1 e q p m = 28 4 q = 5 p = q = 37 p = q = 101 p = q = 229 p = q = 997 p = q = p = m = 26 4 q = 7 p = q = 103 p = q = 487 p = q = 8167 p = m = 22 4 q = 11 p = q = 43 p = q = 107 p = q = 491 p = q = 2027 p = q = 8171 p = q = p = ) m = 28, p = 107, p = 6491, q = pa. µ 0 = 2 + m = 26 A 0 A 0 = 80, 1184, a = A 0 1 m = A m = 28., A = = 107, , ) m = 26 p = 137 pa. µ 0 = 2 + m = 24 A 0 A 0 = 112, ) m = 22 p = 197, p = 1397, p = 6869, p = pa. µ 0 = 2 + m = 20 A 0 A 0 = 176, 1376, = 1397 =
9 6: m = 20, 16, 14; q = 2 e m, p = 2 e q m 1 e q p m = 20 4 q = 13 p = q = 109 p = q = 2029 p = m = 16 4 q = 17 p = q = 113 p = q = 241 p = q = 1009 p = m = 14 3 q = 3 p = q = 19 p = q = 499 p = q = 8179 p = m = 6 2 q = 3 p = q = 11 p = q = 59 p =
10 7: m = 28; q = 2 e m, p = 2 e q m 1 e q p m = 4 2 q = 5 p = q = 13 p = q = 29 p = q = 61 p = q = 509 p = q = 1021 p = q = 4093 p = q = p = q = p = m = 2 1 q = 3 p = q = 7 p = q = 31 p = q = 127 p = m = 0 1 q = 5 p = q = 17 p = q = 257 p = q = p =
11 8: m = 2, 4, 8; q = 2 e m, p = 2 e q m 1 e q p m = 2 1 q = 7 p = q = 11 p = q = 19 p = q = 67 p = q = 131 p = q = 4099 p = q = p = m = 4 2 q = 13 p = q = 37 p = q = 2053 p = q = p = A m = 8 1 q = 13 p = q = 17 p = q = 41 p = q = 73 p = q = 137 p = q = 521 p = A =
12 9: m = 10; q = 2 e m, p = 2 e q m 1 e q p m = 10 2 q = 19 p = q = 43 p = q = 139 p = q = 523 p = q = p = q = p =
13 10: m = 26 m = *19 11: m = 22 m = * *13* *19* µ 0 = 2 + m =
14 12: m = 16 m = * *277 14
15 13: m = 18, 14 m = m =
16 14: m = 4, 3, 2 m = m =
17 15: m = 1, 0, 2 m = m = m = m =
18 16: m = 3, 4, 6 m = m = m = (1969 ) α = 2 e q (q: ). a = 2 e m σ(a) = a + 1 σ(a) = a + 1 = 2 e+1 + m A = σ(a) m A = 2 e+1. a + 1 = 2 e+1 + m σ(a) = 2 e+2 1 = 2 2 e+1 1 = 2a 2m + 1 A = σ(a) m, σ(a) = 2 e+2 1 = 2 2 e+1 1 = 2a 2m + 1 a A a m,a. 18
19 17: a factor A factor B factor m = m = : m = m = m =
20 5 m., m, m. m = 9 a = 3p, p( 2, 3) :, A = 3 e. A = σ(a) m, σ(a) = 2a 2m + 1 m = 9. A = σ(a) + 9, σ(a) = 2a a = 3p A = 3 e. A = σ(a) + 9 = 4p + 13 = 3 e. 3 4p + 13 = 3 e a = 3p, (p 2, 3) A = σ(a) + 9, σ(a) = 2a Proof. A = σ(a) + 9 = 4p + 13 = 3 e, 2σ(A) = 3 e+1 1. End 3 e 13 p. 4 19: m = 9 a = 3p, A = 3 e e 3 p 4 51 = = = = =
21 1 m = 9 a = 3p A = 3 e.. Proof. a = 3p A = σ(a) + 9, σ(a) = 2a + 19, A = σ(a) + 9 = 4p + 13, σ(a) = 2a + 19 = 6p A 13 = 4p, σ(a) 19 = 6p (12p =)3A 39 = 2σ(A) 38., 3A 1 = 2σ(A).. 1 p (p 1)σ(a) = ap 1 a p., p = 2, 3,100. p = 5 a = 7 11 p = 7 a = = p = 11 a = 611 = p = 17 a = 1073 = 29 37, a = 2033 =
22 20: a factor A factor B factor m = m =
23 21: a factor A factor B factor m = m =
24 22: a factor A factor B factor m = m = m = m =
25 6 a = p = σ(2 e ) + m = 2 e m. σ(a) = p + 1 = 2 e+1 + m σ(a) m = 2 e+1. A = σ(a) m A = 2 e+1. σ(a) = 2 e+2 1 φ(a) = 2 e a + 1 m = 2 e+1, 2φ(A) = 2 e+1 = a m + 1. A = σ(a) m 2φ(A) = 2 e+1 = a m + 1 a. A. B = φ(a) a A = 2 e. 25
26 23: m = 2 m =
27 24: m = 2 m = m = m =
28 25: m = 2 m = m =
29 26: m = 2 m =
30 m = 3. B = φ(a) + 1, a = 2 p 2, p :, B = p 2, p :.. a, B 2.!... 2 m = 3 a = 2 p 2, p :,, p = 2 e 3 f 1, B = p 2. Proof. m = 3 A = σ(a) m = σ(a) 3, 2φ(A) = a m + 1 = a 2. a = 2 p 2 A = σ(a) 3 = 3p(p + 1). p + 1 p, p + 1 = 2 e 3 f R, (R,2, 3, p. A = 3p(p + 1) = 2 e 3 f+1 pr p + 1 = 2 e 3 f R 2φ(A) = 2 e+1 3 f (p 1)φ(R) = 2p 2 2 = 2(p + 1)(p 1). 2 e+1 3 f (p 1)φ(R) = 2(p + 1)(p 1) = 2(p 1)2 e 3 f R., φ(r) = R., R = 1, p + 1 = 2 e 3 f, A = 2 e 3 f+1 p. X = 2 e 3 f p + 1 = 2 e 3 f = X., p = 2 e 3 f 1, B = φ(a) + 1 = 2 e 3 f p + 1 = Xp + 1 = X(X 2) + 1 = (X 1) 2 = p 2 End, X = 2 e 3 f p = X 1, a = 2 p 2 A = σ(a) m = σ(a) 3, 2φ(A) = a m + 1 = a 2., A = σ(a) 3 = 3p(p + 1) = 2 e 3 f+1 p, 2φ(A) = 2 e+1 3 f (p 1) = 2X(p 1). a 2 = 2(p 2 1) = 2(p + 1)(p 1) = 2X(p 1),, 2φ(A) = a m + 1 = a 2. e, f ( ) p = 2 e 3 f 1,. 30
31 7, P, a = p = σ(p e ) + m., σ(a) = a + 1. σ(p e ) = P e+1 1 a = p = σ(p e ) + m = P e m, P P σ(a) = a + 1 = P e+1 + P 2 + m P P (a m) + 1 = P e+1., A = P (σ(a) m) + 2 P, A = P e+1. P σ A) = P e+2 1. P (a m) + 1 = P e+1, P σ(a) + 1 = P e+2 = P P (a m) + P. P σ(a) = P P (a m) + P. P, A = P (σ(a) m) + 2 P σ(a) = P (a m) + 1, a A,B = σ(a) 1. A = P (σ(a) m) + 2 P σ(a) = P (a m) + 1. P σ(a) P A = P P (a σ(a) + 1) 1. 5 a a σ(a) + 1 = 0., P σ(a) P A = 1., P σ(a) P A = 1 a σ(a) + 1 = 0.,a. 31
32 27: P = 3 P=3 m= m= m=
33 28: P = 3 m= m= m= m= m= m=
34 29: P = 3 m= m= m= m= m= m=
35 30: P = 3 m= P = 5 31: P = 5 m= m= m=
36 32: P = 5 m= m= m= m= m= m=
37 33: P = 5 m= m= m= m= m=
38 9, A = P (σ(a) m) + 2 P,σ(a) φ(a).,. A = P (σ(a) m)+2 P A = P e+1 φ(a) = φ(p e+1 ) = P P e. P φ(a) = P P e+1, A = P (σ(a) m) + 2 P = P e+1. σ(a) = a + 1, P (a + 1 m) + 2 P = P e+1., P e+1 = P (a + 1 m) + 2 P = P (a m) + P + 2 P = P (a m) + 1. P φ(a) = P P e+1 = P 2 (a m) + P., A = P (σ(a) m) + 2 P P φ(a) = P P e+1 = P 2 (a m) + P,. a A,B = σ(a) 1. 6 a A P e. Proof. a A = P (a + 1 m) + 2 P P φ(a) = P 2 (a m) + P. A = P (a + 1 m) + 2 P = P (a m) + 2 P = P + 2 P = P (a m) + 1. P φ(a) = P 2 (a m) + P = P (P (a m) + 1) = P A. P φ(a) = P A. A P A = P η L, (P L). P φ(a) = P φ(p η )φ(l) = P P η φ(l), P A = P P η L, P P η φ(l) = P P η L., φ(l) = L. L = 1, A = P e. End. 38
39 10 P = 5 34: P = 5 m = 21 q = 8p p 3 q 2q m = 9 q = 8p 5 5 p 3 3 q 18q
40 35: P = 5 m = m = m = m =
41 36: P = 5 m = m = m = m = m = m = m = m = m =
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平成20年5月 協会創立50年の歩み 海の安全と環境保全を目指して 友國八郎 海上保安庁 長官 岩崎貞二 日本船主協会 会長 前川弘幸 JF全国漁業協同組合連合会 代表理事会長 服部郁弘 日本船長協会 会長 森本靖之 日本船舶機関士協会 会長 大内博文 航海訓練所 練習船船長 竹本孝弘 第二管区海上保安本部長 梅田宜弘
統計的データ解析
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LCM,GCD LCM GCD..,.. 1 LCM GCD a b a b. a divides b. a b. a, b :, CD(a, b) = {d a, b }, CM(a, b) = {m a, b }... CM(a, b). q > 0, m 1, m 2 CM
LCM,GCD 2017 4 21 LCM GCD..,.. 1 LCM GCD a b a b. a divides b. a b. a, b :, CD(a, b) = {d a, b }, CM(a, b) = {m a, b }... CM(a, b). q > 0, m 1, m 2 CM(a, b) = m 1 + m 2 CM(a, b), qm 1 CM(a, b) m 1, m 2
SAMA- SUKU-RU Contents p-adic families of Eisenstein series (modular form) Hecke Eisenstein Eisenstein p T
SAMA- SUKU-RU Contents 1. 1 2. 7.1. p-adic families of Eisenstein series 3 2.1. modular form Hecke 3 2.2. Eisenstein 5 2.3. Eisenstein p 7 3. 7.2. The projection to the ordinary part 9 3.1. The ordinary
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q π =0 Ez,t =ε σ {e ikz ωt e ikz ωt } i/ = ε σ sinkz ωt 5.6 x σ σ *105 q π =1 Ez,t = 1 ε σ + ε π {e ikz ωt e ikz ωt } i/ = 1 ε σ + ε π sinkz ωt 5.7 σ
H k r,t= η 5 Stokes X k, k, ε, ε σ π X Stokes 5.1 5.1.1 Maxwell H = A A *10 A = 1 c A t 5.1 A kη r,t=ε η e ik r ωt 5. k ω ε η k η = σ, π ε σ, ε π σ π A k r,t= q η A kη r,t+qηa kηr,t 5.3 η q η E = 1 c A
x, y x 3 y xy 3 x 2 y + xy 2 x 3 + y 3 = x 3 y xy 3 x 2 y + xy 2 x 3 + y 3 = 15 xy (x y) (x + y) xy (x y) (x y) ( x 2 + xy + y 2) = 15 (x y)
x, y x 3 y xy 3 x 2 y + xy 2 x 3 + y 3 = 15 1 1977 x 3 y xy 3 x 2 y + xy 2 x 3 + y 3 = 15 xy (x y) (x + y) xy (x y) (x y) ( x 2 + xy + y 2) = 15 (x y) ( x 2 y + xy 2 x 2 2xy y 2) = 15 (x y) (x + y) (xy
EPSON LP-8900ユーザーズガイド
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W u = u(x, t) u tt = a 2 u xx, a > 0 (1) D := {(x, t) : 0 x l, t 0} u (0, t) = 0, u (l, t) = 0, t 0 (2)
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C p (.2 C p [[T ]] Bernoull B n,χ C p p q p 2 q = p p = 2 q = 4 ω Techmüller a Z p ω(a a ( mod q φ(q ω(a Z p a pz p ω(a = 0 Z p φ Euler Techmüller ω Q
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p- L- [Iwa] [Iwa2] -Leopoldt [KL] p- L-. Kummer Remann ζ(s Bernoull B n (. ζ( n = B n n, ( n Z p a = Kummer [Kum] ( Kummer p m n 0 ( mod p m n a m n ( mod (p p a ( p m B m m ( pn B n n ( mod pa Z p Kummer
, 1. x 2 1 = (x 1)(x + 1) x 3 1 = (x 1)(x 2 + x + 1). a 2 b 2 = (a b)(a + b) a 3 b 3 = (a b)(a 2 + ab + b 2 ) 2 2, 2.. x a b b 2. b {( 2 a } b )2 1 =
x n 1 1.,,.,. 2..... 4 = 2 2 12 = 2 2 3 6 = 2 3 14 = 2 7 8 = 2 2 2 15 = 3 5 9 = 3 3 16 = 2 2 2 2 10 = 2 5 18 = 2 3 3 2, 3, 5, 7, 11, 13, 17, 19.,, 2,.,.,.,?.,,. 1 , 1. x 2 1 = (x 1)(x + 1) x 3 1 = (x 1)(x
7 π L int = gψ(x)ψ(x)φ(x) + (7.4) [ ] p ψ N = n (7.5) π (π +,π 0,π ) ψ (σ, σ, σ )ψ ( A) σ τ ( L int = gψψφ g N τ ) N π * ) (7.6) π π = (π, π, π ) π ±
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7 7. ( ) SU() SU() 9 ( MeV) p 98.8 π + π 0 n 99.57 9.57 97.4 497.70 δm m 0.4%.% 0.% 0.8% π 9.57 4.96 Σ + Σ 0 Σ 89.6 9.46 K + K 0 49.67 (7.) p p = αp + βn, n n = γp + δn (7.a) [ ] p ψ ψ = Uψ, U = n [ α
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応用数学III-4.ppt
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