2014 x n 1 : : :
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- ふさこ ねぎたや
- 4 years ago
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1 2014 x n 1 : : :
2 x n 1 n x 2 1 = (x 1)(x+1) x 3 1 = (x 1)(x 2 +x+1) x 4 1 = (x 1)(x + 1)(x 2 + 1) x 5 1 = (x 1)(x 4 + x 3 + x 2 + x + 1) 1, 1,0 n = n x n 1 Maple 1, 1,0 n 2
3 1 4 2 x n : x
4 1 x 2 y 2 = (x + y)(x y) x 3 y 3 = (x y)(x 2 + xy + y 2 ) y = 1 x 2 1 = (x + 1)(x 1) x 3 1 = (x 1)(x 2 + x + 1) x x 4 1, x 5 1, x 6 1 x n 1 where n Z, n 2 x n 1 n [7] 4
5 [1, 2, 3, 4, 5, 6, 8, 9] 2 x n 1 x n 1 where n N, n n x 2 1 = (x + 1)(x 1) x 3 1 = (x 1)(x 2 + x + 1) x 4 1 = (x + 1)(x 1)(x 2 + 1) x 5 1 = (x 1)(x 4 + x 3 + x 2 + x + 1) x 6 1 = (x + 1)(x 1)(x 2 + x + 1)(x 2 x + 1) x 7 1 = (x 1) ( x 6 + x 5 + x 4 + x 3 + x 2 + x + 1 ) x 8 1 = (x 1) (x + 1) ( x ) ( x ) x 9 1 = (x 1) ( x 2 + x + 1 ) ( x 6 + x ) x 10 1 = (x 1) (x + 1) ( x 4 + x 3 + x 2 + x + 1 ) ( x 4 x 3 + x 2 x + 1 ) 1 5
6 1. x n 1 x n 1 x x x x x x x x x x n 1 x x n 1 n
7 1 1 2 n f n (x) def = x n 1 P (x) x a P (a) P (a) = 0 P (a) = 0 P (x) x a S(x) P (x) = (x a)s(x) + P (a) = (x a)s(x) P (x) x a P (x) x a P (a) = 0 ( ) ( ) n Z (n 2), f n (1) = 1 n 1 = 0 ( ) P (x) x 1 x n 1 x 1 1 7
8 2 2 n f n (x) def = x n 1 n = 105 Maple factor(x 105 1) //factor (x 1) ( x 6 + x 5 + x 4 + x 3 + x 2 + x + 1 ) ( x 4 + x 3 + x 2 + x + 1 ) ( 1 x + x 5 x 6 + x 7 x 8 + x 10 x 11 + x 12 x 13 + x 14 x 16 +x 17 x 18 + x 19 x 23 + x 24) ( x 2 + x + 1 ) ( 1 x + x 3 x 4 +x 6 x 8 + x 9 x 11 + x 12) ( 1 x + x 3 x 4 + x 5 x 7 + x 8) ( 1 + x + x 2 x 6 x 5 2 x 7 x 24 + x 12 x 8 + x 13 + x 14 + x 16 +x 17 + x 15 + x 48 x 20 x 22 x 26 x 28 + x 31 + x 32 + x 33 + x 34 +x 35 + x 36 x 39 x 40 2 x 41 x 42 x 43 + x 46 + x 47 x 9) f 105 (x)
9 factor(x ) 3 n x n 1 1 n [3] 1 n 1 (n 1) 1 1 n 9
10 2 n n Φ n (x) Φ n (x) = ζ 1 n (x ζ) n = 2, 4 Φ 2 (x) = x + 1, Φ 4 (x) = x p Φ p (x) = xp 1 x 1 = xp 1 + x p x + 1 n 3 Φ n (x) = d n (x d 1) µ(n/d) µ N { 1, 0, 1} 10
11 µ(n) = 1 n = 1 0 n ( 1) k n k [ ] x n 1 = (x ζ) ζ n =1 ζ 1 x n 1 = Φ d (x) (1) d n n f(m) = x m 1, g(m) = Φ m (x) ( ) 1 n f(x), g(x) n m f(m) = g(d) d m 11
12 n m g(m) = d m f(d) µ(m/d) Φ n (x) Z 4 Φ n (x) Z [ ] 3 Φ n (x) = f(x) where f(x), g(x) Z[x] g(x) deg g(x) 1 f(x), g(x) deg g(x) = 0 g(x) = 1 Φ n (x) = f(x) Z[x] 12
13 5 Φ n (x) Z[x] [ ] ζ 1 n f(x) Z[x] ζ Q g(x) Z[x] f(x)g(x) = x n 1 (1) f(x), g(x) Z f (x)g(x) + f(x)g (x) = nx n 1 (2) p n (1) ζ p f(ζ p )g(ζ p ) = 0 f(ζ p ) = 0 g(ζ p ) 0 (2) ζ p f (ζ p )g(ζ p ) + f(ζ p )g (ζ p ) = nζ p(n 1) f(x) p = f(x p ) + p h(x) for some h(x) Z[x] ζ 0 = f(ζ p ) + p h(ζ) 13
14 f (ζ p )g(ζ p ) p h(ζ)g (ζ p ) = nζ p(n 1) g(ζ p ) = 0 p h(ζ)g (ζ p ) = nζ p(n 1) ζ p p ζ p h(ζ)g (ζ p ) + n = 0 f(x) Z q(x), r(x) Z[x] s.t. x p h(x)g (x p ) = q(x)f(x) + r(x) deg r(x) < deg f(x) ζ ζ p h(ζ)g (ζ p ) = r(ζ) p r(ζ) + n = 0 f(x) p r(x) + n 0 r(x) = n/p p n r(x) Z g(ζ p ) 0 f(ζ p ) = 0 q n = p 14
15 ζ q f(ζ pq ) = 0 n m f(ζ m ) = 0 f(x) 1 n f(x) = Φ n (x) Φ n (x) n Φ n (x) Z Φ n (x) = (x d 1) µ(n/d) d n 3 3 (1) Z (1) x n 1 x n 1 n n
16 5 2 Suzuki [7] s Z n s Φ n (x) t 3 t p 1 + p 2 > p t p 1 < p 2 < < p t [ ] t 3 p 1 < p 2 < < p t p 1 + p 2 p t 2p 1 < p t k 2 k 1 2 k t π(2 k ) < kt π(s) s ( 16
17 [1] π(x) > cx ln x c c2k ln 2 k < kt c2k < k 2 t ln 2 k [ 2 ] t 3 p 1 + p 2 > p t p 1 < p 2 < < p t p = p t n = p 1 p t 1 Φ n (x) = d n (x d 1) µ(n/d) t Φ n (x) = (xp 1 1) (x p i 1) x 1 (x p i p j p k 1) (x p i p j 1) mod x p+1 (p i 1)(p j 1) 2 p i p j p i + p j + 1 > p t + 1 = p + 1 x p ip j 0 (mod x p+1 ), x p ip j p k 0 (mod x p+1 ),.... Φ n (x) ± (1 xp 1) (1 x p i) 1 x (mod x p+1 ) 17
18 Φ n (0) = 1 + p i + p j p 1 + p 2 > p (1 x p i ) (1 x p i ) (1 x p 1 x p i ) (mod x p+1 ) (1 x) 1 (1 + x + + x p ) (mod x p+1 ) Φ n (x) (1+x+ +x p )(1 x p 1 x p i ) (mod x p+1 ) x p i x pi+1... x pi+p x p = x p t i x p 1 x p 2 t i Φ n x p t + 1 x p 2 (t 1) + 1 = t + 2 t 3 t + 1 t + 2 p 1 3 p 1 + p 2 > p t Φ 2p1 p t n = p 1 p t 18
19 Φ 2n (x) = Φ n ( x) * 1 x p x p 2 Φ 2n Φ n t 1 t x n 1 0, 1 1 n Input: ( Output: x n 1 0, 1 1 n function MAKE_EXCEPTIONAL_NUMBER(t) Primes Array(1..t) // t flag false while(flag false ) *1 n 3 [5] 19
20 c for(i 1 t 1 ) Primes[i] c c c + endfor if(primes[1]+primes[2] > Primes[t]) flag true endif endwhile d ARRAY_MULTI(Primes) // print( -t+1 ) return(d) endfunction function ARRAY_MULTI(A) n num A[1] for(i 2 n 1 ) num num * A[i] endfor return(num) endfunction 20
21 Maple n 6.2 Maple 1 >MAKE_EXCEPTIONAL_NUMBER(3) // 3 x n = 429 p 1 = 3 < p 2 = 11 < p 3 = 13 p 1 + p 2 = 14 > 13 = p 3 n = p 1 p 2 p 3 x = 2 21
22 factor(x 429 1) (x 1) ( 1 + x 12 + x 11 + x 10 + x 9 + x 8 + x 7 + x 6 + x 5 + x 4 + x 3 + x 2 + x ) ( 1 + x 10 + x 9 + x 8 + x 7 + x 6 + x 5 + x 4 + x 3 + x 2 + x ) ( 1 x x 12 + x 11 + x 22 x 14 + x 13 x 38 + x 57 + x 76 x 95 x 23 + x 46 x 69 + x 24 + x 39 x 47 x 58 + x 61 x 62 + x 65 + x 70 x 80 + x 81 x 84 + x 85 x 93 + x 96 + x 107 x 108 x x 33 + x 44 + x 55 + x 35 x 34 x 45 x 49 + x 50 + x 52 x 53 x 56 x 60 +x 63 x 64 x 67 + x 68 x 71 + x 72 + x 74 x 75 x 82 + x 83 x 86 + x 87 + x 94 x 97 + x 98 x x x 120 x 25 + x 26 x 27 + x 48 x 51 x 73 x 40 +x 59 x 36 + x 37 ) ( 1 + x 2 + x ) ( 1 x + x 3 x 4 + x 6 x 7 + x 9 x 10 + x 12 x 14 + x 15 x 17 + x 18 x 20 + x 21 x 23 + x 24 ) ( 1 x + x 3 x 4 + x 6 x 7 +x 9 x 10 + x 11 x 13 + x 14 x 16 + x 17 x 19 + x 20 ) ( 1 x 124 x x 136 x 147 x x x 216 x x x x x 144 x 151 x x 155 x 159 x x x x + x 172 x 54 x x x 201 +x x 207 x x x 182 x 186 x 189 x 193 x 12 x 11 + x 2 x 14 2 x 13 + x 57 2 x 46 + x 24 + x 39 x 47 + x 58 + x 65 x 81 x 84 2 x 85 x 93 +x 96 + x x 33 x 44 + x 35 + x 34 x 45 x 50 2 x 52 x 53 + x 63 + x 64 +x 67 + x 68 + x 72 + x 74 x 83 x 86 x 87 + x 97 + x 98 + x 106 x x 25 + x 26 x 48 x 51 + x 73 + x 40 + x 59 x 15 x x 133 x x 138 x x 66 x 77 x 89 + x 99 + x x x 104 x 112 x 116 x 118 x 122 x 126 x 130 +x x x x x 142 x 149 x 154 x x x x 175 +x x x 183 x x 188 x x 194 x x x x 215 x x 227 x x x 240 x x x x 41 x 79 x 91 +x x x 105 ) 2 22
23 2 >MAKE_EXCEPTIONAL_NUMBER(5) // 5 x n = p 1 = 11 < p 2 = 13 < p 3 = 17 < p 4 = 19 < p 5 = 23 p 1 + p 2 = 24 > 23 = p 5 n = p 1 p 2 p 3 p 4 p 5 x = x n = 4807 p 1 = 11 < p 2 = 19 < p 3 = 23 p 1 + p 2 = 30 > 23 = p 3 n = p 1 p 2 p 3 x =
24 7 x n 1 x 1 0, n n 0, 1 1 ( n n ( ) x n 1 t t p 1 = 11 < p 2 = 13 < p 3 = 19 < p 4 = 23 p 1 + p 2 = 24 > 23 = p 4 n = p 1 p 2 p 3 p 4 = = 3 24
25 1. t 4 n 2. n 3. n 25
26 8 26
27 [1] H. Davenport, Multiplicative Number Thoery, Markham Publ. Co., Chicago, [2], I,, [3],,, [4],,, [5] V.V. Prasolov, Polynomials, Springer, [6] D.J.S. Robinson, An Introduction to Abstract Algebra, Walter de Gruyter, [7] J. Suzuki, On Coefficients of Cyclotomic Polynomials, Proc. Japan Acad., 63, Ser. A [8], 1,, [9], 2,,
28 9 : x factor(x ) (x 1) ( 1 + x 22 + x 21 + x 20 + x 19 + x 18 + x 17 + x 16 + x 15 + x 14 +x 13 + x 12 + x 11 + x 10 + x 9 + x 8 + x 7 + x 6 + x 5 + x 4 + x 3 +x 2 + x ) ( 1 + x 18 + x 17 + x 16 + x 15 + x 14 + x 13 + x 12 + x 11 +x 10 + x 9 + x 8 + x 7 + x 6 + x 5 + x 4 + x 3 + x 2 + x ) (1 x +x 396 x x 377 x x 373 x x 358 x x 354 x x 350 x x 339 x x 335 x x 331 x 330 +x 327 x x 320 x x 316 x x 312 x x 308 x x 304 x x 301 x x 297 x x 293 x 292 +x 289 x x 285 x x 282 x x 278 x x 274 x x 270 x x 266 x x 263 x x 259 x 257 +x 255 x x 251 x x 247 x x 244 x x 240 x x 236 x x 232 x x 228 x x 225 x 223 +x 221 x x 217 x x 213 x x 209 x x 206 x x 202 x x 198 x x 194 x x 190 x
29 +x 187 x x 183 x x 179 x x 175 x x 171 x x 168 x x 164 x x 160 x x 156 x 154 +x 152 x x 149 x x 145 x x 141 x x 137 x x 133 x x 130 x x 126 x x 122 x 119 +x 118 x x 114 x x 111 x x 107 x x 103 x x 99 x 96 + x 95 x 93 + x 92 x 89 + x 88 x 85 + x 84 x 81 + x 80 x 77 + x 76 x 70 + x 69 x 66 + x 65 x 62 + x 61 x 58 + x 57 x 47 + x 46 x 43 + x 42 x 39 + x 38 x 24 + x 23 x 20 + x 19 ) ( 1 + x 10 + x 9 + x 8 + x 7 + x 6 + x 5 + x 4 + x 3 + x 2 +x) ( 1 x + x 220 x x 209 x x 198 x x 187 x x 176 x x 165 x x 154 x x 143 x 139 +x 132 x x 121 x x 110 x x 99 x 93 + x 88 x 81 + x 77 x 70 + x 66 x 58 + x 55 x 47 + x 44 x 35 + x 33 x 24 + x 22 x 12 + x 11 ) ( 1 x + x 180 x x 169 x 168 +x 161 x x 158 x x 150 x x 147 x x 142 x x 139 x x 136 x x 131 x x 128 x 127 +x 125 x x 123 x x 120 x x 117 x x 114 x x 112 x x 109 x x 106 x x 104 x 102 +x 101 x x 98 x 97 + x 95 x 94 + x 93 x 91 + x 90 x 89 + x 87 x 86 + x 85 x 83 + x 82 x 80 + x 79 x 78 + x 76 x 75 + x 74 x 72 + x 71 x 69 + x 68 x 67 + x 66 x 64 + x 63 29
30 x 61 + x 60 x 58 + x 57 x 56 + x 55 x 53 + x 52 x 50 + x 49 x 45 + x 44 x 42 + x 41 x 39 + x 38 x 34 + x 33 x 31 + x 30 x 23 + x 22 x 20 + x 19 x 12 + x 11 ) ( 1 + x x x x 557 +x x x x x x x x x 548 x 539 x 538 x 537 x x x x x x x x 529 x 528 x 527 x 526 x x 516 +x x x x x x x x x x x x 504 x 495 x 494 x 493 x x x 490 x 489 x 488 x 487 x 486 x x x 479 +x x x x x x 467 x 461 x 460 x 459 x 458 x 457 x 456 x 451 x 450 x 449 x 448 x 447 x 446 x 445 x 444 x 443 x 442 x x x x x 425 +x x x x x x x x x 304 +x x x x x x x x x 295 x 286 x 285 x 284 x x x x x x x x 276 x 275 x 274 x 273 x x 263 +x x x x x x x x x x x x 251 x 242 x 241 x 240 x x x x x x x x
31 x 231 x 230 x 229 x x x x x x 215 +x x x x x x x 52 + x 51 + x 50 +x 49 + x 48 + x 47 + x 46 + x 45 + x 44 + x 43 + x 42 x 33 x 32 x 31 x 30 2 x 29 2 x 28 2 x 27 2 x 26 2 x 25 2 x 24 2 x 23 x 22 x 21 x 20 x 19 + x 10 + x 9 + x 8 + x 7 + x 6 +x 5 + x 4 + x 3 + x 2 + x x 635 x 650 x 655 x 660 x 665 x x x 690 x 700 x x x 725 x 740 x x x 765 x x 785 x x x x 840 +x 845 x 860 x x x x 890 x 895 x 900 x 905 x x x x 945 x 955 x x x 985 x 995 x x x x x x x x x 1060 x 1070 x x x 1095 x 1105 x 1110 x 1115 x x x x x 1140 x 1150 x 1155 x 1160 x x x x 1195 x 1205 x 1210 x 1215 x x 1230 x x x 2242 x 2246 x 2247 x 2261 x x x x x x x x x x 2287 x 2291 x 2292 x 2296 x 2297 x 2301 x 2302 x 2306 x x x x x x x x x 2327 x 2332 x 2336 x 2337 x x x x x x 2362 x 2367 x 2371 x x x
32 x 2381 x 2382 x x x x x x x x x x x x x x x 2422 x 2426 x x x x x x x x x x 2452 x x 2457 x 2461 x 2462 x 2466 x 2467 x 2471 x 1661 x 1663 x 1666 x 1668 x x x x x x x x x 1693 x 1696 x 1698 x 1711 x x x x 1736 x 1741 x 1743 x 1746 x 1748 x 1751 x x 1756 x x x x x x x x x 1781 x 1783 x 1786 x 1788 x x x x x 1801 x x x x x x x x x 1823 x 1828 x 1831 x x 1836 x 1838 x 1841 x 1843 x x x x x x x 1871 x 1872 x 1876 x x x x x x x x x x x x x x 1912 x 1916 x 1917 x 1921 x x x x x x x x x x 1946 x 1951 x 1952 x 1966 x 1967 x x 1331 x 1333 x 1336 x x x x x x 1353 x 1366 x 1368 x 1371 x x x x
33 +2 x x x x x 1396 x 1401 x 1403 x 1406 x 1408 x 1411 x 1413 x 1416 x x x x x x x 1451 x 1458 x x x x 1483 x 1486 x 1488 x 1493 x 1496 x 1498 x x 1503 x x x x x x x x x x 1528 x 1531 x 1533 x x x x x x x x x x x x x x 1571 x 1576 x 1578 x x 1583 x 1586 x 1588 x 1591 x x x x x x x 1618 x 1621 x 1623 x 1626 x x x x x x x x x 1648 x 1653 x 1656 x x x 1972 x 1976 x 1977 x 1981 x x 1987 x 1992 x 1996 x 2011 x x x x x x x x 2032 x 2036 x 2037 x 2041 x x x x x 2052 x x x x x x x x x 2077 x 2081 x 2082 x 2086 x x x x x x x 2112 x 2116 x 2117 x 2121 x 2122 x 2126 x 2127 x 2131 x x x x x x x x x 2152 x x x x x
34 x 2171 x 2172 x 2176 x x x x x x x x 2197 x 2201 x 2202 x 2206 x 2207 x 2211 x 2212 x 2217 x x x 2227 x 861 x 864 x 866 x x x x x 889 x 894 x 896 x 899 x 901 x 904 x 906 x 909 x x x x x x 931 +x x x x 946 x 954 x 956 x 964 x x 974 +x x 986 x 991 x x 996 x 999 x x x x x x 1021 x 1031 x x x x 1041 x x x x x x x x x 1064 x 1069 x 1071 x 1074 x x x x x x 1099 x 1104 x 1106 x 1109 x 1111 x 1114 x 1116 x 1119 x x x x x x x x 1141 x 1146 x 1149 x 1151 x x 1156 x 1159 x 1161 x 1164 x x x x x x x x 1199 x 1206 x 1209 x 1216 x x x x x 1238 x 1246 x 1248 x 1251 x x x x x x x x x 1273 x 1278 x 1281 x x x x x 1293 x x x x x x x x 1316 x 1321 x 1323 x x x x x 636 x
35 x 654 x 656 x 659 x 666 x x x x x 689 +x 691 x 699 x 701 x 704 x 709 x 711 x x x 724 x 739 x x 744 x x x x x 766 +x 769 x 779 x x x 786 x 789 x x x 804 +x x x x x x x x 846 x 859 x 3094 x x x x x x 843 x 862 x 863 +x x 883 x 893 x 897 x 907 x 908 x x x 942 x 952 x 953 x 957 x x 997 x x x x x 1018 x 1032 x x 1038 x 1042 x x x x x 1058 x 1117 x 1118 x x x x x 1138 x 1147 x x x x 633 +x 637 x 652 x 653 x 667 x x x 678 x 703 x 712 x x x 733 x 738 x x 743 x x x x x x x 768 x x x x x 788 x 792 x 3093 x x x 722 x x x 692 +x 688 x 658 x x x 1022 x 993 x x x 977 +x 973 x 963 x x x x 932 x x 903 x 902 x x x 887 x 868 x x x x x x x x x x x x x 2033 x 2038 x 2039 x x x x x
36 2 x x x x 1945 x 1948 x 1949 x x x 1973 x 1978 x 1979 x x x 1990 x 1993 x 1994 x 2008 x 2009 x 1840 x x x x x x x 1859 x 1870 x 1873 x 1874 x 1879 x x x x x x x x x 1900 x x x x x 1915 x 1918 x 1919 x x x x x 1934 x 1697 x 1699 x 1712 x 1714 x x x x x 1735 x 1739 x 1742 x 1747 x 1750 x 1754 x 1757 x x x x x x x x x 1780 x 1784 x 1787 x 1789 x x x x x x x x x x x x x x 1825 x 1829 x 1832 x 1834 x x 1837 x x 1502 x 1504 x x x x x x x x 1525 x 1532 x 1534 x x x x x x 1549 x x x x x x x x x 1570 x 1574 x 1577 x 1579 x 1580 x 1582 x 1585 x 1589 x x x x x x 1607 x 1619 x 1622 x 1624 x 1625 x x x x x x x x 1649 x 1652 x
37 x 1655 x 1657 x 1660 x 1664 x 1667 x 1669 x x x x x x x 1690 x x 1228 x 1244 x x x 1250 x x x x x x x x x 1274 x 1277 x 1279 x 1280 x x x x x 1294 x 1295 x x x x x x x x 1315 x 1322 x 1324 x 1325 x x 1330 x 1334 x 1337 x 1339 x x x x x x 1354 x 1364 x 1367 x 1369 x 1370 x 1372 x x x x x x x x 1394 x 1399 x 1400 x 1402 x x 1409 x 1412 x 1414 x 1415 x x x x x x x 1450 x 1459 x 1460 x x x x 1484 x 1487 x 1489 x 1490 x x x x 1063 x 1072 x 1073 x x x 1097 x 1102 x 1103 x x x x 1142 x 1153 x 1157 x 1162 x x x x x x 1198 x 1207 x x 3960 x x x x x 1317 x 1329 x 1332 x x x x x 1365 x 1068 x x x 1098 x 1107 x x x 1143 x x x
38 x x 1422 x x x 1239 x x x x x x x x x x x x x x 1395 x 1404 x 1407 x 1410 x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x 2676 x 2677 x 2678 x 2679 x 2680 x 2681 x 2682 x x x x x x x x x x x x x x x x x x x x x 2661 x 2663 x 2664 x x x 2667 x x x x x x x x x x x x x x 2618 x 2620 x 2621 x 2622 x 2623 x 2624 x 2625 x 2626 x 2627 x x x 2630 x 2631 x 2632 x 2633 x 2634 x 2635 x 2636 x 2637 x 2638 x 2639 x 2544 x 2545 x 2546 x 2547 x 2548 x 2549 x x 2551 x 2552 x 2553 x 2554 x 2555 x 2556 x 2557 x 2558 x 2559 x 2560 x x x
39 +x x x x x x x x x x x x x x x x x 2582 x 2585 x 2586 x 2587 x 2588 x 2589 x 2590 x 2591 x 2592 x 2593 x x x x x x x x x 2450 x 2454 x x 2458 x 2459 x 2460 x 2463 x 2464 x 2465 x 2470 x 2473 x x x x x x x x x x x 2496 x 2498 x 2499 x 2500 x 2501 x x x x x x x x x x x x x x x x x x 2538 x 2541 x 2542 x 2543 x 2333 x 2334 x 2335 x 2338 x 2339 x 2340 x x x x x x x x x x 2365 x 2368 x 2369 x 2370 x 2373 x 2374 x 2375 x 2378 x 2379 x 2380 x 2383 x 2384 x x x x x x x x x x x x x x 2408 x x x x x x x x 2423 x 2424 x 2425 x 2428 x 2429 x x x x x x
40 +x x x 2034 x 2040 x x x x x x x x x x x x x x 2078 x 2080 x 2083 x 2084 x 2085 x x 2089 x x x x x x x x 2110 x 2114 x 2115 x 2118 x 2119 x x x 2124 x 2125 x 2128 x 2129 x 2130 x x x x x x x x x x x x x x x x x x x x x 2168 x 2169 x x x x x x x x 1824 x 1827 x 1830 x 1839 x 1842 x x x 1860 x 1875 x x x x x x x x 1914 x 1920 x x x x x 1944 x 1950 x 1965 x 1968 x x 1989 x 1995 x x x x x x 1482 x 1494 x 1497 x x x x x x 1527 x x x x x x x x x 1572 x x 1584 x 1587 x x x x 1605 x 1617 x x x x x 1647 x x 1662 x
41 +x x x x x 1692 x x x x 1737 x 1740 x 1749 x x x x x 1779 x x x x x 1452 x 2707 x 2708 x x x 2711 x 2712 x 2713 x 2714 x 2715 x x x x x x x x x 2734 x 2740 x 2741 x 2742 x 2743 x 2744 x 2745 x 2750 x 2751 x 2752 x 2753 x 2754 x x x x x x x x x x x x x x x x x x 2791 x 2794 x 2795 x 2796 x 2797 x 2798 x 2799 x 2800 x 2801 x 2802 x x 2804 x 2805 x 2806 x 2807 x 2808 x 2809 x 2810 x 2811 x 2812 x 2813 x x x x x x x x x x x x x x x x x x x x 2835 x 2838 x 2839 x 2840 x 2841 x 2842 x 2843 x 2844 x 2845 x 2846 x x 2848 x 2849 x 2850 x 2851 x 2852 x 2853 x 2854 x 2855 x 2856 x 2857 x x x x x x x x x x x x
42 x 2882 x 2883 x 2884 x 2885 x 2886 x 2887 x 2888 x 2889 x 2890 x 2891 x x x x x x x x x x x x x x x x x x x x 2914 x 2916 x 2917 x x x x x x x x 2925 x 2926 x 2927 x 2173 x 2174 x x x x x x x x x x x x x 2199 x x x 2205 x 2208 x 2209 x 2210 x 2213 x 2214 x 2218 x 2219 x x x x x x x 2243 x 2245 x 2248 x 2249 x 2263 x 2264 x x x x x x x x x x x 2285 x 2289 x 2290 x 2293 x 2294 x x 2298 x 2299 x 2300 x 2303 x 2304 x 2305 x x x x x x x x x x x x 2329 x 2928 x x x x x x x x x x x x x x 2950 x 2959 x 2960 x 2961 x x x 2964 x 2965 x 2966 x 2967 x 2968 x x
43 +x x x x x x x 2987 x 2993 x 2994 x 2995 x 2996 x 2997 x 2998 x 3003 x 3004 x 3005 x 3006 x 3007 x x x x x x x x x x x x x x x x x x 3044 x 3047 x 3048 x 3049 x 3050 x 3051 x 3052 x 3053 x 3054 x 3055 x x 3057 x 3058 x 3059 x 3060 x 3061 x 3062 x 3063 x 3064 x 3065 x 3066 x x x x x x x x x x x x 3080 x 3091 x 3092 x 3096 x 3097 x 3098 x 3099 x 3100 x x x x x x x x x x x x x x x x x x x x x x x 3159 x 3168 x 3169 x 3170 x x x x x x x x 3178 x 3179 x 3180 x 3181 x x x x x x x x x x x x x x 3203 x 3212 x 3213 x 3214 x x x 3217 x 3218 x 3219 x 3220 x 3221 x x x x
44 +x x x x x 3240 x 3246 x 3247 x 3248 x 3249 x 3250 x 3251 x 3256 x 3257 x 3258 x 3259 x 3260 x x x x x x x x x x x x x 3284 x 3290 x 3291 x 3292 x 3293 x 3294 x 3295 x 3300 x 3301 x 3302 x 3303 x 3304 x 3305 x 3306 x 3307 x 3308 x 3309 x x x x x x x x x x x x x x x x x x x x x x x 3412 x 3421 x 3422 x 3423 x x x x x x x x 3431 x 3432 x 3433 x 3434 x x x x x x x x x x x x x x 3456 x 3465 x 3466 x 3467 x x x 3470 x 3471 x 3472 x 3473 x 3474 x x x x x x x x x 3493 x 3499 x 3500 x 3501 x 3502 x 3503 x 3504 x 3509 x 3510 x 3511 x 3512 x 3513 x 3514 x 3515 x 3516 x 3517 x 3518 x x x x x x x x x
45 +x x x x x x x x x x x x x x 3665 x 3674 x 3675 x 3676 x x x x x x x x 3684 x 3685 x 3686 x 3687 x x x x x x x x x x x x x x 3709 x 3718 x 3719 x 3720 x x x x x x x x 3728 x 3729 x 3730 x 3731 x x x x x x x x x x x x x x x x x x x x x x x 3918 x 3927 x 3928 x 3929 x x x x x x x x 3937 x 3938 x 3939 x 3940 x x x x x x x x x x x 3959 ) 45
DVIOUT
A. A. A-- [ ] f(x) x = f 00 (x) f 0 () =0 f 00 () > 0= f(x) x = f 00 () < 0= f(x) x = A--2 [ ] f(x) D f 00 (x) > 0= y = f(x) f 00 (x) < 0= y = f(x) P (, f()) f 00 () =0 A--3 [ ] y = f(x) [, b] x = f (y)
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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
1. A0 A B A0 A : A1,...,A5 B : B1,...,B
1. A0 A B A0 A : A1,...,A5 B : B1,...,B12 2. 3. 4. 5. A0 A, B Z Z m, n Z m n m, n A m, n B m=n (1) A, B (2) A B = A B = Z/ π : Z Z/ (3) A B Z/ (4) Z/ A, B (5) f : Z Z f(n) = n f = g π g : Z/ Z A, B (6)
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8 ( ) 8 5 4 I II III A B C( ),,, 5 I II A B ( ),, I II A B (8 ) 6 8 I II III A B C(8 ) n ( + x) n () n C + n C + + n C n = 7 n () 7 9 C : y = x x A(, 6) () A C () C P AP Q () () () 4 A(,, ) B(,, ) C(,,
基礎数学I
I & II ii ii........... 22................. 25 12............... 28.................. 28.................... 31............. 32.................. 34 3 1 9.................... 1....................... 1............
No2 4 y =sinx (5) y = p sin(2x +3) (6) y = 1 tan(3x 2) (7) y =cos 2 (4x +5) (8) y = cos x 1+sinx 5 (1) y =sinx cos x 6 f(x) = sin(sin x) f 0 (π) (2) y
No1 1 (1) 2 f(x) =1+x + x 2 + + x n, g(x) = 1 (n +1)xn + nx n+1 (1 x) 2 x 6= 1 f 0 (x) =g(x) y = f(x)g(x) y 0 = f 0 (x)g(x)+f(x)g 0 (x) 3 (1) y = x2 x +1 x (2) y = 1 g(x) y0 = g0 (x) {g(x)} 2 (2) y = µ
f(x) = f(x ) + α(x)(x x ) α(x) x = x. x = f (y), x = f (y ) y = f f (y) = f f (y ) + α(f (y))(f (y) f (y )) f (y) = f (y ) + α(f (y)) (y y ) ( (2) ) f
22 A 3,4 No.3 () (2) (3) (4), (5) (6) (7) (8) () n x = (x,, x n ), = (,, n ), x = ( (x i i ) 2 ) /2 f(x) R n f(x) = f() + i α i (x ) i + o( x ) α,, α n g(x) = o( x )) lim x g(x) x = y = f() + i α i(x )
1. A0 A B A0 A : A1,...,A5 B : B1,...,B12 2. 5 3. 4. 5. A0 (1) A, B A B f K K A ϕ 1, ϕ 2 f ϕ 1 = f ϕ 2 ϕ 1 = ϕ 2 (2) N A 1, A 2, A 3,... N A n X N n X N, A n N n=1 1 A1 d (d 2) A (, k A k = O), A O. f
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() 1.1.. 1. 1.1. (1) L K (i) 0 K 1 K (ii) x, y K x + y K, x y K (iii) x, y K xy K (iv) x K \ {0} x 1 K K L L K ( 0 L 1 L ) L K L/K (2) K M L M K L 1.1. C C 1.2. R K = {a + b 3 i a, b Q} Q( 2, 3) = Q( 2
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数論 サンプルページ この本の定価 判型などは, 以下の URL からご覧いただけます. http://www.morikita.co.jp/books/mid/008142 このサンプルページの内容は, 第 2 版 1 刷発行当時のものです. Daniel DUVERNEY: THÉORIE DES NOMBRES c Dunod, Paris, 1998, This book is published
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![Z[i] Z[i] π 4,1 (x) π 4,3 (x) 1 x (x ) 2 log x π m,a (x) 1 x ϕ(m) log x 1.1 ( ). π(x) x (a, m) = 1 π m,a (x) x modm a 1 π m,a (x) 1 ϕ(m) π(x)](/thumbs/94/118373946.jpg "Z[i] Z[i] π 4,1 (x) π 4,3 (x) 1 x (x ) 2 log x π m,a (x) 1 x ϕ(m) log x 1.1 ( ). π(x) x (a, m) = 1 π m,a (x) x modm a 1 π m,a (x) 1 ϕ(m) π(x)")
3 3 22 Z[i] Z[i] π 4, (x) π 4,3 (x) x (x ) 2 log x π m,a (x) x ϕ(m) log x. ( ). π(x) x (a, m) = π m,a (x) x modm a π m,a (x) ϕ(m) π(x) ϕ(m) x log x ϕ(m) m f(x) g(x) (x α) lim f(x)/g(x) = x α mod m (a,
2012 A, N, Z, Q, R, C
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1 1 n 0, 1, 2,, n n 2 a, b a n b n a, b n a b (mod n) 1 1. n = (mod 10) 2. n = (mod 9) n II Z n := {0, 1, 2,, n 1} 1.
1 1 n 0, 1, 2,, n 1 1.1 n 2 a, b a n b n a, b n a b (mod n) 1 1. n = 10 1567 237 (mod 10) 2. n = 9 1567 1826578 (mod 9) n II Z n := {0, 1, 2,, n 1} 1.2 a b a = bq + r (0 r < b) q, r q a b r 2 1. a = 456,
ORIGINAL TEXT I II A B 1 4 13 21 27 44 54 64 84 98 113 126 138 146 165 175 181 188 198 213 225 234 244 261 268 273 2 281 I II A B 292 3 I II A B c 1 1 (1) x 2 + 4xy + 4y 2 x 2y 2 (2) 8x 2 + 16xy + 6y 2
1 1.1 R (ring) R1 R4 R1 R (commutative [abelian] group) R2 a, b, c R (ab)c = a(bc) (associative law) R3 a, b, c R a(b + c) = ab + ac, (a + b)c = ac +
![1 1.1 R (ring) R1 R4 R1 R (commutative [abelian] group) R2 a, b, c R (ab)c = a(bc) (associative law) R3 a, b, c R a(b + c) = ab + ac, (a + b)c = ac +](/thumbs/74/70287522.jpg "1 1.1 R (ring) R1 R4 R1 R (commutative [abelian] group) R2 a, b, c R (ab)c = a(bc) (associative law) R3 a, b, c R a(b + c) = ab + ac, (a + b)c = ac +")
ALGEBRA II Hiroshi SUZUKI Department of Mathematics International Christian University 2004 1 1 1 2 2 1 3 3 1 4 4 1 5 5 1 6 6 1 7 7 1 7.1....................... 7 1 7.2........................... 7 4 8
r 1 m A r/m i) t ii) m i) t B(t; m) ( B(t; m) = A 1 + r ) mt m ii) B(t; m) ( B(t; m) = A 1 + r ) mt m { ( = A 1 + r ) m } rt r m n = m r m n B
1 1.1 1 r 1 m A r/m i) t ii) m i) t Bt; m) Bt; m) = A 1 + r ) mt m ii) Bt; m) Bt; m) = A 1 + r ) mt m { = A 1 + r ) m } rt r m n = m r m n Bt; m) Aert e lim 1 + 1 n 1.1) n!1 n) e a 1, a 2, a 3,... {a n
() Remrk I = [0, ] [x i, x i ]. (x : ) f(x) = 0 (x : ) ξ i, (f) = f(ξ i )(x i x i ) = (x i x i ) = ξ i, (f) = f(ξ i )(x i x i ) = 0 (f) 0.
![() Remrk I = [0, ] [x i, x i ]. (x : ) f(x) = 0 (x : ) ξ i, (f) = f(ξ i )(x i x i ) = (x i x i ) = ξ i, (f) = f(ξ i )(x i x i ) = 0 (f) 0.](/thumbs/89/98384840.jpg "() Remrk I = [0, ] [x i, x i ]. (x : ) f(x) = 0 (x : ) ξ i, (f) = f(ξ i )(x i x i ) = (x i x i ) = ξ i, (f) = f(ξ i )(x i x i ) = 0 (f) 0.")
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I, II 1, A = A 4 : 6 = max{ A, } A A 10 10%
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18 5 10 1 1 1.1 1.1.1 P Q P Q, P, Q P Q P Q P Q, P, Q 2 1 1.1.2 P.Q T F Z R 0 1 x, y x + y x y x y = y x x (y z) = (x y) z x + y = y + x x + (y + z) = (x + y) + z P.Q V = {T, F } V P.Q P.Q T F T F 1.1.3
1 No.1 5 C 1 I III F 1 F 2 F 1 F 2 2 Φ 2 (t) = Φ 1 (t) Φ 1 (t t). = Φ 1(t) t = ( 1.5e 0.5t 2.4e 4t 2e 10t ) τ < 0 t > τ Φ 2 (t) < 0 lim t Φ 2 (t) = 0
1 No.1 5 C 1 I III F 1 F 2 F 1 F 2 2 Φ 2 (t) = Φ 1 (t) Φ 1 (t t). = Φ 1(t) t = ( 1.5e 0.5t 2.4e 4t 2e 10t ) τ < 0 t > τ Φ 2 (t) < 0 lim t Φ 2 (t) = 0 0 < t < τ I II 0 No.2 2 C x y x y > 0 x 0 x > b a dx
ax 2 + bx + c = n 8 (n ) a n x n + a n 1 x n a 1 x + a 0 = 0 ( a n, a n 1,, a 1, a 0 a n 0) n n ( ) ( ) ax 3 + bx 2 + cx + d = 0 4
20 20.0 ( ) 8 y = ax 2 + bx + c 443 ax 2 + bx + c = 0 20.1 20.1.1 n 8 (n ) a n x n + a n 1 x n 1 + + a 1 x + a 0 = 0 ( a n, a n 1,, a 1, a 0 a n 0) n n ( ) ( ) ax 3 + bx 2 + cx + d = 0 444 ( a, b, c, d
7-12.dvi
26 12 1 23. xyz ϕ f(x, y, z) Φ F (x, y, z) = F (x, y, z) G(x, y, z) rot(grad ϕ) rot(grad f) H(x, y, z) div(rot Φ) div(rot F ) (x, y, z) rot(grad f) = rot f x f y f z = (f z ) y (f y ) z (f x ) z (f z )
0 1-4. 1-5. (1) + b = b +, (2) b = b, (3) + 0 =, (4) 1 =, (5) ( + b) + c = + (b + c), (6) ( b) c = (b c), (7) (b + c) = b + c, (8) ( + b)c = c + bc (9
1-1. 1, 2, 3, 4, 5, 6, 7,, 100,, 1000, n, m m m n n 0 n, m m n 1-2. 0 m n m n 0 2 = 1.41421356 π = 3.141516 1-3. 1 0 1-4. 1-5. (1) + b = b +, (2) b = b, (3) + 0 =, (4) 1 =, (5) ( + b) + c = + (b + c),
2 1,384,000 2,000,000 1,296,211 1,793,925 38,000 54,500 27,804 43,187 41,000 60,000 31,776 49,017 8,781 18,663 25,000 35,300 3 4 5 6 1,296,211 1,793,925 27,804 43,187 1,275,648 1,753,306 29,387 43,025
, = = 7 6 = 42, =
http://www.ss.u-tokai.ac.jp/~mahoro/2016autumn/alg_intro/ 1 1 2016.9.26, http://www.ss.u-tokai.ac.jp/~mahoro/2016autumn/alg_intro/ 1.1 1 214 132 = 28258 2 + 1 + 4 1 + 3 + 2 = 7 6 = 42, 4 + 2 = 6 2 + 8
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V 0 = + r pv (H) + qv (T ) = + r ps (H) + qs (T ) = S 0 X n+ (T ) = n S n+ (T ) + ( + r)(x n n S n ) = ( + r)x n + n (d r)s n = ( + r)v n + V n+(h) V
I (..2) (0 < d < + r < u) X 0, X X = 0 S + ( + r)(x 0 0 S 0 ) () X 0 = 0, P (X 0) =, P (X > 0) > 0 0 H, T () X 0 = 0, X (H) = 0 us 0 ( + r) 0 S 0 = 0 S 0 (u r) X (T ) = 0 ds 0 ( + r) 0 S 0 = 0 S 0 (d r)
1. A0 A B A0 A : A1,...,A5 B : B1,...,B
1. A0 A B A0 A : A1,...,A5 B : B1,...,B12 2. 3. 4. 5. A0 A B f : A B 4 (i) f (ii) f (iii) C 2 g, h: C A f g = f h g = h (iv) C 2 g, h: B C g f = h f g = h 4 (1) (i) (iii) (2) (iii) (i) (3) (ii) (iv) (4)
応用数学特論.dvi
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dy + P (x)y = Q(x) (1) dx dy dx = P (x)y + Q(x) P (x), Q(x) dy y dx Q(x) 0 homogeneous dy dx = P (x)y 1 y dy = P (x) dx log y = P (x) dx + C y = C exp
+ P (x)y = Q(x) (1) = P (x)y + Q(x) P (x), Q(x) y Q(x) 0 homogeneous = P (x)y 1 y = P (x) log y = P (x) + C y = C exp{ P (x) } = C e R P (x) 5.1 + P (x)y = 0 (2) y = C exp{ P (x) } = Ce R P (x) (3) αy
40 6 y mx x, y 0, 0 x 0. x,y 0,0 y x + y x 0 mx x + mx m + m m 7 sin y x, x x sin y x x. x sin y x,y 0,0 x 0. 8 x r cos θ y r sin θ x, y 0, 0, r 0. x,
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