Z = X 2 + ky 2 Z
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- あきみ なかじゅく
- 5 years ago
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1 Z = X 2 + ky 2 Z Z = X 2 + ky mod (α) (α) X,Y Z = X 2 + ky 2 Z X k Z = X 2 + ky 2 k = 2, 3, 5, 6, 7 9 = =
2 2 2.1 prolog X, Y Z = X 2 + ky 2 Z 2.2 for(i=<j,i):- I=<J. for(i=<j,k):- I=<J, I1 is I+1,for(I1=<J,K). gcd(a=(a,0)):-!. gcd(d=(a,b)):- B1 is A mod B,A1=B, gcd(d=(a1,b1)). gcd(a=a*1+0*0). gcd(d=a*x+b*y):- res_q(a=b*q+r), (A1,B1)=(B,R), gcd(d=a1*x1+b1*y1), T is X1-Y1*Q,(X,Y)=(Y1,T) ?- gcd(d=144*x+39*y). D = 3, X = 3, Y = -11 2
3 factor(p/2):- Q is P//2,P=:=2*Q,!. factor(p/i):- P1 is floor(sqrt(p)), for(1=<p1,j), J1 is 2*J+1, Q is P//J1, P=:=J1*Q,I=J1,!. factor(p/p):-!. factorize(p,[p]):- factor(p/i),p==i,!. factorize(p,list):- factor(p/i), P1 is P//I, List=[I List1], factorize(p1,list1),!. 48 1?- factorize(48,l). L = [2,2,2,2,3]. Z = X 2 + 2Y 2 bunkai2:- for(1=<100,x),for(1=<100,y), gcd(d=(x,y)),x mod 2\==0,Z is X*X+2*Y*Y,Z<2000,Z mod 2=\=0, D==1,factorize(Z,L),write(Z),tab(9),write(L),tab(9),write( X*X+2*Y*Y),nl,fail. bunkai2. 3 [3] 1*1+2*1*1 9 [3, 3] 1*1+2*2*2 19 [19] 1*1+2*3*3 33 [3, 11] 1*1+2*4*4 51 [3, 17] 1*1+2*5*5 73 [73] 1*1+2*6*6 99 [3, 3, 11] 1*1+2*7*7 129 [3, 43] 1*1+2*8*8 163 [163] 1*1+2*9*9 201 [3, 67] 1*1+2*10*10 Z =
4 Z = X 2 + 3Y 2 bunkai3:- for(1=<100,x),for(1=<100,y), gcd(d=(x,y)),x mod 3\==0,Z is X*X+3*Y*Y,Z<2000,Z mod 3=\=0, D==1,factorize(Z,L),write(Z),tab(9),write(L),tab(9),write( X*X+3*Y*Y),nl,fail. bunkai3. 4 [2, 2] 1*1+3*1*1 13 [13] 1*1+3*2*2 28 [2, 2, 7] 1*1+3*3*3 49 [7, 7] 1*1+3*4*4 76 [2, 2, 19] 1*1+3*5*5 109 [109] 1*1+3*6*6 148 [2, 2, 37] 1*1+3*7*7 193 [193] 1*1+3*8*8 244 [2, 2, 61] 1*1+3*9*9 301 [7, 43] 1*1+3*10*10 Z = 2000 Z = X 2 + 5Y 2 bunkai5:- for(1=<100,x),for(1=<100,y), gcd(d=(x,y)),x mod 5\==0,Z is X*X+5*Y*Y,Z<2000,Z mod 5=\=0, D==1,factorize(Z,L),write(Z),tab(9),write(L),tab(9),write( X*X+5*Y*Y),nl,fail. bunkai5. 6 [2, 3] 1*1+5*1*1 21 [3, 7] 1*1+5*2*2 46 [2, 23] 1*1+5*3*3 81 [3, 3, 3, 3] 1*1+5*4*4 126 [2, 3, 3, 7] 1*1+5*5*5 181 [181] 1*1+5*6*6 246 [2, 3, 41] 1*1+5*7*7 321 [3, 107] 1*1+5*8*8 406 [2, 7, 29] 1*1+5*9*9 501 [3, 167] 1*1+5*10*10 Z =
5 Z = X 2 + 6Y 2 bunkai6:- for(1=<100,x),for(1=<100,y), gcd(d=(x,y)),x mod 6\==0,Z is X*X+6*Y*Y,Z<2000,Z mod 6=\=0, D==1,factorize(Z,L),write(Z),tab(9),write(L),tab(9),write( X*X+6*Y*Y),nl,fail. bunkai6. 7 [7] 1*1+6*1*1 25 [5, 5] 1*1+6*2*2 55 [5, 11] 1*1+6*3*3 97 [97] 1*1+6*4*4 151 [151] 1*1+6*5*5 217 [7, 31] 1*1+6*6*6 295 [5, 59] 1*1+6*7*7 385 [5, 7, 11] 1*1+6*8*8 487 [487] 1*1+6*9*9 601 [601] 1*1+6*10*10 Z = 2000 Z = X 2 + 7Y 2 bunkai7:- for(1=<100,x),for(1=<100,y), gcd(d=(x,y)),x mod 7\==0,Z is X*X+7*Y*Y,Z<2000,Z mod 7=\=0, D==1,factorize(Z,L),write(Z),tab(9),write(L),tab(9),write( X*X+7*Y*Y),nl,fail. bunkai7. 8 [2, 2, 2] 1*1+7*1*1 29 [29] 1*1+7*2*2 64 [2, 2, 2, 2, 2, 2] 1*1+7*3*3 113 [113] 1*1+7*4*4 176 [2, 2, 2, 2, 11] 1*1+7*5*5 253 [11, 23] 1*1+7*6*6 344 [2, 2, 2, 43] 1*1+7*7*7 449 [449] 1*1+7*8*8 568 [2, 2, 2, 71] 1*1+7*9*9 701 [701] 1*1+7*10*10 Z =
6 3 3.1 k = 2, 3, 7 Z 2 X 2 + ky Z n X 2 + ky 2 2 n 1 k = 5, 6 n 2 n 1 k = 5 Z 2 k = Z k = 2 Z 3, 11, 17, 19, 41, 43, 59, 67, 73, Z 1, 3 mod 8 k = 3 Z 7, 13, 19, 31, 37, 43, 61, 67, 73, Z 1, 7 mod 12 k = 5 Z 29, 41, 61, 89, 101, 109, 149, 181, 229, Z 1, 9 mod 20 k = 6 Z 7, 31, 73, 79, 97, 103, 127, 151, 193, Z 1, 7 mod 24 k = 7 Z 11, 23, 29, 37, 43, 53, 67, 71, 79, Z 1, 9, 11, 15, 23, 25 mod 28 6
7 3.2 Z = X 2 + ky 2 1: Excel X 2 + 2Y 2 X 2 + 2Y 2 X 2 + 2Y 2 3 [3] [3, 3] [11] [17] [19] [3, 3, 3] [3, 11] [3, 11] [41] [43] [3, 17] [3, 17] [3, 19] [3, 19] [59] [67] [73] [3, 3, 3, 3] [83] [89] [97] [3, 3, 11] [3, 3, 11] [107] [113] [11, 11] [3, 41] [3, 41] [3, 43] [3, 43] [131] [137] [139] [3, 3, 17] [3, 3, 17] [163] [3, 3, 19] [3, 3, 19]
8 1: Excel X 2 + 2Y 2 X 2 + 2Y 2 X 2 + 2Y [3, 59] [3, 59] [179] [11, 17] [11, 17] [193] [3, 67] [3, 67] [11, 19] [11, 19] [211] [3, 73] [3, 73] [227] [233] [241] [3, 3, 3, 3, 3] [3, 83] [3, 83] [251] [257] [3, 89] [3, 89] [281] [283] [17, 17] [3, 97] [3, 97] [3, 3, 3, 11] [3, 3, 3, 11] [307] [313] [3, 107] [3, 107] [17, 19] [17, 19] [331] [337] [3, 113] [3, 113]
9 1: Excel X 2 + 2Y 2 X 2 + 2Y 2 X 2 + 2Y [347] [353] [19, 19] [3, 11, 11] [3, 11, 11] [3, 3, 41] [3, 3, 41] [379] [3, 3, 43] [3, 3, 43] [3, 131] [3, 131] [401] [409] [3, 137] [3, 137] [3, 139] [3, 139] [419] [433] [443] [449] [11, 41] [11, 41] [457] [3, 3, 3, 17] [3, 3, 3, 17] [467] [11, 43] [11, 43] [3, 163] [3, 163] [491] [499] [3, 3, 3, 19] [3, 3, 3, 19] [521] [523] [3, 3, 59] [3, 3, 59]
10 1: Excel X 2 + 2Y 2 X 2 + 2Y 2 X 2 + 2Y [3, 179] [3, 179] [547] [3, 11, 17] [3, 11, 17] [3, 11, 17] [3, 11, 17] [563] [569] [571] [577] [3, 193] [3, 193] [587] [593] [601] [3, 3, 67] [3, 3, 67] [617] [619] [3, 11, 19] [3, 11, 19] [3, 11, 19] [3, 11, 19] [3, 211] [3, 211] [641] [643] [11, 59] [11, 59] [3, 3, 73] [3, 3, 73] [659] [673] [3, 227] [3, 227] [683] [691] [17, 41] [17, 41]
11 1: Excel X 2 + 2Y 2 X 2 + 2Y 2 X 2 + 2Y [3, 233] [3, 233] [3, 241] [3, 241] [3, 3, 3, 3, 3, 3] [17, 43] [17, 43] [11, 67] [11, 67] [739] [3, 3, 83] [3, 3, 83] [3, 251] [3, 251] [761] [769] [3, 257] [3, 257] [19, 41] [19, 41] [787] [3, 3, 89] [3, 3, 89] [11, 73] [11, 73] [809] [811] [19, 43] [19, 43] [827] [3, 281] [3, 281] [3, 283] [3, 283] [857] [859] [3, 17, 17] [3, 17, 17] [3, 3, 97] [3, 3, 97]
12 1: Excel X 2 + 2Y 2 X 2 + 2Y 2 X 2 + 2Y [881] [883] [3, 3, 3, 3, 11] [3, 3, 3, 3, 11] [907] [11, 83] [11, 83] [3, 307] [3, 307] [929] [937] [3, 313] [3, 313] [947] [953] [3, 3, 107] [3, 3, 107] [3, 17, 19] [3, 17, 19] [3, 17, 19] [3, 17, 19] [971] [977] [11, 89] [11, 89] [3, 331] [3, 331]
13 X 2 + 3Y 2 2: X 2 + 3Y 2 X 2 + 3Y 2 X 2 + 3Y 2 4 [2, 2] [7] [13] [19] [2, 2, 7] [2, 2, 7] [31] [37] [43] [7, 7] [2, 2, 13] [2, 2, 13] [61] [67] [73] [2, 2, 19] [2, 2, 19] [79] [7, 13] [7, 13] [97] [103] [109] [2, 2, 31] [2, 2, 31] [127] [7, 19] [7, 19] [139] [2, 2, 37] [2, 2, 37] [151] [157] [163] [13, 13] [2, 2, 43] [2, 2, 43] [181]
14 2: X 2 + 3Y 2 X 2 + 3Y 2 X 2 + 3Y [193] [2, 2, 7, 7] [2, 2, 7, 7] [199] [211] [7, 31] [7, 31] [223] [229] [241] [2, 2, 61] [2, 2, 61] [13, 19] [13, 19] [7, 37] [7, 37] [2, 2, 67] [2, 2, 67] [271] [277] [283] [2, 2, 73] [2, 2, 73] [7, 43] [7, 43] [307] [313] [2, 2, 79] [2, 2, 79] [331] [337] [7, 7, 7] [349] [19, 19] [2, 2, 7, 13] [2, 2, 7, 13] [2, 2, 7, 13] [2, 2, 7, 13] [367] [373]
15 2: X 2 + 3Y 2 X 2 + 3Y 2 X 2 + 3Y [379] [2, 2, 97] [2, 2, 97] [397] [13, 31] [13, 31] [409] [2, 2, 103] [2, 2, 103] [421] [7, 61] [7, 61] [433] [2, 2, 109] [2, 2, 109] [439] [457] [463] [7, 67] [7, 67] [13, 37] [13, 37] [487] [499] [2, 2, 127] [2, 2, 127] [7, 73] [7, 73] [523] [2, 2, 7, 19] [2, 2, 7, 19] [2, 2, 7, 19] [2, 2, 7, 19] [541] [547] [7, 79] [7, 79] [2, 2, 139] [2, 2, 139] [13, 43]
16 2: X 2 + 3Y 2 X 2 + 3Y 2 X 2 + 3Y [13, 43] [571] [577] [19, 31] [19, 31] [601] [2, 2, 151] [2, 2, 151] [607] [613] [619] [2, 2, 157] [2, 2, 157] [631] [7, 7, 13] [7, 7, 13] [643] [2, 2, 163] [2, 2, 163] [661] [673] [2, 2, 13, 13] [2, 2, 13, 13] [7, 97] [7, 97] [691] [19, 37] [19, 37] [709] [7, 103] [7, 103] [2, 2, 181] [2, 2, 181] [727] [733] [739] [751] [757] [7, 109] [7, 109]
17 2: X 2 + 3Y 2 X 2 + 3Y 2 X 2 + 3Y [769] [2, 2, 193] [2, 2, 193] [787] [13, 61] [13, 61] [2, 2, 199] [2, 2, 199] [811] [19, 43] [19, 43] [823] [829] [2, 2, 211] [2, 2, 211] [853] [859] [2, 2, 7, 31] [2, 2, 7, 31] [2, 2, 7, 31] [2, 2, 7, 31] [13, 67] [13, 67] [877] [883] [7, 127] [7, 127] [2, 2, 223] [2, 2, 223] [907] [2, 2, 229] [2, 2, 229] [919] [7, 7, 19] [7, 7, 19] [937] [13, 73] [13, 73] [31, 31] [2, 2, 241]
18 2: X 2 + 3Y 2 X 2 + 3Y 2 X 2 + 3Y [2, 2, 241] [967] [7, 139] [7, 139] [2, 2, 13, 19] [2, 2, 13, 19] [2, 2, 13, 19] [2, 2, 13, 19] [991] [997]
19 X 2 + 5Y 2 3: X 2 + 5Y 2 X 2 + 5Y 2 X 2 + 5Y 2 6 [2, 3] [3, 3] [2, 7] [3, 7] [3, 7] [29] [41] [2, 23] [7, 7] [2, 3, 3, 3] [61] [3, 23] [3, 23] [3, 3, 3, 3] [2, 43] [89] [2, 47] [101] [109] [2, 3, 3, 7] [2, 3, 3, 7] [3, 43] [3, 43] [2, 67] [3, 47] [3, 47] [149] [7, 23] [7, 23] [2, 83] [2, 3, 29] [2, 3, 29] [181] [3, 3, 3, 7] [3, 3, 3, 7] [3, 67] [3, 67] [2, 103]
20 3: X 2 + 5Y 2 X 2 + 5Y 2 X 2 + 5Y [2, 107] [229] [241] [2, 3, 41] [2, 3, 41] [3, 83] [3, 83] [2, 127] [3, 3, 29] [3, 3, 29] [269] [281] [2, 3, 7, 7] [2, 3, 7, 7] [7, 43] [7, 43] [3, 103] [3, 103] [3, 107] [3, 107] [2, 163] [7, 47] [7, 47] [2, 167] [349] [2, 3, 61] [2, 3, 61] [3, 3, 41] [3, 3, 41] [3, 127] [3, 127] [389] [401] [2, 7, 29] [2, 7, 29] [409] [2, 3, 3, 23] [2, 3, 3, 23] [421] [3, 3, 7, 7]
21 3: X 2 + 5Y 2 X 2 + 5Y 2 X 2 + 5Y [3, 3, 7, 7] [2, 223] [449] [2, 227] [461] [7, 67] [7, 67] [2, 3, 3, 3, 3, 3] [3, 163] [3, 163] [3, 167] [3, 167] [509] [521] [2, 263] [23, 23] [2, 3, 89] [2, 3, 89] [541] [3, 3, 61] [3, 3, 61] [2, 283] [569] [2, 7, 41] [2, 7, 41] [7, 83] [7, 83] [601] [2, 3, 101] [2, 3, 101] [3, 7, 29] [3, 7, 29] [3, 7, 29] [3, 7, 29] [2, 307] [3, 3, 3, 23] [3, 3, 3, 23] [641] [2, 3, 109] [2, 3, 109]
22 3: X 2 + 5Y 2 X 2 + 5Y 2 X 2 + 5Y [661] [3, 223] [3, 223] [3, 227] [3, 227] [2, 7, 7, 7] [2, 347] [701] [709] [7, 103] [7, 103] [3, 3, 3, 3, 3, 3] [2, 367] [7, 107] [7, 107] [761] [2, 383] [769] [2, 3, 3, 43] [2, 3, 3, 43] [3, 263] [3, 263] [3, 3, 89] [3, 3, 89] [809] [821] [829] [29, 29] [2, 3, 3, 47] [2, 3, 3, 47] [3, 283] [3, 283] [2, 7, 61] [2, 7, 61] [3, 7, 41] [3, 7, 41] [3, 7, 41] [3, 7, 41] [881] [2, 443]
23 3: X 2 + 5Y 2 X 2 + 5Y 2 X 2 + 5Y [7, 127] [7, 127] [2, 3, 149] [2, 3, 149] [3, 3, 101] [3, 3, 101] [3, 307] [3, 307] [2, 463] [929] [2, 467] [941] [2, 3, 7, 23] [2, 3, 7, 23] [2, 3, 7, 23] [2, 3, 7, 23] [2, 487] [3, 3, 109] [3, 3, 109] [23, 43] [23, 43]
24 X 2 + 6Y 2 4: X 2 + 6Y 2 X 2 + 6Y 2 X 2 + 6Y 2 7 [7] [2, 5] [3, 5] [2, 11] [5, 5] [31] [3, 11] [7, 7] [5, 11] [5, 11] [2, 29] [2, 5, 7] [2, 5, 7] [73] [79] [3, 29] [97] [103] [3, 5, 7] [3, 5, 7] [2, 53] [2, 59] [11, 11] [127] [5, 29] [5, 29] [151] [2, 7, 11] [2, 7, 11] [3, 53] [2, 83] [5, 5, 7] [5, 5, 7] [3, 59] [193] [199] [2, 101] [2, 107]
25 4: X 2 + 6Y 2 X 2 + 6Y 2 X 2 + 6Y [7, 31] [7, 31] [223] [3, 7, 11] [3, 7, 11] [241] [3, 83] [2, 5, 5, 5] [2, 131] [5, 53] [5, 53] [271] [5, 59] [5, 59] [2, 149] [3, 101] [2, 5, 31] [2, 5, 31] [313] [11, 29] [11, 29] [3, 107] [337] [7, 7, 7] [2, 173] [2, 179] [367] [3, 5, 5, 5] [5, 7, 11] [5, 7, 11] [5, 7, 11] [5, 7, 11] [3, 131] [2, 197] [2, 7, 29] [2, 7, 29] [409] [5, 83] [5, 83] [433]
26 4: X 2 + 6Y 2 X 2 + 6Y 2 X 2 + 6Y [439] [3, 149] [2, 227] [457] [463] [3, 5, 31] [3, 5, 31] [487] [2, 5, 7, 7] [2, 5, 7, 7] [2, 251] [5, 101] [5, 101] [7, 73] [7, 73] [3, 173] [5, 107] [5, 107] [3, 179] [2, 269] [2, 5, 5, 11] [2, 5, 5, 11] [7, 79] [7, 79] [577] [11, 53] [11, 53] [2, 293] [3, 197] [601] [607] [3, 7, 29] [3, 7, 29] [5, 5, 5, 5] [631] [2, 317] [11, 59] [11, 59] [5, 131] [5, 131]
27 4: X 2 + 6Y 2 X 2 + 6Y 2 X 2 + 6Y [673] [7, 97] [7, 97] [3, 227] [2, 11, 31] [2, 11, 31] [2, 347] [7, 103] [7, 103] [727] [2, 5, 73] [2, 5, 73] [3, 5, 7, 7] [3, 5, 7, 7] [2, 7, 53] [2, 7, 53] [5, 149] [5, 149] [751] [3, 251] [769] [5, 5, 31] [5, 5, 31] [2, 389] [2, 5, 79] [2, 5, 79] [3, 269] [823] [3, 5, 5, 11] [3, 5, 5, 11] [2, 7, 59] [2, 7, 59] [2, 419] [29, 29] [7, 11, 11] [7, 11, 11] [5, 173] [5, 173] [3, 293] [2, 443]
28 4: X 2 + 6Y 2 X 2 + 6Y 2 X 2 + 6Y [7, 127] [7, 127] [5, 179] [5, 179] [11, 83] [11, 83] [919] [2, 461] [2, 467] [937] [3, 317] [31, 31] [967] [2, 5, 97] [2, 5, 97] [2, 491] [5, 197] [5, 197] [991]
29 X 2 + 7Y 2 5: X 2 + 7Y 2 X 2 + 7Y 2 X 2 + 7Y 2 8 [2, 2, 2] [11] [2, 2, 2, 2] [23] [29] [2, 2, 2, 2, 2] [37] [43] [53] [2, 2, 2, 2, 2, 2] [67] [71] [79] [2, 2, 2, 11] [2, 2, 2, 11] [107] [109] [113] [11, 11] [127] [2, 2, 2, 2, 2, 2, 2] [137] [149] [151] [163] [2, 2, 2, 2, 11] [2, 2, 2, 2, 11] [179] [2, 2, 2, 23] [2, 2, 2, 23] [191] [193] [197] [211] [2, 2, 2, 29] [2, 2, 2, 29] [233] [239]
30 5: X 2 + 7Y 2 X 2 + 7Y 2 X 2 + 7Y [11, 23] [11, 23] [2, 2, 2, 2, 2, 2, 2, 2] [263] [277] [281] [2, 2, 2, 37] [2, 2, 2, 37] [317] [11, 29] [11, 29] [331] [337] [2, 2, 2, 43] [2, 2, 2, 43] [347] [2, 2, 2, 2, 2, 11] [2, 2, 2, 2, 2, 11] [359] [2, 2, 2, 2, 23] [2, 2, 2, 2, 23] [373] [379] [389] [401] [11, 37] [11, 37] [421] [2, 2, 2, 53] [2, 2, 2, 53] [431] [443] [449] [457] [463] [2, 2, 2, 2, 29] [2, 2, 2, 2, 29] [11, 43] [11, 43] [487]
31 5: X 2 + 7Y 2 X 2 + 7Y 2 X 2 + 7Y [491] [499] [2, 2, 2, 2, 2, 2, 2, 2, 2] [23, 23] [2, 2, 2, 67] [2, 2, 2, 67] [541] [547] [557] [2, 2, 2, 71] [2, 2, 2, 71] [569] [571] [11, 53] [11, 53] [2, 2, 2, 2, 37] [2, 2, 2, 2, 37] [599] [613] [617] [631] [2, 2, 2, 79] [2, 2, 2, 79] [641] [653] [659] [23, 29] [23, 29] [673] [683] [2, 2, 2, 2, 43] [2, 2, 2, 2, 43] [701] [2, 2, 2, 2, 2, 2, 11] [2, 2, 2, 2, 2, 2, 11] [709] [2, 2, 2, 2, 2, 23] [2, 2, 2, 2, 2, 23] [11, 67] [11, 67]
32 5: X 2 + 7Y 2 X 2 + 7Y 2 X 2 + 7Y [739] [743] [751] [757] [11, 71] [11, 71] [809] [821] [823] [827] [29, 29] [2, 2, 2, 2, 53] [2, 2, 2, 2, 53] [23, 37] [23, 37] [2, 2, 2, 107] [2, 2, 2, 107] [863] [11, 79] [11, 79] [2, 2, 2, 109] [2, 2, 2, 109] [877] [883] [2, 2, 2, 113] [2, 2, 2, 113] [907] [911] [919] [2, 2, 2, 2, 2, 29] [2, 2, 2, 2, 2, 29] [947] [953] [967] [2, 2, 2, 11, 11] [2, 2, 2, 11, 11] [977] [23, 43] [23, 43] [991]
33 3.3 k = 2 Z = = 3 17 = = Z = = = = = = k = 3 Z = = 7 13 = = z = = = = = = k = 5 z = = 3 7 = = k = 6 z = = 5 11 = = k = 7 z = = = =
34 3.4 k = 5 z = = 2 43 = z = = = = k = 6 z = = 2 29 = z = = = =
35 4 4.1 mod a p x 2 x 2 a mod p x a p ( a p ) = 1 x a p ( a p ) = 1 a p p, q ( p q )( q p 1 q 1 p ) = ( 1)( 2 )( 2 ) p ( 1 p 1 p ) = ( 1) 2 p ( 2 p ) = ( 1) p2 1 8 ( )p a, b ( ab p ) = ( a p )( b p ) ( )ab a b ( )ab a b k = 7 Z 1, 9, 11, 15, 23, 25 mod 28 35
36 Z = X 2 + 7X 2 Z p X 0 mod p Z 7 Z = X 2 + 7Y 2 mod p X 2 7Y 2 mod p Y mod p U (UX) 2 7 mod p ( 7 p ) = 1 p ( 7 p ) = ( 1 p )( 7 p 1 p ) = ( 1) 2 ( 7 p ) n Z = 4n + 1, 4n + 3 p = 4n + 1, 4n + 3 ( 7 p ) = ( 1)2n ( 7 p ) = ( 7 p ) (p = 4n + 1) ( 7 p ) = ( 1)2n+1 ( 7 p ) = ( 7 p ) (p = 4n + 3) ( 7 p )( p p 1 7 ) = ( 1) 2 3 = ( 1) 3 2n = 1 (p = 4n + 1) ( 7 p )( p p 1 7 ) = ( 1) 2 3 = ( 1) 3(2n+1) = 1 (p = 4n + 3) ( p 7 ) = 1 p p 1, 2, 4 mod 7 ( 1 p ) = 1 p p 1 mod 4 ) = 1 p p 3 mod 4 ( 1 p p 1, 9, 11, 15, 23, 25 mod 28 36
37 4.2 Z = X 2 + 5Y 2 α (α) R = Z[ 5]/(α) (α) Z[ 5] = a + b 5 a, b Z 1 Z = = ( )(3 2 5) α = Z[ 5]/( ) = Z[X]/(X 2 + 5, 3 + 2X) J = (X 2 + 5, 3 + 2X) 2(X 2 + 5) (X 1)(3 + 2X) = 13 X = Y X = 13 Y X = (13 Y ) = Y 2 26Y X = 3 + 2(13 Y ) = 2Y + 29 J = (Y 2 26Y + 174, 2Y + 29) (Y ) 174 = 6 29 Z[X]/J = (Z[Y ]/(Y ))/J = F 29 α 2 Z = = (6 + 5)(6 5) α = Z[ 5]/(6 + 5) = Z[X]/(X 2 + 5, 6 + X) J = (X 2 + 5, 6 + X) 6 + X = Y X = (Y 6) = Y 2 12Y + 41 J = (Y, Y 2 12Y + 41) (Y ) Z[X]/J = (Z[Y ]/(Y ))/J = F 41 α 37
38 3 Z = = ( )(4 3 5) α = Z[ 5]/( ) = Z[X]/(X 2 + 5, 4 + 3X) J = (X 2 + 5, 4 + 3X) 3(X 2 + 5) (X 1)(4 + 3X) = 19 X = Y X = 19 Y X = (19 Y ) = Y 2 38Y X = 4 + 3(19 Y ) = 3Y + 61 J = (Y 2 38Y + 366, 3Y + 61) (Y ) 366 = Z[X]/J = (Z[Y ]/(Y ))/J = F 61 α 4 Z = = ( )(3 4 5) α = Z[ 5]/( ) = Z[X]/(X 2 + 5, 3 + 4X) J = (X 2 + 5, 3 + 4X) 4(X 2 + 5) (X 1)(3 + 4X) = X + 23 = Y X = Y 23 X = (Y 23) = Y 2 46Y X = 3 + 4(Y 23) = 4Y 89 J = (Y 2 46Y + 534, 4Y 89) (Y ) 534 = 5 89 Z[X]/J = (Z[Y ]/(Y ))/J = F 89 α 38
39 5 Z = = ( )(9 2 5) α = Z[ 5]/( ) = Z[X]/(X 2 + 5, 9 + 2X) J = (X 2 + 5, 9 + 2X) 2(X 2 + 5) (X 4)(9 + 2X) = 46 X = Y X = 46 Y X = (46 Y ) = Y 2 92Y X = 9 + 2(46 Y ) = 2Y J = (Y 2 92Y , 2Y + 101) (Y ) 2121 = Z[X]/J = (Z[Y ]/(Y ))/J = F 101 α 6 Z = = ( )(8 3 5) α = Z[ 5]/( ) = Z[X]/(X 2 + 5, 8 + 3X) J = (X 2 + 5, 8 + 3X) 3(X 2 + 5) (X 3)(8 + 3X) = X + 39 = Y X = Y 39 X = (Y 39) = Y 2 78Y X = 8 + 3(Y 39) = 3Y 109 J = (Y 2 78Y , 3Y 109) (Y ) 1526 = Z[X]/J = (Z[Y ]/(Y ))/J = F 109 α 39
40 7 Z = = (12 + 5)(12 5) α = Z[ 5]/(12 + 5) = Z[X]/(X 2 + 5, 12 + X) J = (X 2 + 5, 12 + X) 12 + X = Y X = (Y 12) = Y 2 24Y J = (Y, Y 2 24Y + 149) (Y ) Z[X]/J = (Z[Y ]/(Y ))/J = F 149 α 8 Z = = ( )(1 6 5) α = Z[ 5]/( ) = Z[X]/(X 2 + 5, 1 + 6X) J = (X 2 + 5, 1 + 6X) 30(X 2 + 5) (5X 1)(1 + 6X) = X = Y X = Y 151 X = (Y 151) = Y 2 302Y X = 1 + 6(Y 151) = 6Y 905 J = (Y 2 302Y , 6Y 905) (Y ) = = Z[X]/J = (Z[Y ]/(Y ))/J = F 181 α 40
41 9 Z = = ( )(7 6 5) α = Z[ 5]/( ) = Z[X]/(X 2 + 5, 7 + 6X) J = (X 2 + 5, 7 + 6X) 6(X 2 + 5) (X 1)(7 + 6X) = 37 X = Y X = 37 Y X = (37 Y ) = Y 2 74Y X = 7 + 6(37 Y ) = 6Y J = (Y 2 74Y , 6Y + 229) (Y ) 1374 = Z[X]/J = (Z[Y ]/(Y ))/J = F 229 α 10 Z = = ( )(14 3 5) α = Z[ 5]/( ) = Z[X]/(X 2 + 5, X) J = (X 2 + 5, X) 3(X 2 + 5) (X 5)(14 + 3X) = X + 85 = Y X = Y 85 X = (Y 85) = Y 2 170Y X = (Y 85) = 3Y 241 J = (Y 2 170Y , 3Y 241) (Y ) 7230 = Z[X]/J = (Z[Y ]/(Y ))/J = F 241 α 41
42 11 Z = = ( )(12 5 5) α = Z[ 5]/( ) = Z[X]/(X 2 + 5, X) J = (X 2 + 5, X) 10(X 2 + 5) (2X 5)(12 + 5X) = X = Y X = Y 110 X = (Y 110) = Y 2 220Y X = (Y 110) = 5Y 538 J = (Y 2 220Y , 5Y 538) (Y ) = = Z[X]/J = (Z[Y ]/(Y ))/J = F 269 α 12 Z = = ( )(6 7 5) α = Z[ 5]/( ) = Z[X]/(X 2 + 5, 6 + 7X) J = (X 2 + 5, 6 + 7X) 7(X 2 + 5) (X 1)(6 + 7X) = X + 41 = Y X = Y 41 X = (Y 41) = Y 2 82Y X = 6 + 7(Y 41) = 7Y 281 J = (Y 2 82Y , 7Y 281) (Y ) 1686 = Z[X]/J = (Z[Y ]/(Y ))/J = F 281 α 42
43 4.2.2 (α) (α) 1 Z = 6 6 = 2 3 = (1 + 5)(1 5) Z[ 5]/(2) = Z[X]/(X 2 + 5, 2) J 1 = (X 2 + 5, 2) X = X 2 1 = (X + 1)(X 1) Y = X 1 J 1 (2) X 1 = X + 1 = Y Z[ 5]/(3) = Z[Y ]/J 1 = (Z[Y ]/(2))/(Y 2 ) = F 2 /(Y 2 ) Z[ 5]/(3) = Z[X]/(X 2 + 5, 3) J 2 = (X 2 + 5, 3) X = X 2 1 = (X + 1)(X 1) Y = X 1 J 2 = (3, Y (Y + 2)) (3) Z[ 5]/(3) = Z[Y ]/J 2 = (Z[Y ]/(3))/(Y (Y + 2)) = F3 /(Y (Y + 2)) 2x 1 mod 3 x 3a + 2b = 1 a, b a = 1, b = 1 x = 1 A = (Y + 2), B = Y AB = Y (Y + 2) (Y (Y + 2)) F mod 3 A + B = 2 = 1 Z[ 5]/(3) = F 3 [Y ]/(A) F 3 [Y ]/(B) F 3 [Y ]/(A) = F 3, F 3 [Y ]/(B) = F 3 Z[ 5]/(3) = F 3 F 3 (3) 43
44 α = Z[ 5]/(1 + 5) = Z[X]/(X 2 + 5, 1 + X) I = (X 2 + 5, 1 + X) 1 + X = Y X = (Y 1) = Y 2 2Y + 6 I = (Y, Y 2 2Y + 6) (Y ) Z[X]/I = (Z[Y ]/(Y ))/I = Z/(6) = Z/(2) Z/(3) = F 2 F 3 (α) = (1 + 5) j 1 = (1 + 5, 2), j 2 = (1 + 5, 3) j 1 j 2 = ( , , , 6) = 3(1 + 5), = 2(1 + 5), 6 = (1 5)(1 + 5), 6 + ( ) = 2(1 + 5) j 1 j 2 = (1 + 5) 6 = 2 3 = (1 + 5)(1 5) = j 1 j 2 j 1 j 2 44
45 2 Z = = 2 7 = (3 + 5)(3 5) 1 (2) Z[ 5]/(7) = Z[X]/(X 2 + 5, 7) J = (X 2 + 5, 7) X = X 2 9 = (X + 3)(X 3) Y = X 3 J = (7, Y (Y + 6)) (7) Z[ 5]/(7) = Z[Y ]/J = (Z[Y ]/(7))/(Y (Y + 6)) = F 7 /(Y (Y + 6)) 6x 1 mod 7 x 7a + 6b = 1 a, b a = 1, b = 1 x = 1 A = (Y + 6), B = Y AB = Y (Y + 6) (Y (Y + 6)) F mod 7 A + B = 6 = 1 Z[ 5]/(7) = F 7 [Y ]/(A) F 7 [Y ]/(B) F 7 [Y ]/(A) = F 7, F 7 [Y ]/(B) = F 7 Z[ 5]/(7) = F 7 F 7 (7) α = Z[ 5]/(3 + 5) = Z[X]/(X 2 + 5, 3 + X) I = (X 2 + 5, 3 + X) 3 + X = Y X = (Y 3) = Y 2 6Y + 14 I = (Y, Y 2 6Y + 14) (Y ) Z[X]/I = (Z[Y ]/(Y ))/I = Z/(14) = Z/(2) Z/(7) = F 2 F 7 (α) = (3 + 5) 45
46 j 1 = (3 + 5, 2), j 2 = (3 + 5, 7) j 1 j 2 = ( , , , 14) = 7(3 + 5), = 2(3 + 5), 14 = (3 5)(3 + 5), 14 + ( ) = 6(3 + 5) j 1 j 2 = (3 + 5) 14 = 2 7 = (3 + 5)(3 5) = j 1 j 2 j 1 j 2 3 Z = = 2 23 = ( )(1 3 5) 1 (2) Z[ 5]/(23) = Z[X]/(X 2 + 5, 23) J = (X 2 + 5, 23) X = X 2 64 = (X + 8)(X 8) Y = X 8 J = (23, Y (Y + 16)) (23) Z[ 5]/(23) = Z[Y ]/J = (Z[Y ]/(23))/(Y (Y + 16)) = F 23 /(Y (Y + 16)) 16x 1 mod 23 x 23a + 16b = 1 a, b a = 7, b = 10 x = 10 A = 10(Y + 16), B = 10Y AB = 100Y (Y + 16) (Y (Y + 16)) F mod 23 A + B = 160 = 1 Z[ 5]/(23) = F 23 [Y ]/(A) F 23 [Y ]/(B) F 23 [Y ]/(A) = F 23, F 23 [Y ]/(B) = F 23 Z[ 5]/(23) = F 23 F 23 (23) 46
47 α = Z[ 5]/( ) = Z[X]/(X 2 + 5, 1 + 3X) I = (X 2 + 5, 1 + 3X) 6(X 2 + 5) (2X 1)(1 + 3X) = X + 31 = Y X = Y 31 X = (Y 31) = Y 2 62Y X = 1 + 3(Y 31) = 3Y 92 I = (Y 2 62Y + 966, 3Y 92) (Y ) 966 = = Z[X]/I = (Z[Y ]/(Y ))/I = Z/(46) = Z/(2) Z/(23) = F 2 F 23 (α) = ( ) j 1 = ( , 2), j 2 = ( , 23) j 1 j 2 = ( , , , 46) = 23( ), = 2( ), 46 = (1 3 5)( ), 46 + ( ) = 2( ) j 1 j 2 = ( ) 46 = 2 23 = ( )(1 3 5) = j 1 j 2 j 1 j 2 47
48 4 Z = = 2 43 = (9 + 5)(9 5) 1 (2) Z[ 5]/(43) = Z[X]/(X 2 + 5, 43) J = (X 2 + 5, 43) X = X 2 81 = (X + 9)(X 9) Y = X 9 J = (43, Y (Y + 18)) (43) Z[ 5]/(43) = Z[Y ]/J = (Z[Y ]/(43))/(Y (Y + 18)) = F 43 /(Y (Y + 18)) 18x 1 mod 43 x 43a + 18b = 1 a, b a = 5, b = 12 x = 12 A = 12(Y + 18), B = 12Y AB = 144Y (Y + 18) (Y (Y + 18)) F mod 43 A + B = 216 = 1 Z[ 5]/(43) = F 43 [Y ]/(A) F 43 [Y ]/(B) F 43 [Y ]/(A) = F 43, F 43 [Y ]/(B) = F 43 Z[ 5]/(43) = F 43 F 43 (43) α = Z[ 5]/(9 + 5) = Z[X]/(X 2 + 5, 9 + X) I = (X 2 + 5, 9 + X) 9 + X = Y X = (Y 9) = Y 2 18Y + 86 I = (Y, Y 2 18Y + 86) (Y ) Z[X]/I = (Z[Y ]/(Y ))/I = Z/(86) = Z/(2) Z/(43) = F 2 F 43 (α) = (9 + 5) 48
49 j 1 = (9 + 5, 2), j 2 = (9 + 5, 43) j 1 j 2 = ( , , , 86) = 43(9 + 5), = 2(9 + 5), 86 = (9 5)(9 + 5), 86 + ( ) = 18(9 + 5) j 1 j 2 = (9 + 5) 86 = 2 43 = (9 + 5)(9 5) = j 1 j 2 j 1 j 2 5 Z = = 2 47 = ( )(7 3 5) 1 (2) Z[ 5]/(47) = Z[X]/(X 2 + 5, 47) J = (X 2 + 5, 47) X = X = (X + 18)(X 18) Y = X 18 J = (47, Y (Y + 36)) (47) Z[ 5]/(47) = Z[Y ]/J = (Z[Y ]/(47))/(Y (Y + 36)) = F 47 /(Y (Y + 36)) 36x 1 mod 47 x 47a + 36b = 1 a, b a = 13, b = 17 x = 17 A = 17(Y + 36), B = 17Y AB = 289Y (Y + 36) (Y (Y + 36)) F mod 47 A + B = 612 = 1 Z[ 5]/(47) = F 47 [Y ]/(A) F 47 [Y ]/(B) F 47 [Y ]/(A) = F 47, F 47 [Y ]/(B) = F 47 Z[ 5]/(47) = F 47 F 47 (47) 49
50 α = Z[ 5]/( ) = Z[X]/(X 2 + 5, 7 + 3X) I = (X 2 + 5, 7 + 3X) 3(X 2 + 5) (X 2)(7 + 3X) = 29 X = Y X = 29 Y X = (29 Y ) = Y 2 58Y X = 7 + 3(29 Y ) = 3Y + 94 I = (Y 2 58Y + 846, 3Y + 94) (Y ) 846 = = 2 47 Z[X]/I = (Z[Y ]/(Y ))/I = Z/(94) = Z/(2) Z/(47) = F 2 F 47 (α) = ( ) j 1 = (7 + sqrt 5, 2), j 2 = ( , 47) j 1 j 2 = ( , , , 94) = 47( ), = 2( ), 94 = (3 7 5)( ), 94 + ( ) = 14( ) j 1 j 2 = ( ) 94 = 2 47 = ( )(7 3 5) = j 1 j 2 j 1 j 2 50
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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
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EPSON エプソンプリンタ共通 取扱説明書 ネットワーク編
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公務員人件費のシミュレーション分析
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PowerPoint プレゼンテーション
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システムの概要
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AHPを用いた大相撲の新しい番付編成
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変 位 変位とは 物体中のある点が変形後に 別の点に異動したときの位置の変化で あり ベクトル量である 変位には 物体の変形の他に剛体運動 剛体変位 が含まれている 剛体変位 P(x, y, z) 平行移動と回転 P! (x + u, y + v, z + w) Q(x + d x, y + dy, z + dz) Q! (x + d x + u + du, y + dy + v + dv, z +
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II 16 16.0 2 1 15 x α 16 x n 1 17 (x α) 2 16.1 16.1.1 2 x P (x) P (x) = 3x 3 4x + 4 369 Q(x) = x 4 ax + b ( ) 1 P (x) x Q(x) x P (x) x P (x) x = a P (a) P (x) = x 3 7x + 4 P (2) = 2 3 7 2 + 4 = 8 14 +
15 mod 12 = 3, 3 mod 12 = 3, 9 mod 12 = N N 0 x, y x y N x y (mod N) x y N mod N mod N N, x, y N > 0 (1) x x (mod N) (2) x y (mod N) y x
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合併後の交付税について
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Fermat s Last Theorem Hajime Mashima November 19, 2018 Abstract About 380 years ago, Pierre de Fermat wrote the following idea to Diophantus s Arithmetica. Cubum autem in duos cubos, aut quadratoquadratum
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