数论-北大-1811

[china_sjc]

#include <stdio.h>
#include <math.h>
#include <stdlib.h>
#include <float.h>

unsigned __int64 mulmod(unsigned __int64 a, unsigned __int64 b,
unsigned __int64 m)
{
static const double _2_64 = 18446744073709551616.0;
unsigned __int64 rh, rl;
__asm
{
xor edi,edi
mov eax,dword ptr [a]
mov ebx,dword ptr [b]
push eax
mov ecx,dword ptr [b+4]
mul ebx
mov dword ptr [rl],eax
mov dword ptr [rl+4],edx
pop eax
mul ecx
mov dword ptr [rh],edx
mov dword ptr [rh+4],edi
add dword ptr [rl+4],eax
adc dword ptr [rh],edi
mov eax,dword ptr [a+4]
mul ebx
add dword ptr [rh],edx
adc dword ptr [rh+4],edi
add dword ptr [rl+4],eax
adc dword ptr [rh],edi
adc dword ptr [rh+4],edi
mov eax,dword ptr [a+4]
mov ebx,dword ptr [b+4]
mul ebx
add dword ptr [rh],eax
adc dword ptr [rh+4],edx
}
rh %= m;
rl %= m;
__asm
{
fild qword ptr [m]
fild qword ptr [rh]
fmul qword ptr [_2_64]
fprem
fild qword ptr [rl]
faddp st(1), st
fprem
fistp qword ptr [m]
ffree st
}
return m;
}

unsigned __int64 powermod(unsigned int a, unsigned __int64 n, unsigned
__int64 m)
{
unsigned __int64 p = 1;
unsigned __int64 b = a;
if(m <= 0x7FFFFFFF)
{
while(n > 0)
{
if(n & 1 != 0)
{
p *= b;
if(p >= m)
{
p %= m;
}
}
n >>= 1;
b *= b;
if(b >= m)
{
b %= m;
}
}
}
else
{
while(n > 0)
{
if(n & 1 != 0)
{
p = mulmod(p, b, m);
}
n >>= 1;
b = mulmod(b, b, m);
}
}
return p;
}

static const unsigned int prime[] =
{
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61,
67, 71, \
73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149,
151, \
157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229,
233, \
239, 241, 251, 257, 263, 269, 271, 277, 281, 283, 293, 307, 311, 313,
317, \
331, 337, 347, 349, 353, 359, 367, 373, 379, 383, 389, 397, 401, 409,
419, \
421, 431, 433, 439, 443, 449, 457, 461, 463, 467, 479, 487, 491, 499,
503, \
509, 521, 523, 541, 547, 557, 563, 569, 571, 577, 587, 593, 599, 601,
607, \
613, 617, 619, 631, 641, 643, 647, 653, 659, 661, 673, 677, 683, 691,
701, \
709, 719, 727, 733, 739, 743, 751, 757, 761, 769, 773, 787, 797, 809,
811, \
821, 823, 827, 829, 839, 853, 857, 859, 863, 877, 881, 883, 887, 907,
911, \
919, 929, 937, 941, 947, 953, 967, 971, 977, 983, 991, 997, 1009,
1013, 1019, \
1021, 1031, 1033, 1039, 1049, 1051, 1061, 1063, 1069, 1087, 1091,
1093, 1097, \
1103, 1109, 1117, 1123, 1129, 1151, 1153, 1163, 1171, 1181, 1187,
1193, 1201, \
1213, 1217, 1223, 1229, 1231, 1237, 1249, 1259, 1277, 1279, 1283,
1289, 1291, \
1297, 1301, 1303, 1307, 1319, 1321, 1327, 1361, 1367, 1373, 1381,
1399, 1409, \
1423, 1427, 1429, 1433, 1439, 1447, 1451, 1453, 1459, 1471, 1481,
1483, 1487, \
1489, 1493, 1499, 1511, 1523, 1531, 1543, 1549, 1553, 1559, 1567,
1571, 1579, \
1583, 1597, 1601, 1607, 1609, 1613, 1619, 1621, 1627, 1637, 1657,
1663, 1667, \
1669, 1693, 1697, 1699, 1709, 1721, 1723, 1733, 1741, 1747, 1753,
1759, 1777, \
1783, 1787, 1789, 1801, 1811, 1823, 1831, 1847, 1861, 1867, 1871,
1873, 1877, \
1879, 1889, 1901, 1907, 1913, 1931, 1933, 1949, 1951, 1973, 1979,
1987, 1993, \
1997, 1999, 2003, 2011, 2017, 2027, 2029, 2039, 2053, 2063, 2069,
2081, 2083, \
2087, 2089, 2099, 2111, 2113, 2129, 2131, 2137, 2141, 2143, 2153,
2161, 2179, \
2203, 2207, 2213, 2221, 2237, 2239, 2243, 2251, 2267, 2269, 2273,
2281, 2287, \
2293, 2297, 2309, 2311, 2333, 2339, 2341, 2347, 2351, 2357, 2371,
2377, 2381, \
2383, 2389, 2393, 2399, 2411, 2417, 2423, 2437, 2441, 2447, 2459,
2467, 2473, \
2477, 2503, 2521, 2531, 2539, 2543, 2549, 2551, 2557, 2579, 2591,
2593, 2609, \
2617, 2621, 2633, 2647, 2657, 2659, 2663, 2671, 2677, 2683, 2687,
2689, 2693, \
2699, 2707, 2711, 2713, 2719, 2729, 2731, 2741, 2749, 2753, 2767,
2777, 2789, \
2791, 2797, 2801, 2803, 2819, 2833, 2837, 2843, 2851, 2857, 2861,
2879, 2887, \
2897, 2903, 2909, 2917, 2927, 2939, 2953, 2957, 2963, 2969, 2971,
2999, 3001, \
3011, 3019, 3023, 3037, 3041, 3049, 3061, 3067, 3079, 3083, 3089,
3109, 3119, \
3121, 3137, 3163, 3167, 3169, 3181, 3187, 3191, 3203, 3209, 3217,
3221, 3229, \
3251, 3253, 3257, 3259, 3271, 3299, 3301, 3307, 3313, 3319, 3323,
3329, 3331, \
3343, 3347, 3359, 3361, 3371, 3373, 3389, 3391, 3407, 3413, 3433,
3449, 3457, \
3461, 3463, 3467, 3469, 3491, 3499, 3511, 3517, 3527, 3529, 3533,
3539, 3541, \
3547, 3557, 3559, 3571, 3581, 3583, 3593, 3607, 3613, 3617, 3623,
3631, 3637, \
3643, 3659, 3671, 3673, 3677, 3691, 3697, 3701, 3709, 3719, 3727,
3733, 3739, \
3761, 3767, 3769, 3779, 3793, 3797, 3803, 3821, 3823, 3833, 3847,
3851, 3853, \
3863, 3877, 3881, 3889, 3907, 3911, 3917, 3919, 3923, 3929, 3931,
3943, 3947, \
3967, 3989, 4001, 4003, 4007, 4013, 4019, 4021, 4027, 4049, 4051,
4057, 4073, \
4079, 4091, 4093, 4099, 4111, 4127, 4129, 4133, 4139, 4153, 4157,
4159, 4177, \
4201, 4211, 4217, 4219, 4229, 4231, 4241, 4243, 4253, 4259, 4261,
4271, 4273, \
4283, 4289, 4297, 4327, 4337, 4339, 4349, 4357, 4363, 4373, 4391,
4397, 4409, \
4421, 4423, 4441, 4447, 4451, 4457, 4463, 4481, 4483, 4493, 4507,
4513, 4517, \
4519, 4523, 4547, 4549, 4561, 4567, 4583, 4591, 4597, 4603, 4621,
4637, 4639, \
4643, 4649, 4651, 4657, 4663, 4673, 4679, 4691, 4703, 4721, 4723,
4729, 4733, \
4751, 4759, 4783, 4787, 4789, 4793, 4799, 4801, 4813, 4817, 4831,
4861, 4871, \
4877, 4889, 4903, 4909, 4919, 4931, 4933, 4937, 4943, 4951, 4957,
4967, 4969, \
4973, 4987, 4993, 4999, 5003, 5009, 5011, 5021, 5023, 5039, 5051,
5059, 5077, \
5081, 5087, 5099, 5101, 5107, 5113, 5119, 5147, 5153, 5167, 5171,
5179, 5189, \
5197, 5209, 5227, 5231, 5233, 5237, 5261, 5273, 5279, 5281, 5297,
5303, 5309, \
5323, 5333, 5347, 5351, 5381, 5387, 5393, 5399, 5407, 5413, 5417,
5419, 5431, \
5437, 5441, 5443, 5449, 5471, 5477, 5479, 5483, 5501, 5503, 5507,
5519, 5521, \
5527, 5531, 5557, 5563, 5569, 5573, 5581, 5591, 5623, 5639, 5641,
5647, 5651, \
5653, 5657, 5659, 5669, 5683, 5689, 5693, 5701, 5711, 5717, 5737,
5741, 5743, \
5749, 5779, 5783, 5791, 5801, 5807, 5813, 5821, 5827, 5839, 5843,
5849, 5851, \
5857, 5861, 5867, 5869, 5879, 5881, 5897, 5903, 5923, 5927, 5939,
5953, 5981, \
5987, 6007, 6011, 6029, 6037, 6043, 6047, 6053, 6067, 6073, 6079,
6089, 6091, \
6101, 6113, 6121, 6131, 6133, 6143, 6151, 6163, 6173, 6197, 6199,
6203, 6211, \
6217, 6221, 6229, 6247, 6257, 6263, 6269, 6271, 6277, 6287, 6299,
6301, 6311, \
6317, 6323, 6329, 6337, 6343, 6353, 6359, 6361, 6367, 6373, 6379,
6389, 6397, \
6421, 6427, 6449, 6451, 6469, 6473, 6481, 6491, 6521, 6529, 6547,
6551, 6553, \
6563, 6569, 6571, 6577, 6581, 6599, 6607, 6619, 6637, 6653, 6659,
6661, 6673, \
6679, 6689, 6691, 6701, 6703, 6709, 6719, 6733, 6737, 6761, 6763,
6779, 6781, \
6791, 6793, 6803, 6823, 6827, 6829, 6833, 6841, 6857, 6863, 6869,
6871, 6883, \
6899, 6907, 6911, 6917, 6947, 6949, 6959, 6961, 6967, 6971, 6977,
6983, 6991, \
6997, 7001, 7013, 7019, 7027, 7039, 7043, 7057, 7069, 7079, 7103,
7109, 7121, \
7127, 7129, 7151, 7159, 7177, 7187, 7193, 7207, 7211, 7213, 7219,
7229, 7237, \
7243, 7247, 7253, 7283, 7297, 7307, 7309, 7321, 7331, 7333, 7349,
7351, 7369, \
7393, 7411, 7417, 7433, 7451, 7457, 7459, 7477, 7481, 7487, 7489,
7499, 7507, \
7517, 7523, 7529, 7537, 7541, 7547, 7549, 7559, 7561, 7573, 7577,
7583, 7589, \
7591, 7603, 7607, 7621, 7639, 7643, 7649, 7669, 7673, 7681, 7687,
7691, 7699, \
7703, 7717, 7723, 7727, 7741, 7753, 7757, 7759, 7789, 7793, 7817,
7823, 7829, \
7841, 7853, 7867, 7873, 7877, 7879, 7883, 7901, 7907, 7919, 7927,
7933, 7937, \
7949, 7951, 7963, 7993, 8009, 8011, 8017, 8039, 8053, 8059, 8069,
8081, 8087, \
8089, 8093, 8101, 8111, 8117, 8123, 8147, 8161, 8167, 8171, 8179,
8191, 8209, \
8219, 8221, 8231, 8233, 8237, 8243, 8263, 8269, 8273, 8287, 8291,
8293, 8297, \
8311, 8317, 8329, 8353, 8363, 8369, 8377, 8387, 8389, 8419, 8423,
8429, 8431, \
8443, 8447, 8461, 8467, 8501, 8513, 8521, 8527, 8537, 8539, 8543,
8563, 8573, \
8581, 8597, 8599, 8609, 8623, 8627, 8629, 8641, 8647, 8663, 8669,
8677, 8681, \
8689, 8693, 8699, 8707, 8713, 8719, 8731, 8737, 8741, 8747, 8753,
8761, 8779, \
8783, 8803, 8807, 8819, 8821, 8831, 8837, 8839, 8849, 8861, 8863,
8867, 8887, \
8893, 8923, 8929, 8933, 8941, 8951, 8963, 8969, 8971, 8999, 9001,
9007, 9011, \
9013, 9029, 9041, 9043, 9049, 9059, 9067, 9091, 9103, 9109, 9127,
9133, 9137, \
9151, 9157, 9161, 9173, 9181, 9187, 9199, 9203, 9209, 9221, 9227,
9239, 9241, \
9257, 9277, 9281, 9283, 9293, 9311, 9319, 9323, 9337, 9341, 9343,
9349, 9371, \
9377, 9391, 9397, 9403, 9413, 9419, 9421, 9431, 9433, 9437, 9439,
9461, 9463, \
9467, 9473, 9479, 9491, 9497, 9511, 9521, 9533, 9539, 9547, 9551,
9587, 9601, \
9613, 9619, 9623, 9629, 9631, 9643, 9649, 9661, 9677, 9679, 9689,
9697, 9719, \
9721, 9733, 9739, 9743, 9749, 9767, 9769, 9781, 9787, 9791, 9803,
9811, 9817, \
9829, 9833, 9839, 9851, 9857, 9859, 9871, 9883, 9887, 9901, 9907,
9923, 9929, \
9931, 9941, 9949, 9967, 9973, 10007, 10009, 10037, 10039, 10061,
10067, \
10069, 10079, 10091, 10093, 10099, 10103, 10111, 10133, 10139, 10141,
10151, \
10159, 10163, 10169, 10177, 10181, 10193, 10211, 10223, 10243, 10247,
10253, \
10259, 10267, 10271, 10273, 10289, 10301, 10303, 10313, 10321, 10331,
10333, \
10337, 10343, 10357, 10369, 10391, 10399, 10427, 10429, 10433, 10453,
10457, \
10459, 10463, 10477, 10487, 10499, 10501, 10513, 10529, 10531, 10559,
10567, \
10589, 10597, 10601, 10607, 10613, 10627, 10631, 10639, 10651, 10657,
10663, \
10667, 10687, 10691, 10709, 10711, 10723, 10729, 10733, 10739, 10753,
10771, \
10781, 10789, 10799, 10831, 10837, 10847, 10853, 10859, 10861, 10867,
10883, \
10889, 10891, 10903, 10909, 10937, 10939, 10949, 10957, 10973, 10979,
10987, \
10993, 11003, 11027, 11047, 11057, 11059, 11069, 11071, 11083, 11087,
11093, \
11113, 11117, 11119, 11131, 11149, 11159, 11161, 11171, 11173, 11177,
11197, \
11213, 11239, 11243, 11251, 11257, 11261, 11273, 11279, 11287, 11299,
11311, \
11317, 11321, 11329, 11351, 11353, 11369, 11383, 11393, 11399, 11411,
11423, \
11437, 11443, 11447, 11467, 11471, 11483, 11489, 11491, 11497, 11503,
11519, \
11527, 11549, 11551, 11579, 11587, 11593, 11597, 11617, 11621, 11633,
11657, \
11677, 11681, 11689, 11699, 11701, 11717, 11719, 11731, 11743, 11777,
11779, \
11783, 11789, 11801, 11807, 11813, 11821, 11827, 11831, 11833, 11839,
11863, \
11867, 11887, 11897, 11903, 11909, 11923, 11927, 11933, 11939, 11941,
11953, \
11959, 11969, 11971, 11981, 11987, 12007, 12011, 12037, 12041, 12043,
12049, \
12071, 12073, 12097, 12101, 12107, 12109, 12113, 12119, 12143, 12149,
12157, \
12161, 12163, 12197, 12203, 12211, 12227, 12239, 12241, 12251, 12253,
12263, \
12269, 12277, 12281, 12289, 12301, 12323, 12329, 12343, 12347, 12373,
12377, \
12379, 12391, 12401, 12409, 12413, 12421, 12433, 12437, 12451, 12457,
12473, \
12479, 12487, 12491, 12497, 12503, 12511, 12517, 12527, 12539, 12541,
12547, \
12553, 12569, 12577, 12583, 12589, 12601, 12611, 12613, 12619, 12637,
12641, \
12647, 12653, 12659, 12671, 12689, 12697, 12703, 12713, 12721, 12739,
12743, \
12757, 12763, 12781, 12791, 12799, 12809, 12821, 12823, 12829, 12841,
12853, \
12889, 12893, 12899, 12907, 12911, 12917, 12919, 12923, 12941, 12953,
12959, \
12967, 12973, 12979, 12983, 13001, 13003, 13007, 13009, 13033, 13037,
13043, \
13049, 13063, 13093, 13099, 13103, 13109, 13121, 13127, 13147, 13151,
13159, \
13163, 13171, 13177, 13183, 13187, 13217, 13219, 13229, 13241, 13249,
13259, \
13267, 13291, 13297, 13309, 13313, 13327, 13331, 13337, 13339, 13367,
13381, \
13397, 13399, 13411, 13417, 13421, 13441, 13451, 13457, 13463, 13469,
13477, \
13487, 13499, 13513, 13523, 13537, 13553, 13567, 13577, 13591, 13597,
13613, \
13619, 13627, 13633, 13649, 13669, 13679, 13681, 13687, 13691, 13693,
13697, \
13709, 13711, 13721, 13723, 13729, 13751, 13757, 13759, 13763, 13781,
13789, \
13799, 13807, 13829, 13831, 13841, 13859, 13873, 13877, 13879, 13883,
13901, \
13903, 13907, 13913, 13921, 13931, 13933, 13963, 13967, 13997, 13999,
14009, \
14011, 14029, 14033, 14051, 14057, 14071, 14081, 14083, 14087, 14107,
14143, \
14149, 14153, 14159, 14173, 14177, 14197, 14207, 14221, 14243, 14249,
14251, \
14281, 14293, 14303, 14321, 14323, 14327, 14341, 14347, 14369, 14387,
14389, \
14401, 14407, 14411, 14419, 14423, 14431, 14437, 14447, 14449, 14461,
14479, \
14489, 14503, 14519, 14533, 14537, 14543, 14549, 14551, 14557, 14561,
14563, \
14591, 14593, 14621, 14627, 14629, 14633, 14639, 14653, 14657, 14669,
14683, \
14699, 14713, 14717, 14723, 14731, 14737, 14741, 14747, 14753, 14759,
14767, \
14771, 14779, 14783, 14797, 14813, 14821, 14827, 14831, 14843, 14851,
14867, \
14869, 14879, 14887, 14891, 14897, 14923, 14929, 14939, 14947, 14951,
14957, \
14969, 14983, 15013, 15017, 15031, 15053, 15061, 15073, 15077, 15083,
15091, \
15101, 15107, 15121, 15131, 15137, 15139, 15149, 15161, 15173, 15187,
15193, \
15199, 15217, 15227, 15233, 15241, 15259, 15263, 15269, 15271, 15277,
15287, \
15289, 15299, 15307, 15313, 15319, 15329, 15331, 15349, 15359, 15361,
15373, \
15377, 15383, 15391, 15401, 15413, 15427, 15439, 15443, 15451, 15461,
15467, \
15473, 15493, 15497, 15511, 15527, 15541, 15551, 15559, 15569, 15581,
15583, \
15601, 15607, 15619, 15629, 15641, 15643, 15647, 15649, 15661, 15667,
15671, \
15679, 15683, 15727, 15731, 15733, 15737, 15739, 15749, 15761, 15767,
15773, \
15787, 15791, 15797, 15803, 15809, 15817, 15823, 15859, 15877, 15881,
15887, \
15889, 15901, 15907, 15913, 15919, 15923, 15937, 15959, 15971, 15973,
15991, \
16001, 16007, 16033, 16057, 16061, 16063, 16067, 16069, 16073, 16087,
16091, \
16097, 16103, 16111, 16127, 16139, 16141, 16183, 16187, 16189, 16193,
16217, \
16223, 16229, 16231, 16249, 16253, 16267, 16273, 16301, 16319, 16333,
16339, \
16349, 16361, 16363, 16369, 16381
};

int miller_rabin(unsigned __int64 m)
{
if(m > 2 && powermod(2, m - 1, m) != 1)
{
return 0;
}
if(m > 3 && powermod(3, m - 1, m) != 1)
{
return 0;
}
if(m > 5 && powermod(5, m - 1, m) != 1)
{
return 0;
}
if(m > 7 && powermod(7, m - 1, m) != 1)
{
return 0;
}
if(m > 11 && powermod(11, m - 1, m) != 1)
{
return 0;
}
if(m > 13 && powermod(13, m - 1, m) != 1)
{
return 0;
}
if(m > 17 && powermod(17, m - 1, m) != 1)
{
return 0;
}
if(m > 19 && powermod(19, m - 1, m) != 1)
{
return 0;
}
if(m > 23 && powermod(23, m - 1, m) != 1)
{
return 0;
}
return 1;
}

unsigned __int64 gcd(unsigned __int64 a, unsigned __int64 b)
{
int z = 0;
while(b != 0 && a != 0)
{
while((a & 1) == 0 && (b & 1) == 0)
{
z++;
a >>= 1;
b >>= 1;
}
while((a & 1) == 0)
{
a >>= 1;
}
while((b & 1) == 0)
{
b >>= 1;
}
if(a > b)
{
a -= b;
}
else
{
b -= a;
}
}
if(b == 0)
{
return a << z;
}
return b << z;
}

unsigned int brute_force64(unsigned __int64 m)
{
int i;
for(i = 0; prime[i] * prime[i] <= m && i < 1900; i++)
{
if(m % prime[i] == 0)
{
return prime[i];
}
}
return -1;
}

unsigned int brute_force(unsigned int m)
{
int i;
for(i = 0; prime[i] * prime[i] <= m && i < 1900; i++)
{
if(m % prime[i] == 0)
{
return prime[i];
}
}
return -1;
}

unsigned int pollard_rho(unsigned __int64 m)
{
unsigned __int64 d = 1;
unsigned __int64 fd, fm, y;
int i, k;
unsigned __int64 x = 43;
y = x;
k = 2;
i = 1;
for(;;)
{
__int64 diff;
++i;
x = (mulmod(x, x, m) + m - 1) % m;
if(x > y)
{
diff = x - y;
}
else
{
diff = y - x;
}
d = gcd(diff, m);
if(d != 1 && d != m)
{
break;
}
else
{
if(i == k)
{
y = x;
k <<= 1;
}
}
}
m /= d;
if(miller_rabin(d) == 0)
{
if(d <= 4294967295)
{
fd = brute_force((unsigned int)d);
}
else
{
fd = brute_force64(d);
}
if(fd == 4294967295)
{
fd = pollard_rho(d);
}
}
else
{
fd = d;
}
if(miller_rabin(m) == 0)
{
if(m <= 4294967295)
{
fm = brute_force((unsigned int)m);
}
else
{
fm = brute_force64(m);
}
if(fm == 4294967295)
{
fm = pollard_rho(m);
}
}
else
{
fm = m;
}
return (unsigned int)(fd < fm ? fd : fm);
}


unsigned int factor(unsigned __int64 m)
{
unsigned int f;
if(m > 4294967296)
{
f = brute_force64(m);
}
else
{
f = brute_force((unsigned int)m);
}
if(f != -1)
{
return f;
}
if(m <= 4294967295)
{
return (unsigned int)m;
}
return pollard_rho(m);
}

int main(void)
{
unsigned __int64 m;
int n;
_controlfp(_PC_64, _MCW_PC);
scanf("%d", &n);
while(n > 0)
{
scanf("%I64u", &m);
if(miller_rabin(m) != 0)
{
puts("Prime");
}
else
{
printf("%u\n", factor(m));
}
n--;
}
return 0;
}

[china_sjc]

http://acm.pku.edu.cn/JudgeOnline/problem?id=1811
题意只有一行：素数测试，给你一个数，判断一下它是否是素数。
难点：数非常巨大，最大到2的54次方。

[china_sjc]

上面的程序太难了，居然用到了汇编语言。大家想想。