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The "permanent" is similar to the "determinant" of a matrix
perm|a b| = ad + bc
|c d|
det|a b| = ad - bc
|c d|
perm|a b c| = aei + bfg + cdh + ceg + bdi + afh.
|d e f|
|g h i|
det|a b c| = aei + bfg + cdh - ceg - bdi - afh.
|d e f|
|g h i|
Notice the cyclic pattern. Clockwise is positive,
counter clockwise is negative.
The 3!=6 permutations of 1, 2, 3 with matrix subscripts are:
1 2 3 A(1,1) * A(2,2) * A(3,3) aei
2 3 1 A(1,2) * A(2,3) * A(3,1) bfg
3 1 2 A(1,3) * A(2,1) * A(3,2) cdh
3 2 1 A(1,3) * A(2,2) * A(3,1) ceg
2 1 3 A(1,2) * A(2,1) * A(3,3) bdi
1 3 2 A(1,1) * A(2,3) * A(3,2) afh
A(i,sigma(i))
Thus:
For a diagonal matrix, just one nonzero term,
the permanent is equal to the determinant.
Same for any square matrix with all zeros either
above or below the diagonal.
let sigma(n) represent all n! permutations of 1, 2, ..., n
,sigma(i) will be the ith element of the current permutation
perm of n by n matrix A with elements A(i,j) is
perm A = sum over sigma(n) product i=1..n A(i,sigma(i))
Direct computation requires n * n! floating point operations, flops.
Starting with the output file, permanent of random matrices
initializing, n= 1
A( 1 , 1 )= 0.79107778276832674
permanent = 0.79107778276832674 in 0.0000000000000000 seconds
initializing, n= 2
A( 1 , 1 )= 0.64429498726702061
A( 1 , 2 )= 0.66585099681553006
A( 2 , 1 )= 0.95770347861466165
A( 2 , 2 )= 0.12236507661750777
permanent = 0.71652702137047830 in 0.0000000000000000 seconds
initializing, n= 3
A( 1 , 1 )= 0.58984271045301240
A( 1 , 2 )= 0.48643458377869547
A( 1 , 3 )= 0.50604956853485183
A( 2 , 1 )= 0.17509836525428032
A( 2 , 2 )= 0.87822482868946383
A( 2 , 3 )= 0.32469578381846464
A( 3 , 1 )= 0.16203863693496148
A( 3 , 2 )= 0.38337096589774405
A( 3 , 3 )= 0.31582384338407954
permanent = 0.39550116362032195 in 0.0000000000000000 seconds
initializing, n= 4
permanent = 1.6972004238748366 in 0.0000000000000000 seconds
initializing, n= 5
permanent = 1.1995565139206676 in 0.0000000000000000 seconds
initializing, n= 6
permanent = 8.9015066849645361 in 0.0000000000000000 seconds
initializing, n= 7
permanent = 17.050252850698129 in 1.99999997857958078E-003 seconds
initializing, n= 8
permanent = 178.10765667301948 in 1.30000000353902578E-002 seconds
initializing, n= 9
permanent = 516.72987129756325 in 8.60000000102445483E-002 seconds
initializing, n= 10
permanent = 1020.0838023705367 in 0.68199999991338700 seconds
initializing, n= 11
permanent = 19478.337547346702 in 8.0130000000353903 seconds
initializing, n= 12
permanent = 57979.479713982662 in 102.07500000006985 seconds
flops are n * n!
finished test_permanent.f90
test_permanent_f90.out output
test_permanent.f90 source code
permanent.f90 compute permanent source code
permute.f90 compute permutations source code
test_permute.f90 test permute source code
test_permute_f90.out output
udrnrt.f90 compute random numbers source code
makefile.permanent for above/a>
Compute Combinations and Permutations in other languages
combinations.c "C" source code
combinations.h header file
combinations.adb Ada source code
combinations.ads specification
combinations.java Java source code
permute.c "C" source code
permute.h header file
test_permute.c "C" source code
test_permute_c.out output
permute.adb Ada source code
permute.ads specification
test_permute.adb Ada source code
test_permute_ada.out output
permute.java Java source code
test_permute.java Java source code
test_permute_java.out output
prime numbers are numbers with no remainder equal 0
prime numbers are generated by testing that no number less than half
that number has a remainder equal zero.
Sample python3 code below finds the first 1000 prime numbers,
then 10,000 , 100,000 , 1,000,000 prime numbers.
Only the first 1000 prime numbers are printed.
Data on spacing and counts is covered up to 1 million primes.
The 1,000 th prime number is 7,919
The 10,000 th prime number is 104,729
the 100,000 th prime number is 1,299,709
the 1,000,000 th prime number is
the number of primes less than 1,000 is 168
the number of primes less than 10,000 is 1,229
the number of primes less than 100,000 is 78,498
the number of primes less than 1,000,000 is
prime1.py3 source code
prime1_py3.out output
prime1.py3 running, computing n= 1000 primes
1000 primes =
[ 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]
number of adjacent primes with difference
difference number
1 1
2 174
4 169
6 244
8 83
10 100
12 75
14 42
16 26
18 34
20 10
22 12
24 12
26 3
28 4
30 8
32 1
number of primes less than value
number value
95 500
168 1000
239 1500
303 2000
367 2500
430 3000
489 3500
550 4000
610 4500
669 5000
prime1.py3 finished in 0.37908172607421875 seconds
prime10.py3 source code
prime10_py3.out output
prime10.py3 running, n= 10000
number of adjacent primes with difference
difference number
1 1
2 1270
4 1263
6 2012
8 801
10 953
12 1008
14 512
16 353
18 537
20 249
22 235
24 222
26 91
28 102
30 154
32 35
34 36
36 55
38 20
40 28
42 20
44 5
46 6
48 3
50 5
52 8
54 5
56 1
58 4
60 2
62 1
64 1
10000 th prime number is 104729
number of primes less than value
number value
168 1000
303 2000
430 3000
550 4000
669 5000
783 6000
900 7000
1007 8000
1117 9000
1229 10000
prime10.py3 finished in 30.235355854034424 seconds
prime100.py3 source code
prime100_py3.out output
prime100.py3 running, computing n= 100000 primes
number of adjacent primes with difference
difference number
1 1
2 10250
4 10213
6 16989
8 7067
10 8873
12 10158
14 5353
16 3661
18 6304
20 3043
22 2826
24 3538
26 1543
28 1631
30 2507
32 742
34 756
36 1032
38 455
40 563
42 661
44 250
46 219
48 290
50 154
52 110
54 197
56 84
58 81
60 131
62 29
64 34
66 65
68 19
70 35
72 26
74 17
76 10
78 21
80 15
82 9
84 10
86 5
88 2
90 7
92 2
94 1
96 3
98 2
100 2
106 1
112 1
100000 th prime number is 1299709
number of primes less than value
number value
9592 10000
17984 20000
25997 30000
33860 40000
41538 50000
49098 60000
56543 70000
63951 80000
71274 90000
78498 100000
prime100.py3 finished in 3643.3456246852875 seconds
prime1000.py3 source code
prime1000_py3.out output
The 1,000,000 th prime is 15,485,863
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