Carmichael 3 strong pseudoprimes

Carmichael 3 strong pseudoprimes is a draft programming task. It is not yet considered ready to be promoted as a complete task, for reasons that should be found in its talk page.

A lot of composite numbers can be detected by the Miller Rabin Test, but there are some that evade it.

The purpose of this task is to investigate such numbers. The method suggested is based on Notes by G.J.O Jameson March 2010

The objective is to find Carmichael numbers of the form Prime1 X Prime2 X Prime3.

Prime1 < Prime2 < Prime3 for all Prime1 upto 61 see page 7 of Notes by G.J.O Jameson March 2010.

For a given Prime1

for 1 < h3 < Prime1

for 0 < d < h3+Prime1

if (h3+Prime1)*(Prime1-1) mod d == 0 and -Prime1 squared mod h3 == d mod h3

then

Prime2 = 1 + ((Prime1-1) * g/d)

next d if Prime2 is not prime

Prime3 = 1 + (Prime1*Prime2/h3)

next d if Prime3 is not prime

next d if (Prime2*Prime3) mod (Prime1-1) not equal 1

Prime1 * Prime2 * Prime3 is a Carmichael Number

Ruby

<lang ruby>

1. Generate Charmichael Numbers
3. Nigel_Galloway
4. November 30th., 2012.

require 'prime' Integer.each_prime(61) {|p|

 (2...p).each {|h3|
   g = h3+p
   (1...g).each {|d|
     if (g*(p-1)) % d == 0 and (-1*p*p) % h3 == d % h3 then
       q = 1+((p-1)*g/d)
       next if not q.prime?
       r = 1+(p*q/h3)
       next if not r.prime? or not (q*r) % (p-1) == 1
       puts "#{p} X #{q} X #{r}" 
     end
   }
 }
 puts ""

} </lang>

Output:

3 X 11 X 17

5 X 29 X 73
5 X 17 X 29
5 X 13 X 17

7 X 19 X 67
7 X 31 X 73
7 X 13 X 31
7 X 23 X 41
7 X 73 X 103
7 X 13 X 19


13 X 61 X 397
13 X 37 X 241
13 X 97 X 421
13 X 37 X 97
13 X 37 X 61

17 X 41 X 233
17 X 353 X 1201

19 X 43 X 409
19 X 199 X 271

23 X 199 X 353

29 X 113 X 1093
29 X 197 X 953

31 X 991 X 15361
31 X 61 X 631
31 X 151 X 1171
31 X 61 X 271
31 X 61 X 211
31 X 271 X 601
31 X 181 X 331

37 X 109 X 2017
37 X 73 X 541
37 X 613 X 1621
37 X 73 X 181
37 X 73 X 109

41 X 1721 X 35281
41 X 881 X 12041
41 X 101 X 461
41 X 241 X 761
41 X 241 X 521
41 X 73 X 137
41 X 61 X 101

43 X 631 X 13567
43 X 271 X 5827
43 X 127 X 2731
43 X 127 X 1093
43 X 211 X 757
43 X 631 X 1597
43 X 127 X 211
43 X 211 X 337
43 X 433 X 643
43 X 547 X 673
43 X 3361 X 3907

47 X 3359 X 6073
47 X 1151 X 1933
47 X 3727 X 5153

53 X 157 X 2081
53 X 79 X 599
53 X 157 X 521

59 X 1451 X 2089

61 X 421 X 12841
61 X 181 X 5521
61 X 1301 X 19841
61 X 277 X 2113
61 X 181 X 1381
61 X 541 X 3001
61 X 661 X 2521
61 X 271 X 571
61 X 241 X 421
61 X 3361 X 4021