Carmichael 3 strong pseudoprimes

A lot of composite numbers can be separated from primes by Fermat's Little Theorem, but there are some that completely confound it.

The Miller Rabin Test uses a combination of Fermat's Little Theorem and Chinese Division Theorem to overcome this.

The purpose of this task is to investigate such numbers using a method based on Carmichael numbers, as suggested in Notes by G.J.O Jameson March 2010.

Task

Find Carmichael numbers of the form:

Prime₁ × Prime₂ × Prime₃

where (Prime₁ < Prime₂ < Prime₃) for all Prime₁ up to 61.
(See page 7 of Notes by G.J.O Jameson March 2010 for solutions.)

Pseudocode

For a given $Prime_{1}$

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) * (h3+Prime₁)/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

Ada

Uses the Miller_Rabin package from Miller-Rabin primality test#ordinary integers. <lang Ada>with Ada.Text_IO, Miller_Rabin;

procedure Nemesis is

  type Number is range 0 .. 2**40-1; -- sufficiently large for the task

  function Is_Prime(N: Number) return Boolean is
     package MR is new Miller_Rabin(Number); use MR;
  begin
     return MR.Is_Prime(N) = Probably_Prime;
  end Is_Prime;

begin

  for P1 in Number(2) .. 61 loop
     if Is_Prime(P1) then
        for H3 in Number(1) .. P1 loop
           declare
              G: Number := H3 + P1;
              P2, P3: Number;
           begin
              Inner:
              for D in 1 .. G-1 loop
                 if ((H3+P1) * (P1-1)) mod D = 0 and then
                   (-(P1 * P1)) mod H3 = D mod H3
                 then
                    P2 := 1 + ((P1-1) * G / D);
                    P3 := 1 +(P1*P2/H3);
                    if Is_Prime(P2) and then Is_Prime(P3)
                      and then (P2*P3) mod (P1-1) = 1
                    then
                      Ada.Text_IO.Put_Line
                       ( Number'Image(P1) & " *"   & Number'Image(P2) & " *" &
                         Number'Image(P3) & "  = " & Number'Image(P1*P2*P3) );
                    end if;
                 end if;
              end loop Inner;
           end;
        end loop;
     end if;
  end loop;

end Nemesis;</lang>

Output:

 3 * 11 * 17  =  561
 5 * 29 * 73  =  10585
 5 * 17 * 29  =  2465
 5 * 13 * 17  =  1105
 7 * 19 * 67  =  8911

... (the full output is 69 lines long) ...

 61 * 271 * 571  =  9439201
 61 * 241 * 421  =  6189121
 61 * 3361 * 4021  =  824389441

Uses the Sieve of Eratosthenes code from the Smith Numbers task with an increased upper-bound (included here for convenience). <lang algol68># sieve of Eratosthene: sets s[i] to TRUE if i is prime, FALSE otherwise # PROC sieve = ( REF[]BOOL s )VOID:

    BEGIN
       # start with everything flagged as prime                             # 
       FOR i TO UPB s DO s[ i ] := TRUE OD;
       # sieve out the non-primes                                           #
       s[ 1 ] := FALSE;
       FOR i FROM 2 TO ENTIER sqrt( UPB s ) DO
           IF s[ i ] THEN FOR p FROM i * i BY i TO UPB s DO s[ p ] := FALSE OD FI
       OD
    END # sieve # ;

construct a sieve of primes up to the maximum number required for the task #
For Prime1, we need to check numbers up to around 120 000 #

INT max number = 200 000; [ 1 : max number ]BOOL is prime; sieve( is prime );

Find the Carmichael 3 Stromg Pseudoprimes for Prime1 up to 61 #

FOR prime1 FROM 2 TO 61 DO

   IF is prime[ prime 1 ] THEN
       FOR h3 TO prime1 - 1 DO
           FOR d TO ( h3 + prime1 ) - 1 DO
               IF   ( h3 + prime1 ) * ( prime1 - 1 ) MOD d = 0
               AND ( - ( prime1 * prime1 ) ) MOD h3 = d MOD h3
               THEN
                   INT prime2 = 1 + ( ( prime1 - 1 ) * ( h3 + prime1 ) OVER d );
                   IF is prime[ prime2 ] THEN
                       INT prime3 = 1 + ( prime1 * prime2 OVER h3 );
                       IF is prime[ prime3 ] THEN 
                           IF ( prime2 * prime3 ) MOD ( prime1 - 1 ) = 1 THEN
                               print( ( whole( prime1, 0 ), " ", whole( prime2, 0 ), " ", whole( prime3, 0 ), newline ) )
                           FI
                       FI
                   FI
               FI
           OD
       OD
   FI

OD</lang>

Output:

3 11 17
5 29 73
5 17 29
5 13 17
7 19 67
7 31 73
7 13 31
7 23 41
7 73 103
7 13 19
13 61 397
13 37 241
13 97 421
13 37 97
13 37 61
...
59 1451 2089
61 421 12841
61 181 5521
61 1301 19841
61 277 2113
61 181 1381
61 541 3001
61 661 2521
61 271 571
61 241 421
61 3361 4021

C

include <stdio.h>

/* C's % operator actually calculates the remainder of a / b so we need a

* small adjustment so it works as expected for negative values */

define mod(n,m) ((((n) % (m)) + (m)) % (m))

int is_prime(unsigned int n) {

   if (n <= 3) {
       return n > 1;
   }
   else if (!(n % 2) || !(n % 3)) {
       return 0;
   }
   else {
       unsigned int i;
       for (i = 5; i*i <= n; i += 6)
           if (!(n % i) || !(n % (i + 2)))
               return 0;
       return 1;
   }

}

void carmichael3(int p1) {

   if (!is_prime(p1)) return;

   int h3, d, p2, p3;
   for (h3 = 1; h3 < p1; ++h3) {
       for (d = 1; d < h3 + p1; ++d) {
           if ((h3 + p1)*(p1 - 1) % d == 0 && mod(-p1 * p1, h3) == d % h3) {
               p2 = 1 + ((p1 - 1) * (h3 + p1)/d);
               if (!is_prime(p2)) continue;
               p3 = 1 + (p1 * p2 / h3);
               if (!is_prime(p3) || (p2 * p3) % (p1 - 1) != 1) continue;
               printf("%d %d %d\n", p1, p2, p3);
           }
       }
   }

}

int main(void) {

   int p1;
   for (p1 = 2; p1 < 62; ++p1)
       carmichael3(p1);
   return 0;

} </lang>

Output:

3 11 17
5 29 73
5 17 29
5 13 17
7 19 67
7 31 73
.
.
.
61 181 1381
61 541 3001
61 661 2521
61 271 571
61 241 421
61 3361 4021

Clojure

<lang lisp> (ns example

 (:gen-class))

(defn prime? [n]

 " Prime number test (using Java) "
 (.isProbablePrime (biginteger n) 16))

(defn carmichael [p1]

 " Triplets of Carmichael primes, with first element prime p1 "
 (if (prime? p1)
   (into [] (for [h3 (range 2 p1)
         :let [g (+ h3 p1)]
         d (range 1 g)
         :when (and (= (mod (* g (dec p1)) d) 0)
                    (= (mod (- (* p1 p1)) h3) (mod d h3)))
         :let [p2 (inc (quot (* (dec p1) g) d))]
         :when (prime? p2)
         :let [p3 (inc (quot (* p1 p2) h3))]
         :when (prime? p3)
         :when (= (mod (* p2 p3) (dec p1)) 1)]
        [p1 p2 p3]))))

Generate Result

(def numbers (mapcat carmichael (range 2 62))) (println (count numbers) "Carmichael numbers found:") (doseq [t numbers]

 (println (format "%5d x %5d x %5d = %10d" (first t) (second t) (last t) (apply * t))))

</lang>

Output:

69 Carmichael numbers found
    3 x    11 x    17 =        561
    5 x    29 x    73 =      10585
    5 x    17 x    29 =       2465
    5 x    13 x    17 =       1105
    7 x    19 x    67 =       8911
    7 x    31 x    73 =      15841
    7 x    13 x    31 =       2821
    7 x    23 x    41 =       6601
    7 x    73 x   103 =      52633
    7 x    13 x    19 =       1729
   13 x    61 x   397 =     314821
   13 x    37 x   241 =     115921
   13 x    97 x   421 =     530881
   13 x    37 x    97 =      46657
   13 x    37 x    61 =      29341
   17 x    41 x   233 =     162401
   17 x   353 x  1201 =    7207201
   19 x    43 x   409 =     334153
   19 x   199 x   271 =    1024651
   23 x   199 x   353 =    1615681
   29 x   113 x  1093 =    3581761
   29 x   197 x   953 =    5444489
   31 x   991 x 15361 =  471905281
   31 x    61 x   631 =    1193221
   31 x   151 x  1171 =    5481451
   31 x    61 x   271 =     512461
   31 x    61 x   211 =     399001
   31 x   271 x   601 =    5049001
   31 x   181 x   331 =    1857241
   37 x   109 x  2017 =    8134561
   37 x    73 x   541 =    1461241
   37 x   613 x  1621 =   36765901
   37 x    73 x   181 =     488881
   37 x    73 x   109 =     294409
   41 x  1721 x 35281 = 2489462641
   41 x   881 x 12041 =  434932961
   41 x   101 x   461 =    1909001
   41 x   241 x   761 =    7519441
   41 x   241 x   521 =    5148001
   41 x    73 x   137 =     410041
   41 x    61 x   101 =     252601
   43 x   631 x 13567 =  368113411
   43 x   271 x  5827 =   67902031
   43 x   127 x  2731 =   14913991
   43 x   127 x  1093 =    5968873
   43 x   211 x   757 =    6868261
   43 x   631 x  1597 =   43331401
   43 x   127 x   211 =    1152271
   43 x   211 x   337 =    3057601
   43 x   433 x   643 =   11972017
   43 x   547 x   673 =   15829633
   43 x  3361 x  3907 =  564651361
   47 x  3359 x  6073 =  958762729
   47 x  1151 x  1933 =  104569501
   47 x  3727 x  5153 =  902645857
   53 x   157 x  2081 =   17316001
   53 x    79 x   599 =    2508013
   53 x   157 x   521 =    4335241
   59 x  1451 x  2089 =  178837201
   61 x   421 x 12841 =  329769721
   61 x   181 x  5521 =   60957361
   61 x  1301 x 19841 = 1574601601
   61 x   277 x  2113 =   35703361
   61 x   181 x  1381 =   15247621
   61 x   541 x  3001 =   99036001
   61 x   661 x  2521 =  101649241
   61 x   271 x   571 =    9439201
   61 x   241 x   421 =    6189121
   61 x  3361 x  4021 =  824389441

D

<lang d>enum mod = (in int n, in int m) pure nothrow @nogc=> ((n % m) + m) % m;

bool isPrime(in uint n) pure nothrow @nogc {

 if (n == 2 || n == 3)
   return true;
 else if (n < 2 || n % 2 == 0 || n % 3 == 0)
   return false;
 for (uint div = 5, inc = 2; div ^^ 2 <= n;
    div += inc, inc = 6 - inc)
   if (n % div == 0)
     return false;
 return true;

}

void main() {

 import std.stdio;

 foreach (immutable p; 2 .. 62) {
   if (!p.isPrime) continue;
   foreach (immutable h3; 2 .. p) {
     immutable g = h3 + p;
     foreach (immutable d; 1 .. g) {
       if ((g * (p - 1)) % d != 0 || mod(-p * p, h3) != d % h3)
         continue;
       immutable q = 1 + (p - 1) * g / d;
       if (!q.isPrime) continue;
       immutable r = 1 + (p * q / h3);
       if (!r.isPrime || (q * r) % (p - 1) != 1) continue;
       writeln(p, " x ", q, " x ", r);
     }
   }
 }

}</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

EchoLisp

(define (charms (N 18) local: (h31 0) (Prime2 0) (Prime3 0)) (for* ((Prime1 (primes N))

      (h3 (in-range 1 Prime1))
      (d  (+ h3 Prime1)))
     (set! h31 (+ h3 Prime1))
     #:continue (!zero? (modulo (* h31 (1- Prime1)) d))
     #:continue (!= (modulo d h3) (modulo (- (* Prime1 Prime1)) h3))
     (set! Prime2 (1+ ( * (1- Prime1) (quotient h31 d))))
     #:when (prime? Prime2)
     (set! Prime3 (1+ (quotient (*  Prime1  Prime2)  h3)))
     #:when (prime? Prime3)
     #:when (= 1 (modulo (* Prime2 Prime3) (1- Prime1)))
     (printf " 💥 %12d = %d x %d x %d"  (* Prime1 Prime2 Prime3) Prime1 Prime2 Prime3)))

</lang>

Output:

<lang scheme> (charms 3) 💥 561 = 3 x 11 x 17 💥 10585 = 5 x 29 x 73 💥 2465 = 5 x 17 x 29 💥 1105 = 5 x 13 x 17

(charms 18)

💥 902645857 = 47 x 3727 x 5153 💥 2632033 = 53 x 53 x 937 💥 17316001 = 53 x 157 x 2081 💥 4335241 = 53 x 157 x 521 💥 178837201 = 59 x 1451 x 2089 💥 329769721 = 61 x 421 x 12841 💥 60957361 = 61 x 181 x 5521 💥 6924781 = 61 x 61 x 1861 💥 6924781 = 61 x 61 x 1861 💥 15247621 = 61 x 181 x 1381 💥 99036001 = 61 x 541 x 3001 💥 101649241 = 61 x 661 x 2521 💥 6189121 = 61 x 241 x 421 💥 824389441 = 61 x 3361 x 4021 </lang>

Fortran

Plan

This is F77 style, and directly translates the given calculation as per formula translation. It turns out that the normal integers suffice for the demonstration, except for just one of the products of the three primes: 41x1721x35281 = 2489462641, which is bigger than 2147483647, the 32-bit limit. Fortunately, INTEGER*8 variables are also available, so the extension is easy. Otherwise, one would have to mess about with using two integers in a bignum style, one holding say the millions, and the second the number up to a million.

Source

So, using the double MOD approach (see the Discussion) - which gives the same result for either style of MOD... <lang Fortran> LOGICAL FUNCTION ISPRIME(N) !Ad-hoc, since N is not going to be big...

      INTEGER N			!Despite this intimidating allowance of 32 bits...
      INTEGER F			!A possible factor.
       ISPRIME = .FALSE.		!Most numbers aren't prime.
       DO F = 2,SQRT(DFLOAT(N))	!Wince...
         IF (MOD(N,F).EQ.0) RETURN	!Not even avoiding even numbers beyond two.
       END DO				!Nice and brief, though.
       ISPRIME = .TRUE.		!No factor found.
     END FUNCTION ISPRIME		!So, done. Hopefully, not often.

     PROGRAM CHASE
     INTEGER P1,P2,P3	!The three primes to be tested.
     INTEGER H3,D	!Assistants.
     INTEGER MSG	!File unit number.
     MSG = 6		!Standard output.
     WRITE (MSG,1)	!A heading would be good.
   1 FORMAT ("Carmichael numbers that are the product of three primes:"
    & /"    P1  x P2  x P3 =",9X,"C")
     DO P1 = 2,61	!Step through the specified range.
       IF (ISPRIME(P1)) THEN	!Selecting only the primes.
         DO H3 = 2,P1 - 1		!For 1 < H3 < P1.
           DO D = 1,H3 + P1 - 1		!For 0 < D < H3 + P1.
             IF (MOD((H3 + P1)*(P1 - 1),D).EQ.0	!Filter.
    &        .AND. (MOD(H3 + MOD(-P1**2,H3),H3) .EQ. MOD(D,H3))) THEN	!Beware MOD for negative numbers! MOD(-P1**2, may surprise...
               P2 = 1 + (P1 - 1)*(H3 + P1)/D	!Candidate for the second prime.
               IF (ISPRIME(P2)) THEN		!Is it prime?
                 P3 = 1 + P1*P2/H3			!Yes. Candidate for the third prime.
                 IF (ISPRIME(P3)) THEN			!Is it prime?
                   IF (MOD(P2*P3,P1 - 1).EQ.1) THEN		!Yes! Final test.
                     WRITE (MSG,2) P1,P2,P3, INT8(P1)*P2*P3		!Result!
   2                 FORMAT (3I6,I12)
                   END IF
                 END IF
               END IF
             END IF
           END DO
         END DO
       END IF
     END DO
     END</lang>

Output

Carmichael numbers that are the product of three primes:
    P1  x P2  x P3 =         C
     3    11    17         561
     5    29    73       10585
     5    17    29        2465
     5    13    17        1105
     7    19    67        8911
     7    31    73       15841
     7    13    31        2821
     7    23    41        6601
     7    73   103       52633
     7    13    19        1729
    13    61   397      314821
    13    37   241      115921
    13    97   421      530881
    13    37    97       46657
    13    37    61       29341
    17    41   233      162401
    17   353  1201     7207201
    19    43   409      334153
    19   199   271     1024651
    23   199   353     1615681
    29   113  1093     3581761
    29   197   953     5444489
    31   991 15361   471905281
    31    61   631     1193221
    31   151  1171     5481451
    31    61   271      512461
    31    61   211      399001
    31   271   601     5049001
    31   181   331     1857241
    37   109  2017     8134561
    37    73   541     1461241
    37   613  1621    36765901
    37    73   181      488881
    37    73   109      294409
    41  1721 35281  2489462641
    41   881 12041   434932961
    41   101   461     1909001
    41   241   761     7519441
    41   241   521     5148001
    41    73   137      410041
    41    61   101      252601
    43   631 13567   368113411
    43   271  5827    67902031
    43   127  2731    14913991
    43   127  1093     5968873
    43   211   757     6868261
    43   631  1597    43331401
    43   127   211     1152271
    43   211   337     3057601
    43   433   643    11972017
    43   547   673    15829633
    43  3361  3907   564651361
    47  3359  6073   958762729
    47  1151  1933   104569501
    47  3727  5153   902645857
    53   157  2081    17316001
    53    79   599     2508013
    53   157   521     4335241
    59  1451  2089   178837201
    61   421 12841   329769721
    61   181  5521    60957361
    61  1301 19841  1574601601
    61   277  2113    35703361
    61   181  1381    15247621
    61   541  3001    99036001
    61   661  2521   101649241
    61   271   571     9439201
    61   241   421     6189121
    61  3361  4021   824389441

FreeBASIC

<lang freebasic>' version 17-10-2016 ' compile with: fbc -s console

' using a sieve for finding primes

Define max_sieve 10000000 ' 10^7

ReDim Shared As Byte isprime(max_sieve)

' translated the pseudo code to FreeBASIC Sub carmichael3(p1 As Integer)

 If isprime(p1) = 0 Then Exit Sub

 Dim As Integer h3, d, p2, p3, t1, t2

 For h3 = 1 To p1 -1
   t1 = (h3 + p1) * (p1 -1)
   t2 = (-p1 * p1) Mod h3
   If t2 < 0 Then t2 = t2 + h3
   For d = 1 To h3 + p1 -1
     If t1 Mod d = 0 And t2 = (d Mod h3) Then
       p2 = 1 + (t1 \ d)
       If isprime(p2) = 0 Then Continue For
       p3 = 1 + (p1 * p2 \ h3)
       If isprime(p3) = 0 Or ((p2 * p3) Mod (p1 -1)) <> 1 Then Continue For
       Print Using "### * #### * #####"; p1; p2; p3
     End If
   Next d
 Next h3

End Sub

' ------=< MAIN >=------

Dim As UInteger i, j

'set up sieve For i = 3 To max_sieve Step 2

 isprime(i) = 1

Next i

isprime(2) = 1 For i = 3 To Sqr(max_sieve) Step 2

 If isprime(i) = 1 Then
   For j = i * i To max_sieve Step i * 2
     isprime(j) = 0
   Next j
 End If

Next i

For i = 2 To 61

 carmichael3(i)

Next i

' empty keyboard buffer While InKey <> "" : Wend Print : Print "hit any key to end program" Sleep End</lang>

Output:

  3 *   11 *    17
  5 *   29 *    73
  5 *   17 *    29
  5 *   13 *    17
  7 *   19 *    67
  7 *   31 *    73
  7 *   13 *    31
  7 *   23 *    41
  7 *   73 *   103
  7 *   13 *    19
 13 *   61 *   397
 13 *   37 *   241
 13 *   97 *   421
 13 *   37 *    97
 13 *   37 *    61
 17 *   41 *   233
 17 *  353 *  1201
 19 *   43 *   409
 19 *  199 *   271
 23 *  199 *   353
 29 *  113 *  1093
 29 *  197 *   953
 31 *  991 * 15361
 31 *   61 *   631
 31 *  151 *  1171
 31 *   61 *   271
 31 *   61 *   211
 31 *  271 *   601
 31 *  181 *   331
 37 *  109 *  2017
 37 *   73 *   541
 37 *  613 *  1621
 37 *   73 *   181
 37 *   73 *   109
 41 * 1721 * 35281
 41 *  881 * 12041
 41 *  101 *   461
 41 *  241 *   761
 41 *  241 *   521
 41 *   73 *   137
 41 *   61 *   101
 43 *  631 * 13567
 43 *  271 *  5827
 43 *  127 *  2731
 43 *  127 *  1093
 43 *  211 *   757
 43 *  631 *  1597
 43 *  127 *   211
 43 *  211 *   337
 43 *  433 *   643
 43 *  547 *   673
 43 * 3361 *  3907
 47 * 3359 *  6073
 47 * 1151 *  1933
 47 * 3727 *  5153
 53 *  157 *  2081
 53 *   79 *   599
 53 *  157 *   521
 59 * 1451 *  2089
 61 *  421 * 12841
 61 *  181 *  5521
 61 * 1301 * 19841
 61 *  277 *  2113
 61 *  181 *  1381
 61 *  541 *  3001
 61 *  661 *  2521
 61 *  271 *   571
 61 *  241 *   421
 61 * 3361 *  4021

Go

<lang go>package main

import "fmt"

// Use this rather than % for negative integers func mod(n, m int) int {

   return ((n % m) + m) % m

}

func isPrime(n int) bool {

   if n < 2 { return false }
   if n % 2 == 0 { return n == 2 }
   if n % 3 == 0 { return n == 3 }
   d := 5
   for d * d <= n {
       if n % d == 0 { return false }
       d += 2
       if n % d == 0 { return false }
       d += 4
   }
   return true

}

func carmichael(p1 int) {

   for h3 := 2; h3 < p1; h3++ {
       for d := 1; d < h3 + p1; d++ {
           if (h3 + p1) * (p1 - 1) % d == 0 && mod(-p1 * p1, h3) == d % h3 {
               p2 := 1 + (p1 - 1) * (h3 + p1) / d
               if !isPrime(p2) { continue }
               p3 := 1 + p1 * p2 / h3
               if !isPrime(p3) { continue }
               if p2 * p3 % (p1 - 1) != 1 { continue }
               c := p1 * p2 * p3
               fmt.Printf("%2d   %4d   %5d     %d\n", p1, p2, p3, c)
           }
       }
   }

}

func main() {

   fmt.Println("The following are Carmichael munbers for p1 <= 61:\n")
   fmt.Println("p1     p2      p3     product")
   fmt.Println("==     ==      ==     =======")

   for p1 := 2; p1 <= 61; p1++ {
       if isPrime(p1) { carmichael(p1) }
   }

}</lang>

Output:

The following are Carmichael munbers for p1 <= 61:

p1     p2      p3     product
==     ==      ==     =======
 3     11      17     561
 5     29      73     10585
 5     17      29     2465
 5     13      17     1105
 7     19      67     8911
 7     31      73     15841
 7     13      31     2821
 7     23      41     6601
 7     73     103     52633
 7     13      19     1729
13     61     397     314821
13     37     241     115921
13     97     421     530881
13     37      97     46657
13     37      61     29341
17     41     233     162401
17    353    1201     7207201
19     43     409     334153
19    199     271     1024651
23    199     353     1615681
29    113    1093     3581761
29    197     953     5444489
31    991   15361     471905281
31     61     631     1193221
31    151    1171     5481451
31     61     271     512461
31     61     211     399001
31    271     601     5049001
31    181     331     1857241
37    109    2017     8134561
37     73     541     1461241
37    613    1621     36765901
37     73     181     488881
37     73     109     294409
41   1721   35281     2489462641
41    881   12041     434932961
41    101     461     1909001
41    241     761     7519441
41    241     521     5148001
41     73     137     410041
41     61     101     252601
43    631   13567     368113411
43    271    5827     67902031
43    127    2731     14913991
43    127    1093     5968873
43    211     757     6868261
43    631    1597     43331401
43    127     211     1152271
43    211     337     3057601
43    433     643     11972017
43    547     673     15829633
43   3361    3907     564651361
47   3359    6073     958762729
47   1151    1933     104569501
47   3727    5153     902645857
53    157    2081     17316001
53     79     599     2508013
53    157     521     4335241
59   1451    2089     178837201
61    421   12841     329769721
61    181    5521     60957361
61   1301   19841     1574601601
61    277    2113     35703361
61    181    1381     15247621
61    541    3001     99036001
61    661    2521     101649241
61    271     571     9439201
61    241     421     6189121
61   3361    4021     824389441

Haskell

Translation of: Ruby

Library: primes

Works with: GHC version 7.4.1

Works with: primes version 0.2.1.0

<lang haskell>#!/usr/bin/runhaskell

import Data.Numbers.Primes import Control.Monad (guard)

carmichaels = do

 p <- takeWhile (<= 61) primes
 h3 <- [2..(p-1)]
 let g = h3 + p
 d <- [1..(g-1)]
 guard $ (g * (p - 1)) `mod` d == 0 && (-1 * p * p) `mod` h3 == d `mod` h3
 let q = 1 + (((p - 1) * g) `div` d)
 guard $ isPrime q
 let r = 1 + ((p * q) `div` h3)
 guard $ isPrime r && (q * r) `mod` (p - 1) == 1
 return (p, q, r)

main = putStr $ unlines $ map show carmichaels</lang>

Output:

(3,11,17)
(5,29,73)
(5,17,29)
(5,13,17)
(7,19,67)
(7,31,73)
(7,13,31)
(7,23,41)
(7,73,103)
(7,13,19)
(13,61,397)
(13,37,241)
(13,97,421)
(13,37,97)
(13,37,61)
(17,41,233)
(17,353,1201)
(19,43,409)
(19,199,271)
(23,199,353)
(29,113,1093)
(29,197,953)
(31,991,15361)
(31,61,631)
(31,151,1171)
(31,61,271)
(31,61,211)
(31,271,601)
(31,181,331)
(37,109,2017)
(37,73,541)
(37,613,1621)
(37,73,181)
(37,73,109)
(41,1721,35281)
(41,881,12041)
(41,101,461)
(41,241,761)
(41,241,521)
(41,73,137)
(41,61,101)
(43,631,13567)
(43,271,5827)
(43,127,2731)
(43,127,1093)
(43,211,757)
(43,631,1597)
(43,127,211)
(43,211,337)
(43,433,643)
(43,547,673)
(43,3361,3907)
(47,3359,6073)
(47,1151,1933)
(47,3727,5153)
(53,157,2081)
(53,79,599)
(53,157,521)
(59,1451,2089)
(61,421,12841)
(61,181,5521)
(61,1301,19841)
(61,277,2113)
(61,181,1381)
(61,541,3001)
(61,661,2521)
(61,271,571)
(61,241,421)
(61,3361,4021)

Icon and Unicon

The following works in both languages. <lang unicon>link "factors"

procedure main(A)

   n := integer(!A) | 61
   every write(carmichael3(!n))

end

procedure carmichael3(p1)

   every (isprime(p1), (h := 1+!(p1-1)), (d := !(h+p1-1))) do
       if (mod(((h+p1)*(p1-1)),d) = 0, mod((-p1*p1),h) = mod(d,h)) then {
           p2 := 1 + (p1-1)*(h+p1)/d
           p3 := 1 + p1*p2/h
           if (isprime(p2), isprime(p3), mod((p2*p3),(p1-1)) = 1) then
               suspend format(p1,p2,p3)
           }

end

procedure mod(n,d)

  return (d+n%d)%d

end

procedure format(p1,p2,p3)

   return left(p1||" * "||p2||" * "||p3,20)||" = "||(p1*p2*p3)

end</lang>

Output, with middle lines elided:

->c3sp
3 * 11 * 17          = 561
5 * 29 * 73          = 10585
5 * 17 * 29          = 2465
5 * 13 * 17          = 1105
7 * 19 * 67          = 8911
7 * 31 * 73          = 15841
7 * 13 * 31          = 2821
7 * 23 * 41          = 6601
7 * 73 * 103         = 52633
7 * 13 * 19          = 1729
13 * 61 * 397        = 314821
13 * 37 * 241        = 115921
...
53 * 157 * 2081      = 17316001
53 * 79 * 599        = 2508013
53 * 157 * 521       = 4335241
59 * 1451 * 2089     = 178837201
61 * 421 * 12841     = 329769721
61 * 181 * 5521      = 60957361
61 * 1301 * 19841    = 1574601601
61 * 277 * 2113      = 35703361
61 * 181 * 1381      = 15247621
61 * 541 * 3001      = 99036001
61 * 661 * 2521      = 101649241
61 * 271 * 571       = 9439201
61 * 241 * 421       = 6189121
61 * 3361 * 4021     = 824389441
->

J

<lang J> q =: (,"0 1~ >:@i.@<:@+/"1)&.>@(,&.>"0 1~ >:@i.)&.>@I.@(1&p:@i.)@>: f1 =: (0: = {. | <:@{: * 1&{ + {:) *. ((1&{ | -@*:@{:) = 1&{ | {.) f2 =: 1: = <:@{. | ({: * 1&{) p2 =: 0:`((* 1&p:)@(<.@(1: + <:@{: * {. %~ 1&{ + {:)))@.f1 p3 =: 3:$0:`((* 1&p:)@({: , {. , (<.@>:@(1&{ %~ {. * {:))))@.(*@{.)@(p2 , }.) (-. 3:$0:)@(((*"0 f2)@p3"1)@;@;@q) 61 </lang> Output

 3   11    17
 5   29    73
 5   17    29
 5   13    17
 7   19    67
 7   31    73
 7   13    31
 7   23    41
 7   73   103
 7   13    19
13   61   397
13   37   241
13   97   421
13   37    97
13   37    61
17   41   233
17  353  1201
19   43   409
19  199   271
23  199   353
29  113  1093
29  197   953
31  991 15361
31   61   631
31  151  1171
31   61   271
31   61   211
31  271   601
31  181   331
37  109  2017
37   73   541
37  613  1621
37   73   181
37   73   109
41 1721 35281
41  881 12041
41  101   461
41  241   761
41  241   521
41   73   137
41   61   101
43  631 13567
43  271  5827
43  127  2731
43  127  1093
43  211   757
43  631  1597
43  127   211
43  211   337
43  433   643
43  547   673
43 3361  3907
47 3359  6073
47 1151  1933
47 3727  5153
53  157  2081
53   79   599
53  157   521
59 1451  2089
61  421 12841
61  181  5521
61 1301 19841
61  277  2113
61  181  1381
61  541  3001
61  661  2521
61  271   571
61  241   421
61 3361  4021

Java

Translation of: D

<lang java>public class Test {

   static int mod(int n, int m) {
       return ((n % m) + m) % m;
   }

   static boolean isPrime(int n) {
       if (n == 2 || n == 3)
           return true;
       else if (n < 2 || n % 2 == 0 || n % 3 == 0)
           return false;
       for (int div = 5, inc = 2; Math.pow(div, 2) <= n;
               div += inc, inc = 6 - inc)
           if (n % div == 0)
               return false;
       return true;
   }

   public static void main(String[] args) {
       for (int p = 2; p < 62; p++) {
           if (!isPrime(p))
               continue;
           for (int h3 = 2; h3 < p; h3++) {
               int g = h3 + p;
               for (int d = 1; d < g; d++) {
                   if ((g * (p - 1)) % d != 0 || mod(-p * p, h3) != d % h3)
                       continue;
                   int q = 1 + (p - 1) * g / d;
                   if (!isPrime(q))
                       continue;
                   int r = 1 + (p * q / h3);
                   if (!isPrime(r) || (q * r) % (p - 1) != 1)
                       continue;
                   System.out.printf("%d x %d x %d%n", p, q, r);
               }
           }
       }
   }

}</lang>

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

Julia

This solution is a straightforward implementation of the algorithm of the Jameson paper cited in the task description. Just for fun, I use Julia's capacity to accommodate Unicode identifiers to match some of the paper's symbols to the variables used in the carmichael function.

Function <lang Julia> function carmichael{T<:Integer}(pmax::T)

   0 < pmax || throw(DomainError())
   car = T[]
   for p in primes(pmax)
       for h₃ in 2:(p-1)
           m = (p - 1)*(h₃ + p)
           pmh = mod(-p^2, h₃)
           for Δ in 1:(h₃+p-1)
               m%Δ==0 && Δ%h₃==pmh || continue
               q = div(m, Δ) + 1
               isprime(q) || continue
               r = div((p*q - 1), h₃) + 1
               isprime(r) && mod(q*r, (p-1))==1 || continue
               append!(car, [p, q, r])
           end
       end
   end
   reshape(car, 3, div(length(car), 3))

end </lang>

Main <lang Julia> hi = 61 car = carmichael(hi)

curp = 0 tcnt = 0 print("Carmichael 3 (p\u00d7q\u00d7r) Pseudoprimes, up to p = ", hi, ":") for j in sortperm(1:size(car)[2], by=x->(car[1,x], car[2,x], car[3,x]))

   p, q, r = car[:,j]
   c = prod(car[:,j])
   if p != curp
       curp = p
       print(@sprintf("\n\np = %d\n  ", p))
       tcnt = 0
   end
   if tcnt == 4
       print("\n  ")
       tcnt = 1
   else
       tcnt += 1
   end
   print(@sprintf("p\u00d7%d\u00d7%d = %d  ", q, r, c))

end println("\n\n", size(car)[2], " results in total.") </lang>

Output:

Carmichael 3 (p×q×r) Pseudoprimes, up to p = 61:

p = 3
  p×11×17 = 561  

p = 5
  p×13×17 = 1105  p×17×29 = 2465  p×29×73 = 10585  

p = 7
  p×13×19 = 1729  p×13×31 = 2821  p×19×67 = 8911  p×23×41 = 6601  
  p×31×73 = 15841  p×73×103 = 52633  

p = 13
  p×37×61 = 29341  p×37×97 = 46657  p×37×241 = 115921  p×61×397 = 314821  
  p×97×421 = 530881  

p = 17
  p×41×233 = 162401  p×353×1201 = 7207201  

p = 19
  p×43×409 = 334153  p×199×271 = 1024651  

p = 23
  p×199×353 = 1615681  

p = 29
  p×113×1093 = 3581761  p×197×953 = 5444489  

p = 31
  p×61×211 = 399001  p×61×271 = 512461  p×61×631 = 1193221  p×151×1171 = 5481451  
  p×181×331 = 1857241  p×271×601 = 5049001  p×991×15361 = 471905281  

p = 37
  p×73×109 = 294409  p×73×181 = 488881  p×73×541 = 1461241  p×109×2017 = 8134561  
  p×613×1621 = 36765901  

p = 41
  p×61×101 = 252601  p×73×137 = 410041  p×101×461 = 1909001  p×241×521 = 5148001  
  p×241×761 = 7519441  p×881×12041 = 434932961  p×1721×35281 = 2489462641  

p = 43
  p×127×211 = 1152271  p×127×1093 = 5968873  p×127×2731 = 14913991  p×211×337 = 3057601  
  p×211×757 = 6868261  p×271×5827 = 67902031  p×433×643 = 11972017  p×547×673 = 15829633  
  p×631×1597 = 43331401  p×631×13567 = 368113411  p×3361×3907 = 564651361  

p = 47
  p×1151×1933 = 104569501  p×3359×6073 = 958762729  p×3727×5153 = 902645857  

p = 53
  p×79×599 = 2508013  p×157×521 = 4335241  p×157×2081 = 17316001  

p = 59
  p×1451×2089 = 178837201  

p = 61
  p×181×1381 = 15247621  p×181×5521 = 60957361  p×241×421 = 6189121  p×271×571 = 9439201  
  p×277×2113 = 35703361  p×421×12841 = 329769721  p×541×3001 = 99036001  p×661×2521 = 101649241  
  p×1301×19841 = 1574601601  p×3361×4021 = 824389441  

69 results in total.

Kotlin

Translation of: D

<lang scala>fun Int.isPrime(): Boolean {

   return when {
       this == 2 -> true
       this <= 1 || this % 2 == 0 -> false
       else -> {
           val max = Math.sqrt(toDouble()).toInt()
           (3..max step 2)
               .filter { this % it == 0 }
               .forEach { return false }
           true
       }
   }

}

fun mod(n: Int, m: Int) = ((n % m) + m) % m

fun main(args: Array<String>) {

   for (p1 in 3..61) {
       if (p1.isPrime()) {
           for (h3 in 2 until p1) {
               val g = h3 + p1
               for (d in 1 until g) {
                   if ((g * (p1 - 1)) % d == 0 && mod(-p1 * p1, h3) == d % h3) {
                       val q = 1 + (p1 - 1) * g / d
                       if (q.isPrime()) {
                           val r = 1 + (p1 * q / h3)
                           if (r.isPrime() && (q * r) % (p1 - 1) == 1) {
                               println("$p1 x $q x $r")
                           }
                       }
                   }
               }
           }
       }
   }

}</lang>

Output:

See D output.

Mathematica / Wolfram Language

<lang mathematica>Cases[Cases[

 Cases[Table[{p1, h3, d}, {p1, Array[Prime, PrimePi@61]}, {h3, 2, 
    p1 - 1}, {d, 1, h3 + p1 - 1}], {p1_Integer, h3_, d_} /; 
    PrimeQ[1 + (p1 - 1) (h3 + p1)/d] && 
     Divisible[p1^2 + d, h3] :> {p1, 1 + (p1 - 1) (h3 + p1)/d, h3}, 
  Infinity], {p1_, p2_, h3_} /; PrimeQ[1 + Floor[p1 p2/h3]] :> {p1, 
   p2, 1 + Floor[p1 p2/h3]}], {p1_, p2_, p3_} /; 
  Mod[p2 p3, p1 - 1] == 1 :> Print[p1, "*", p2, "*", p3]]</lang>

PARI/GP

<lang parigp>f(p)={

 my(v=List(),q,r);
 for(h=2,p-1,
   for(d=1,h+p-1,
     if((h+p)*(p-1)%d==0 && Mod(p,h)^2==-d && isprime(q=(p-1)*(h+p)/d+1) && isprime(r=p*q\h+1)&&q*r%(p-1)==1,
       listput(v,p*q*r)
     )
   )
 );
 Set(v)

}; forprime(p=3,67,v=f(p); for(i=1,#v,print1(v[i]", ")))</lang>

Output:

561, 1105, 2465, 10585, 1729, 2821, 6601, 8911, 15841, 52633, 29341, 46657, 115921, 314821, 530881, 162401, 7207201, 334153, 1024651, 1615681, 3581761, 5444489, 399001, 512461, 1193221, 1857241, 5049001, 5481451, 471905281, 294409, 488881, 1461241, 8134561, 36765901, 252601, 410041, 1909001, 5148001, 7519441, 434932961, 2489462641, 1152271, 3057601, 5968873, 6868261, 11972017, 14913991, 15829633, 43331401, 67902031, 368113411, 564651361, 104569501, 902645857, 958762729, 2508013, 4335241, 17316001, 178837201, 6189121, 9439201, 15247621, 35703361, 60957361, 99036001, 101649241, 329769721, 824389441, 1574601601, 10267951, 163954561, 7991602081,

Perl

Library: ntheory

<lang perl>use ntheory qw/forprimes is_prime vecprod/;

forprimes { my $p = $_;

  for my $h3 (2 .. $p-1) {
     my $ph3 = $p + $h3;
     for my $d (1 .. $ph3-1) {               # Jameseon procedure page 6
        next if ((-$p*$p) % $h3) != ($d % $h3);
        next if (($p-1)*$ph3) % $d;
        my $q = 1 + ($p-1)*$ph3 / $d;        # Jameson eq 7
        next unless is_prime($q);
        my $r = 1 + ($p*$q-1) / $h3;         # Jameson eq 6
        next unless is_prime($r);
        next unless ($q*$r) % ($p-1) == 1;
        printf "%2d x %5d x %5d = %s\n",$p,$q,$r,vecprod($p,$q,$r);
     }
  }

} 3,61;</lang>

Output:

 3 x    11 x    17 = 561
 5 x    29 x    73 = 10585
 5 x    17 x    29 = 2465
 5 x    13 x    17 = 1105
 ... full output is 69 lines ...
61 x   661 x  2521 = 101649241
61 x   271 x   571 = 9439201
61 x   241 x   421 = 6189121
61 x  3361 x  4021 = 824389441

Perl 6

Works with: Rakudo version 2015.12

An almost direct translation of the pseudocode. We take the liberty of going up to 67 to show we aren't limited to 32-bit integers. (Perl 6 uses arbitrary precision in any case.) <lang perl6>for (2..67).grep: *.is-prime -> \Prime1 {

   for 1 ^..^ Prime1 -> \h3 {
       my \g = h3 + Prime1;
       for 0 ^..^ h3 + Prime1 -> \d {
           if (h3 + Prime1) * (Prime1 - 1) %% d and -Prime1**2 % h3 == d % h3  {
               my \Prime2 = floor 1 + (Prime1 - 1) * g / d;
               next unless Prime2.is-prime;
               my \Prime3 = floor 1 + Prime1 * Prime2 / h3;
               next unless Prime3.is-prime;
               next unless (Prime2 * Prime3) % (Prime1 - 1) == 1;
               say "{Prime1} × {Prime2} × {Prime3} == {Prime1 * Prime2 * Prime3}";
           }
       }
   }

}</lang>

Output:

3 × 11 × 17 == 561
5 × 29 × 73 == 10585
5 × 17 × 29 == 2465
5 × 13 × 17 == 1105
7 × 19 × 67 == 8911
7 × 31 × 73 == 15841
7 × 13 × 31 == 2821
7 × 23 × 41 == 6601
7 × 73 × 103 == 52633
7 × 13 × 19 == 1729
13 × 61 × 397 == 314821
13 × 37 × 241 == 115921
13 × 97 × 421 == 530881
13 × 37 × 97 == 46657
13 × 37 × 61 == 29341
17 × 41 × 233 == 162401
17 × 353 × 1201 == 7207201
19 × 43 × 409 == 334153
19 × 199 × 271 == 1024651
23 × 199 × 353 == 1615681
29 × 113 × 1093 == 3581761
29 × 197 × 953 == 5444489
31 × 991 × 15361 == 471905281
31 × 61 × 631 == 1193221
31 × 151 × 1171 == 5481451
31 × 61 × 271 == 512461
31 × 61 × 211 == 399001
31 × 271 × 601 == 5049001
31 × 181 × 331 == 1857241
37 × 109 × 2017 == 8134561
37 × 73 × 541 == 1461241
37 × 613 × 1621 == 36765901
37 × 73 × 181 == 488881
37 × 73 × 109 == 294409
41 × 1721 × 35281 == 2489462641
41 × 881 × 12041 == 434932961
41 × 101 × 461 == 1909001
41 × 241 × 761 == 7519441
41 × 241 × 521 == 5148001
41 × 73 × 137 == 410041
41 × 61 × 101 == 252601
43 × 631 × 13567 == 368113411
43 × 271 × 5827 == 67902031
43 × 127 × 2731 == 14913991
43 × 127 × 1093 == 5968873
43 × 211 × 757 == 6868261
43 × 631 × 1597 == 43331401
43 × 127 × 211 == 1152271
43 × 211 × 337 == 3057601
43 × 433 × 643 == 11972017
43 × 547 × 673 == 15829633
43 × 3361 × 3907 == 564651361
47 × 3359 × 6073 == 958762729
47 × 1151 × 1933 == 104569501
47 × 3727 × 5153 == 902645857
53 × 157 × 2081 == 17316001
53 × 79 × 599 == 2508013
53 × 157 × 521 == 4335241
59 × 1451 × 2089 == 178837201
61 × 421 × 12841 == 329769721
61 × 181 × 5521 == 60957361
61 × 1301 × 19841 == 1574601601
61 × 277 × 2113 == 35703361
61 × 181 × 1381 == 15247621
61 × 541 × 3001 == 99036001
61 × 661 × 2521 == 101649241
61 × 271 × 571 == 9439201
61 × 241 × 421 == 6189121
61 × 3361 × 4021 == 824389441
67 × 2311 × 51613 == 7991602081
67 × 331 × 7393 == 163954561
67 × 331 × 463 == 10267951

PicoLisp

<lang PicoLisp>(de modulo (X Y)

  (% (+ Y (% X Y)) Y) )

(de prime? (N)

  (let D 0
     (or
        (= N 2)
        (and
           (> N 1)
           (bit? 1 N)
           (for (D 3  T  (+ D 2))
              (T (> D (sqrt N)) T)
              (T (=0 (% N D)) NIL) ) ) ) ) )

(for P1 61

  (when (prime? P1)
     (for (H3 2 (> P1 H3) (inc H3))
        (let G (+ H3 P1)
           (for (D 1 (> G D) (inc D))
              (when
                 (and
                    (=0
                       (% (* G (dec P1)) D) )
                    (=
                       (modulo (* (- P1) P1) H3)
                       (% D H3)) )
                 (let
                    (P2
                       (inc
                          (/ (* (dec P1) G) D) )
                       P3 (inc (/ (* P1 P2) H3)) )
                    (when
                       (and
                          (prime? P2)
                          (prime? P3)
                          (= 1 (modulo (* P2 P3) (dec P1))) )
                       (print (list P1 P2 P3)) ) ) ) ) ) ) ) )

(prinl)

(bye)</lang>

PL/I

<lang PL/I>Carmichael: procedure options (main, reorder); /* 24 January 2014 */

  declare (Prime1, Prime2, Prime3, h3, d) fixed binary (31);

  put ('Carmichael numbers are:');

  do Prime1 = 1 to 61;

     do h3 = 2 to Prime1;

d_loop: do d = 1 to h3+Prime1-1;

           if (mod((h3+Prime1)*(Prime1-1), d) = 0) &
              (mod(-Prime1*Prime1, h3) = mod(d, h3)) then
              do;
                 Prime2 = (Prime1-1) * (h3+Prime1)/d; Prime2 = Prime2 + 1;
                 if ^is_prime(Prime2) then iterate d_loop;
                 Prime3 = Prime1*Prime2/h3; Prime3 = Prime3 + 1;
                 if ^is_prime(Prime3) then iterate d_loop;
                 if mod(Prime2*Prime3, Prime1-1) ^= 1 then iterate d_loop;
                 put skip edit (trim(Prime1), ' x ', trim(Prime2), ' x ', trim(Prime3)) (A);
              end;
        end;
     end;
  end;

  /* Uses is_prime from Rosetta Code PL/I. */

end Carmichael;</lang> Results:

Carmichael numbers are: 
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
9 x 89 x 401
9 x 29 x 53
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
21 x 761 x 941
23 x 199 x 353
27 x 131 x 443
27 x 53 x 131
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
35 x 647 x 7549
35 x 443 x 3877
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
49 x 313 x 5113
49 x 97 x 433
51 x 701 x 7151
53 x 157 x 2081
53 x 79 x 599
53 x 157 x 521
55 x 3079 x 84673
55 x 163 x 4483
55 x 1567 x 28729
55 x 109 x 1999
55 x 433 x 2647
55 x 919 x 3889
55 x 139 x 547
55 x 3889 x 12583
55 x 109 x 163
55 x 433 x 487
57 x 113 x 1289
57 x 113 x 281
57 x 4649 x 10193
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

Python

<lang python>class Isprime():

   
   Extensible sieve of Eratosthenes
   
   >>> isprime.check(11)
   True
   >>> isprime.multiples
   {2, 4, 6, 8, 9, 10}
   >>> isprime.primes
   [2, 3, 5, 7, 11]
   >>> isprime(13)
   True
   >>> isprime.multiples
   {2, 4, 6, 7, 8, 9, 10, 11, 12, 14, 15, 16, 18, 20, 21, 22}
   >>> isprime.primes
   [2, 3, 5, 7, 11, 13, 17, 19]
   >>> isprime.nmax
   22
   >>> 
   
   multiples = {2}
   primes = [2]
   nmax = 2
   
   def __init__(self, nmax):
       if nmax > self.nmax:
           self.check(nmax)

   def check(self, n):
       if type(n) == float:
           if not n.is_integer(): return False
           n = int(n)
       multiples = self.multiples
       if n <= self.nmax:
           return n not in multiples
       else:
           # Extend the sieve
           primes, nmax = self.primes, self.nmax
           newmax = max(nmax*2, n)
           for p in primes:
               multiples.update(range(p*((nmax + p + 1) // p), newmax+1, p))
           for i in range(nmax+1, newmax+1):
               if i not in multiples:
                   primes.append(i)
                   multiples.update(range(i*2, newmax+1, i))
           self.nmax = newmax
           return n not in multiples

   __call__ = check

def carmichael(p1):

   ans = []
   if isprime(p1):
       for h3 in range(2, p1):
           g = h3 + p1
           for d in range(1, g):
               if (g * (p1 - 1)) % d == 0 and (-p1 * p1) % h3 == d % h3:
                   p2 = 1 + ((p1 - 1)* g // d)
                   if isprime(p2):
                       p3 = 1 + (p1 * p2 // h3)
                       if isprime(p3):
                           if (p2 * p3) % (p1 - 1) == 1:
                               #print('%i X %i X %i' % (p1, p2, p3))
                               ans += [tuple(sorted((p1, p2, p3)))]
   return ans

isprime = Isprime(2)

ans = sorted(sum((carmichael(n) for n in range(62) if isprime(n)), [])) print(',\n'.join(repr(ans[i:i+5])[1:-1] for i in range(0, len(ans)+1, 5)))</lang>

Output:

(3, 11, 17), (5, 13, 17), (5, 17, 29), (5, 29, 73), (7, 13, 19),
(7, 13, 31), (7, 19, 67), (7, 23, 41), (7, 31, 73), (7, 73, 103),
(13, 37, 61), (13, 37, 97), (13, 37, 241), (13, 61, 397), (13, 97, 421),
(17, 41, 233), (17, 353, 1201), (19, 43, 409), (19, 199, 271), (23, 199, 353),
(29, 113, 1093), (29, 197, 953), (31, 61, 211), (31, 61, 271), (31, 61, 631),
(31, 151, 1171), (31, 181, 331), (31, 271, 601), (31, 991, 15361), (37, 73, 109),
(37, 73, 181), (37, 73, 541), (37, 109, 2017), (37, 613, 1621), (41, 61, 101),
(41, 73, 137), (41, 101, 461), (41, 241, 521), (41, 241, 761), (41, 881, 12041),
(41, 1721, 35281), (43, 127, 211), (43, 127, 1093), (43, 127, 2731), (43, 211, 337),
(43, 211, 757), (43, 271, 5827), (43, 433, 643), (43, 547, 673), (43, 631, 1597),
(43, 631, 13567), (43, 3361, 3907), (47, 1151, 1933), (47, 3359, 6073), (47, 3727, 5153),
(53, 79, 599), (53, 157, 521), (53, 157, 2081), (59, 1451, 2089), (61, 181, 1381),
(61, 181, 5521), (61, 241, 421), (61, 271, 571), (61, 277, 2113), (61, 421, 12841),
(61, 541, 3001), (61, 661, 2521), (61, 1301, 19841), (61, 3361, 4021)

Racket

lang racket

(require math)

(for ([p1 (in-range 3 62)] #:when (prime? p1))

 (for ([h3 (in-range 2 p1)])
   (define g (+ p1 h3))
   (let next ([d 1])
     (when (< d g)
       (when (and (zero? (modulo (* g (- p1 1)) d))
                  (= (modulo (- (sqr p1)) h3) (modulo d h3)))
         (define p2 (+ 1 (quotient (* g (- p1 1)) d)))
         (when (prime? p2)
           (define p3 (+ 1 (quotient (* p1 p2) h3)))
           (when (and (prime? p3) (= 1 (modulo (* p2 p3) (- p1 1))))
             (displayln (list p1 p2 p3 '=> (* p1 p2 p3))))))
       (next (+ d 1))))))

</lang> Output: <lang racket> (3 11 17 => 561) (5 29 73 => 10585) (5 17 29 => 2465) (5 13 17 => 1105) (7 19 67 => 8911) (7 31 73 => 15841) (7 23 41 => 6601) (7 73 103 => 52633) (13 61 397 => 314821) (13 97 421 => 530881) (13 37 97 => 46657) (13 37 61 => 29341) (17 41 233 => 162401) (17 353 1201 => 7207201) (19 43 409 => 334153) (19 199 271 => 1024651) (23 199 353 => 1615681) (29 113 1093 => 3581761) (29 197 953 => 5444489) (31 991 15361 => 471905281) (31 61 631 => 1193221) (31 151 1171 => 5481451) (31 61 271 => 512461) (31 61 211 => 399001) (31 271 601 => 5049001) (31 181 331 => 1857241) (37 109 2017 => 8134561) (37 73 541 => 1461241) (37 613 1621 => 36765901) (37 73 181 => 488881) (37 73 109 => 294409) (41 1721 35281 => 2489462641) (41 881 12041 => 434932961) (41 101 461 => 1909001) (41 241 761 => 7519441) (41 241 521 => 5148001) (41 73 137 => 410041) (41 61 101 => 252601) (43 631 13567 => 368113411) (43 127 1093 => 5968873) (43 211 757 => 6868261) (43 631 1597 => 43331401) (43 127 211 => 1152271) (43 211 337 => 3057601) (43 433 643 => 11972017) (43 547 673 => 15829633) (43 3361 3907 => 564651361) (47 3359 6073 => 958762729) (47 1151 1933 => 104569501) (47 3727 5153 => 902645857) (53 157 2081 => 17316001) (53 79 599 => 2508013) (53 157 521 => 4335241) (59 1451 2089 => 178837201) (61 421 12841 => 329769721) (61 1301 19841 => 1574601601) (61 277 2113 => 35703361) (61 541 3001 => 99036001) (61 661 2521 => 101649241) (61 271 571 => 9439201) (61 241 421 => 6189121) (61 3361 4021 => 824389441) </lang>

REXX

slightly optimized

Note that REXX's version of modulus (//) is really a remainder function.

The Carmichael numbers are shown in numerical order.

Some code optimization was done, while not necessary for the small default number (61), it was significant for larger numbers. <lang rexx>/*REXX program calculates Carmichael 3─strong pseudoprimes (up to and including N). */ numeric digits 18 /*handle big dig #s (9 is the default).*/ parse arg N .; if N== then N=61 /*allow user to specify for the search.*/ tell= N>0; N=abs(N) /*N>0? Then display Carmichael numbers*/

=0 /*number of Carmichael numbers so far. */

@.=0; @.2=1; @.3=1; @.5=1; @.7=1; @.11=1; @.13=1; @.17=1; @.19=1; @.23=1; @.29=1; @.31=1

                                                /*[↑]  prime number memoization array. */
   do p=3  to N  by 2;  pm=p-1;   bot=0;  top=0 /*step through some (odd) prime numbers*/
   if \isPrime(p)  then iterate;  nps=-p*p      /*is   P   a prime?   No, then skip it.*/
   !.=0                                         /*the list of Carmichael #'s  (so far).*/
            do h3=2  to  pm;  g=h3+p            /*find Carmichael #s  for this prime.  */
            gPM=g*pm;  npsH3=((nps//h3)+h3)//h3 /*define a couple of shortcuts for pgm.*/
                                                /* [↓] perform some weeding of D values*/
                do d=1  for g-1;                   if gPM//d \== 0      then iterate
                                                   if npsH3  \== d//h3  then iterate
                                      q=1+gPM%d;   if \isPrime(q)       then iterate
                                      r=1+p*q%h3;  if q*r//pm\==1       then iterate
                                                   if \isPrime(r)       then iterate
                #=#+1;                !.q=r     /*bump Carmichael counter; add to array*/
                if bot==0  then bot=q;   bot=min(bot,q);    top=max(top,q)
                end   /*d*/
            end       /*h3*/
   $=                                           /*display a list of some Carmichael #s.*/
            do j=bot  to top  by 2  while tell;   if !.j\==0  then $=$  p"∙"j'∙'!.j
            end           /*j*/

   if $\==  then say 'Carmichael number: '      strip($)
   end                /*p*/

say say '──────── ' # " Carmichael numbers found." exit /*stick a fork in it, we're all done. */ /*──────────────────────────────────────────────────────────────────────────────────────*/ isPrime: parse arg x; if @.x then return 1 /*X a known prime?*/

        if x<37  then return 0;  if x//2==0  then return 0; if x// 3==0     then return 0
        parse var x  -1 _;     if _==5     then return 0; if x// 7==0     then return 0
                               do k=11  by 6  until k*k>x;  if x// k   ==0  then return 0
                                                            if x//(k+2)==0  then return 0
                               end  /*i*/
        @.x=1;   return 1</lang>

output when using the default input:

Carmichael number:  3∙11∙17
Carmichael number:  5∙13∙17 5∙17∙29 5∙29∙73
Carmichael number:  7∙13∙19 7∙19∙67 7∙23∙41 7∙31∙73 7∙73∙103
Carmichael number:  13∙37∙61 13∙61∙397 13∙97∙421
Carmichael number:  17∙41∙233 17∙353∙1201
Carmichael number:  19∙43∙409 19∙199∙271
Carmichael number:  23∙199∙353
Carmichael number:  29∙113∙1093 29∙197∙953
Carmichael number:  31∙61∙211 31∙151∙1171 31∙181∙331 31∙271∙601 31∙991∙15361
Carmichael number:  37∙73∙109 37∙109∙2017 37∙613∙1621
Carmichael number:  41∙61∙101 41∙73∙137 41∙101∙461 41∙241∙521 41∙881∙12041 41∙1721∙35281
Carmichael number:  43∙127∙211 43∙211∙337 43∙271∙5827 43∙433∙643 43∙547∙673 43∙631∙1597 43∙3361∙3907
Carmichael number:  47∙1151∙1933 47∙3359∙6073 47∙3727∙5153
Carmichael number:  53∙79∙599 53∙157∙521
Carmichael number:  59∙1451∙2089
Carmichael number:  61∙181∙1381 61∙241∙421 61∙271∙571 61∙277∙2113 61∙421∙12841 61∙541∙3001 61∙661∙2521 61∙1301∙19841 61∙3361∙4021

────────  69  Carmichael numbers found.

output when using the input of: -1000

────────  1038  Carmichael numbers found.

output when using the input of: -10000

────────  8716  Carmichael numbers found.

more optimized

This REXX version (pre-)generates a number of primes to assist the isPrime function. <lang rexx>/*REXX program calculates Carmichael 3─strong pseudoprimes (up to and including N). */ numeric digits 18 /*handle big dig #s (9 is the default).*/ parse arg N .; if N== then N=61 /*allow user to specify for the search.*/ tell= N>0; N=abs(N) /*N>0? Then display Carmichael numbers*/

=0; @.=0 /*number of Carmichael numbers so far. */

@.2=1; @.3=1; @.5=1; @.7=1; @.11=1; @.13=1; @.17=1; @.19=1; @.23=1; @.29=1; @.31=1; @.37=1 HP=37; do i=HP+2 by 2 for N*20; if isPrime(i) then do; @.i=1; HP=i; end; end /*i*/ HP=HP+2

                                                /*[↑]  prime number memoization array. */
   do p=3  to N  by 2;  pm=p-1;   bot=0;  top=0 /*step through some (odd) prime numbers*/
   if \isPrime(p)  then iterate;  nps=-p*p      /*is   P   a prime?   No, then skip it.*/
   !.=0                                         /*the list of Carmichael #'s  (so far).*/
            do h3=2  to  pm;  g=h3+p            /*find Carmichael #s  for this prime.  */
            gPM=g*pm;  npsH3=((nps//h3)+h3)//h3 /*define a couple of shortcuts for pgm.*/
                                                /* [↓] perform some weeding of D values*/
                do d=1  for g-1;                   if gPM//d \== 0      then iterate
                                                   if npsH3  \== d//h3  then iterate
                                      q=1+gPM%d;   if \isPrime(q)       then iterate
                                      r=1+p*q%h3;  if q*r//pm\==1       then iterate
                                                   if \isPrime(r)       then iterate
                #=#+1;                !.q=r     /*bump Carmichael counter; add to array*/
                if bot==0  then bot=q;   bot=min(bot,q);    top=max(top,q)
                end   /*d*/
            end       /*h3*/
   $=                                           /*display a list of some Carmichael #s.*/
            do j=bot  to top  by 2  while tell;   if !.j\==0  then $=$  p"∙"j'∙'!.j
            end           /*j*/

   if $\==  then say 'Carmichael number: '      strip($)
   end                /*p*/

say say '──────── ' # " Carmichael numbers found." exit /*stick a fork in it, we're all done. */ /*──────────────────────────────────────────────────────────────────────────────────────*/ isPrime: parse arg x; if @.x then return 1 /*X a known prime?*/

        if x<HP  then return 0;  if x//2==0  then return 0; if x// 3==0     then return 0
        parse var x  -1 _;     if _==5     then return 0; if x// 7==0     then return 0
        if x//11==0      then return 0;                     if x//13==0     then return 0
        if x//17==0      then return 0;                     if x//19==0     then return 0
        if x//23==0      then return 0;                     if x//29==0     then return 0
        if x//31==0      then return 0;                     if x//37==0     then return 0

c=x /* ___*/ b=1; do while b<=x; b=b*4; end /*these two lines compute integer √ X */ s=0; do while b>1; b=b%4; _=c-s-b; s=s%2; if _>=0 then do; c=_; s=s+b; end; end

                               do k=41  by 6  to s;               parse var  k    -1  _
                               if _\==5  then               if x// k   ==0  then return 0
                               if _\==3  then               if x//(k+2)==0  then return 0
                               end  /*k*/       /*K   will never be divisible by three.*/
        @.x=1;   return 1                       /*Define a new prime (X).  Indicate so.*/</lang>

output when the following us used for input: 1000

Carmichael number:  3∙11∙17
Carmichael number:  5∙13∙17 5∙17∙29 5∙29∙73
Carmichael number:  7∙13∙19 7∙19∙67 7∙23∙41 7∙31∙73 7∙73∙103
Carmichael number:  13∙37∙61 13∙61∙397 13∙97∙421
Carmichael number:  17∙41∙233 17∙353∙1201
Carmichael number:  19∙43∙409 19∙199∙271
Carmichael number:  23∙199∙353
Carmichael number:  29∙113∙1093 29∙197∙953
Carmichael number:  31∙61∙211 31∙151∙1171 31∙181∙331 31∙271∙601 31∙991∙15361
Carmichael number:  37∙73∙109 37∙109∙2017 37∙613∙1621
Carmichael number:  41∙61∙101 41∙73∙137 41∙101∙461 41∙241∙521 41∙881∙12041 41∙1721∙35281
Carmichael number:  43∙127∙211 43∙211∙337 43∙271∙5827 43∙433∙643 43∙547∙673 43∙631∙1597 43∙3361∙3907
Carmichael number:  47∙1151∙1933 47∙3359∙6073 47∙3727∙5153
Carmichael number:  53∙79∙599 53∙157∙521
Carmichael number:  59∙1451∙2089
Carmichael number:  61∙181∙1381 61∙241∙421 61∙271∙571 61∙277∙2113 61∙421∙12841 61∙541∙3001 61∙661∙2521 61∙1301∙19841 61∙3361∙4021
Carmichael number:  67∙331∙463 67∙2311∙51613
Carmichael number:  71∙271∙521 71∙421∙491 71∙631∙701 71∙701∙5531 71∙911∙9241
Carmichael number:  73∙157∙2293 73∙379∙523 73∙601∙21937 73∙937∙13681
Carmichael number:  79∙2237∙25247
Carmichael number:  83∙2953∙4019 83∙6971∙289297
Carmichael number:  89∙353∙617 89∙617∙3433 89∙881∙7129 89∙4049∙120121
Carmichael number:  97∙193∙1249 97∙673∙769 97∙769∙10657 97∙1201∙38833 97∙2113∙5857
Carmichael number:  101∙151∙251 101∙1301∙43801
Carmichael number:  103∙647∙1361 103∙1123∙6427 103∙3571∙183907 103∙3877∙4591 103∙5407∙185641
Carmichael number:  107∙743∙1061 107∙3181∙26183
Carmichael number:  109∙163∙379 109∙229∙4993 109∙241∙2389 109∙379∙919 109∙433∙2053 109∙541∙2269 109∙1297∙12853 109∙1657∙6229 109∙1801∙4789
Carmichael number:  113∙337∙449 113∙827∙18691 113∙6833∙85793 113∙8737∙22961
Carmichael number:  127∙631∙26713 127∙659∙1373 127∙991∙3313 127∙2143∙30241 127∙16633∙422479
Carmichael number:  131∙491∙4021 131∙521∙2731 131∙571∙1871 131∙911∙2341 131∙1171∙38351 131∙1301∙8971 131∙4421∙115831 131∙5851∙191621 131∙17291∙1132561
Carmichael number:  137∙409∙14009 137∙953∙5441 137∙3673∙20129 137∙5441∙11833
Carmichael number:  139∙691∙829 139∙3359∙66701 139∙4003∙92737 139∙4969∙8971 139∙17389∙21391
Carmichael number:  149∙593∙29453 149∙1481∙3109
Carmichael number:  151∙211∙541 151∙331∙3571 151∙601∙751 151∙701∙1451 151∙751∙8101 151∙2551∙192601 151∙4951∙53401
Carmichael number:  157∙313∙1093 157∙937∙6397 157∙1093∙15601 157∙2017∙28789 157∙4993∙11701 157∙26053∙409033
Carmichael number:  163∙379∙2377 163∙487∙811 163∙811∙1297 163∙1297∙42283 163∙1621∙37747
Carmichael number:  167∙4483∙34031 167∙29383∙490697
Carmichael number:  173∙1291∙31907 173∙10321∙255077 173∙36809∙155317
Carmichael number:  179∙1069∙4451 179∙3739∙7121 179∙9257∙57139 179∙10859∙485941
Carmichael number:  181∙271∙9811 181∙541∙3061 181∙631∙811 181∙733∙66337 181∙1693∙43777 181∙2953∙3637 181∙3061∙9721 181∙5881∙9421 181∙11161∙15661 181∙16561∙999181
Carmichael number:  191∙421∙431 191∙571∙15581 191∙1901∙7411 191∙3877∙56963 191∙12541∙342191
Carmichael number:  193∙257∙1601 193∙401∙11057 193∙577∙5569 193∙1249∙2593 193∙2689∙30529 193∙7681∙211777 193∙61057∙94273
Carmichael number:  199∙397∙4159 199∙859∙2311 199∙937∙20719 199∙991∙32869 199∙4159∙8713 199∙8713∙82567
Carmichael number:  211∙281∙491 211∙421∙631 211∙631∙66571 211∙1051∙9241 211∙1741∙4651 211∙1831∙4111 211∙2311∙54181 211∙2521∙3571 211∙3221∙35771 211∙3571∙9661 211∙4019∙11159 211∙4201∙98491 211∙32341∙70351 211∙68041∙127051
Carmichael number:  223∙1777∙23311 223∙2221∙5107 223∙5107∙37963 223∙14653∙79699 223∙25087∙1864801
Carmichael number:  227∙1583∙6781 227∙2713∙5651 227∙7459∙423299 227∙21019∙91757
Carmichael number:  229∙457∙2053 229∙571∙4219 229∙1597∙182857 229∙2243∙73379 229∙7069∙32377
Carmichael number:  233∙5569∙185369
Carmichael number:  239∙409∙3911 239∙1429∙1667 239∙1667∙6427 239∙32369∙234431
Carmichael number:  241∙337∙4513 241∙433∙52177 241∙1201∙2161 241∙1361∙4001 241∙2161∙3361 241∙3361∙32401 241∙5521∙110881 241∙6481∙780961 241∙7321∙588121 241∙84961∙181201
Carmichael number:  251∙751∙3251 251∙2251∙56501 251∙3251∙4001 251∙4751∙22501 251∙16001∙803251 251∙22501∙297251 251∙31751∙2656501
Carmichael number:  257∙641∙1153 257∙769∙49409 257∙67073∙3447553 257∙78593∙403969
Carmichael number:  263∙787∙2621 263∙1049∙3407 263∙6551∙12577 263∙71527∙1881161
Carmichael number:  269∙1877∙126229 269∙4289∙384581 269∙10453∙65393 269∙15277∙21977
Carmichael number:  271∙541∙811 271∙811∙2971 271∙1171∙7741 271∙1801∙16831 271∙2161∙65071 271∙4591∙4861 271∙4861∙77491 271∙8191∙1109881 271∙8641∙47791 271∙9631∙52201 271∙10531∙1426951 271∙14797∙1336663
Carmichael number:  277∙829∙7177 277∙1381∙1933 277∙3313∙9661 277∙6073∙31741 277∙18493∙88321 277∙19597∙36433
Carmichael number:  281∙421∙701 281∙617∙673 281∙1009∙4649 281∙1321∙23201 281∙4201∙9521 281∙7121∙26681 281∙9521∙13721 281∙25621∙84701
Carmichael number:  283∙4231∙598687 283∙17203∙58657
Carmichael number:  293∙877∙4673 293∙1607∙31391 293∙3943∙5987
Carmichael number:  307∙613∙919 307∙919∙141067 307∙1531∙3673 307∙2143∙13159 307∙3673∙225523 307∙6427∙246637 307∙17443∙153001 307∙18973∙1941571
Carmichael number:  311∙1117∙26723 311∙1303∙2357 311∙2791∙21701 311∙3659∙7069 311∙23251∙33791 311∙26041∙323951 311∙28211∙165541 311∙44641∙52391
Carmichael number:  313∙521∙23297 313∙937∙58657 313∙1093∙6709 313∙1249∙55849 313∙3433∙38377 313∙3793∙395737 313∙5449∙12097 313∙6577∙8761 313∙7177∙70201 313∙9049∙472057 313∙12637∙359581 313∙49297∙5143321 313∙51481∙947857 313∙66457∙184081 313∙129169∙400297
Carmichael number:  317∙18013∙41081 317∙104281∙2542853
Carmichael number:  331∙661∙991 331∙947∙24113 331∙991∙4621 331∙1321∙17491 331∙2311∙12541 331∙2971∙49171 331∙3331∙551281 331∙4051∙18121 331∙4621∙14851 331∙37181∙1758131 331∙37951∙897271 331∙41141∙316691
Carmichael number:  337∙421∙47293 337∙449∙21617 337∙673∙1009 337∙953∙8681 337∙1009∙3697 337∙1597∙12517 337∙2473∙11113 337∙3697∙12097 337∙16369∙1379089 337∙19489∙597073 337∙35393∙40433 337∙58129∙2176609
Carmichael number:  347∙3461∙92383 347∙4153∙29411
Carmichael number:  349∙929∙7541 349∙1741∙2089 349∙3191∙12239 349∙4177∙20533 349∙122149∙21315001
Carmichael number:  353∙617∙19801 353∙1153∙5153 353∙1321∙66617 353∙13217∙77761 353∙130241∙2704417
Carmichael number:  359∙43319∙3887881 359∙46183∙592133
Carmichael number:  367∙733∙1831 367∙1831∙9883 367∙5003∙42701 367∙9151∙419803 367∙24889∙51607 367∙28183∙574621
Carmichael number:  373∙1117∙1861 373∙1613∙150413 373∙5581∙1040857 373∙16741∙81097 373∙139501∙26016937
Carmichael number:  379∙631∙9199 379∙757∙4159 379∙2269∙24571 379∙2539∙21871 379∙6427∙202987 379∙9829∙17011 379∙10639∙268813
Carmichael number:  383∙33617∙40111 383∙38201∙860647 383∙74873∙3186263
Carmichael number:  389∙3299∙6791
Carmichael number:  397∙1783∙4951 397∙2971∙51283 397∙4357∙8317 397∙30097∙56629 397∙55837∙852589 397∙79201∙10480933 397∙99793∙370261
Carmichael number:  401∙641∙2161 401∙1201∙1601 401∙2161∙216641 401∙2801∙9601 401∙9521∙19681 401∙9601∙70001 401∙15601∙18401 401∙18401∙567601 401∙161201∙32320801
Carmichael number:  409∙2857∙6529 409∙6121∙96289 409∙6529∙22441 409∙7039∙575791 409∙35089∙683401 409∙36721∙114649
Carmichael number:  419∙15467∙47653 419∙22573∙78167 419∙47653∙539639
Carmichael number:  421∙631∙11551 421∙701∙2381 421∙3851∙85331 421∙7561∙289381 421∙9661∙15121 421∙13441∙209581 421∙18481∙39901 421∙20231∙54251 421∙35533∙7479697 421∙42589∙208489 421∙89041∙12495421
Carmichael number:  431∙1721∙29671 431∙1979∙142159 431∙8171∙55901 431∙13331∙168991
Carmichael number:  433∙937∙11593 433∙1297∙2161 433∙2161∙16417 433∙2593∙48817 433∙2953∙21673 433∙3457∙6481 433∙3697∙55201 433∙6481∙87697
Carmichael number:  439∙3067∙673207 439∙3943∙45553 439∙9199∙2019181 439∙10513∙17959 439∙64679∙7098521 439∙96799∙14164921
Carmichael number:  443∙1327∙4421 443∙2029∙4967 443∙74257∙143651 443∙102103∙2380613 443∙167077∙236471 443∙251057∙889747
Carmichael number:  449∙2689∙3137 449∙50849∙4566241 449∙145601∙325249 449∙202049∙45360001
Carmichael number:  457∙3877∙93253 457∙5701∙8893 457∙7297∙32377 457∙15733∙19381 457∙21433∙163249 457∙28729∙55633 457∙71593∙2337001 457∙73721∙1203233 457∙114001∙1211593
Carmichael number:  461∙691∙1151 461∙1013∙38917 461∙1381∙159161 461∙3541∙23321 461∙5981∙24841 461∙26681∙4099981
Carmichael number:  463∙2927∙15401 463∙6007∙39733 463∙214831∙49733377 463∙218527∙10117801
Carmichael number:  467∙141199∙474389
Carmichael number:  479∙57839∙219881
Carmichael number:  487∙1459∙8263 487∙1531∙2683 487∙1621∙1783 487∙1783∙108541 487∙8263∙9721 487∙12637∙32563 487∙17011∙2761453 487∙26731∙110323 487∙51517∙69499
Carmichael number:  491∙1471∙10781 491∙6959∙569479 491∙16661∙154351 491∙41651∙46061 491∙122501∙6683111 491∙386611∙637001
Carmichael number:  499∙997∙4483 499∙10459∙39841
Carmichael number:  503∙5021∙21587
Carmichael number:  509∙3557∙41149 509∙7621∙23369 509∙11939∙110491 509∙86869∙11054081
Carmichael number:  521∙1301∙8581 521∙21841∙41081
Carmichael number:  523∙1567∙163909 523∙6091∙1592797 523∙9397∙140419 523∙15661∙481807 523∙38629∙69427 523∙155557∙1114471 523∙193663∙462493
Carmichael number:  541∙811∙3511 541∙1621∙7561 541∙6661∙257401 541∙7561∙54541 541∙12421∙197641 541∙16561∙814501
Carmichael number:  547∙1093∙2731 547∙2731∙6553 547∙6553∙35491 547∙7333∙235951 547∙26209∙186187 547∙52963∙827737 547∙158341∙2624623
Carmichael number:  557∙1669∙42257 557∙38921∙7226333
Carmichael number:  563∙28663∙329333
Carmichael number:  569∙2273∙117577 569∙13633∙1108169 569∙17609∙25561 569∙21017∙37489 569∙22153∙787817
Carmichael number:  571∙661∙16411 571∙2281∙2851 571∙2851∙13681 571∙6841∙43891 571∙13681∙1562371 571∙65323∙18649717
Carmichael number:  577∙757∙39709 577∙1153∙6337 577∙5569∙100417 577∙6337∙26497 577∙20161∙646273 577∙32833∙37441 577∙53857∙181729 577∙79777∙86689 577∙339841∙15083713 577∙559297∙819073
Carmichael number:  587∙5861∙33403 587∙9377∙54499 587∙12893∙36919 587∙49811∙3654883
Carmichael number:  593∙21017∙31081 593∙35521∙3009137 593∙176417∙34871761
Carmichael number:  599∙2393∙84319 599∙120199∙17999801 599∙179999∙35939801 599∙266111∙547769 599∙368369∙12979591
Carmichael number:  601∙1201∙1801 601∙1801∙541201 601∙3001∙200401 601∙3121∙38281 601∙3301∙5101 601∙4201∙4801 601∙4801∙412201 601∙5101∙278701 601∙6151∙7951 601∙9001∙386401 601∙19801∙28201 601∙52201∙3921601 601∙99901∙923701
Carmichael number:  607∙1213∙9091 607∙4243∙1287751 607∙21817∙322999 607∙24847∙1885267 607∙61813∙7504099 607∙186649∙12588439 607∙370873∙45023983 607∙373903∙22695913
Carmichael number:  613∙919∙2143 613∙1021∙312937 613∙1327∙73951 613∙1429∙23053 613∙2857∙17341 613∙7549∙87313 613∙9181∙2813977 613∙12241∙111997 613∙51817∙213181 613∙246637∙783361 613∙364753∙386173
Carmichael number:  617∙661∙1013 617∙8009∙705937 617∙16633∙120737 617∙29569∙2606297 617∙59753∙81929 617∙73613∙133981 617∙129361∙6139673 617∙137369∙1629937 617∙383153∙47281081
Carmichael number:  619∙1237∙4327 619∙2267∙26987 619∙5563∙1721749 619∙28429∙703903 619∙53149∙56239 619∙92083∙452377 619∙398611∙9490009
Carmichael number:  631∙1471∙46411 631∙5881∙90511 631∙26209∙82279 631∙32831∙67481
Carmichael number:  641∙4481∙7681 641∙12161∙26881 641∙17921∙370561 641∙19841∙176641
Carmichael number:  643∙107857∙2391451
Carmichael number:  647∙4523∙19381 647∙64601∙75583 647∙188633∙532951 647∙444449∙7013623
Carmichael number:  653∙13367∙2909551 653∙176041∙732197
Carmichael number:  659∙2633∙5923 659∙23689∙624443 659∙27919∙34781 659∙30269∙92779 659∙73039∙6876101 659∙92779∙1329161
Carmichael number:  661∙991∙131011 661∙1321∙4621 661∙2131∙4231 661∙3191∙6491 661∙3301∙12541 661∙4621∙763621 661∙5281∙81181 661∙22111∙1623931 661∙22441∙95701 661∙138821∙152681
Carmichael number:  673∙1009∙14449 673∙2017∙3361 673∙3361∙12097 673∙13441∙1292257 673∙40801∙155137 673∙231841∙9178177
Carmichael number:  677∙2029∙85853 677∙4733∙1602121 677∙6761∙25013 677∙45293∙511057
Carmichael number:  683∙8867∙16369 683∙11161∙206027 683∙15749∙32303 683∙42967∙2934647 683∙94117∙9183131
Carmichael number:  691∙7591∙2622691 691∙16561∙2288731 691∙31051∙71761 691∙34501∙2648911 691∙69691∙3009781 691∙743131∙1330321
Carmichael number:  701∙2801∙10501 701∙3701∙1297201 701∙3851∙899851 701∙6301∙7001 701∙18401∙58901 701∙41651∙2245951 701∙44101∙170801 701∙46901∙319201 701∙52501∙296801 701∙53201∙632101
Carmichael number:  709∙4957∙12037 709∙7789∙16993 709∙9677∙21713 709∙36109∙5120257 709∙210277∙819157
Carmichael number:  719∙97649∙190271
Carmichael number:  727∙1453∙2179 727∙2179∙792067 727∙2663∙193601 727∙3631∙8713 727∙4423∙321553 727∙176903∙32152121 727∙308551∙1823713 727∙651223∙2784937
Carmichael number:  733∙5857∙84181 733∙13177∙47581 733∙18301∙789097 733∙22571∙2363507 733∙25621∙9390097 733∙150427∙1238911 733∙271573∙22118113 733∙631717∙3561913
Carmichael number:  739∙821∙4019 739∙3691∙454609 739∙10333∙2545363 739∙62731∙1783009 739∙152029∙1321759
Carmichael number:  743∙6679∙225569 743∙6997∙9011 743∙596569∙7266407
Carmichael number:  751∙2251∙10501 751∙2851∙237901 751∙21751∙181501 751∙109751∙649001 751∙123001∙1338751 751∙153001∙1767751 751∙191251∙10259251 751∙318751∙2418001
Carmichael number:  757∙2017∙18397 757∙2269∙858817 757∙15121∙3815533 757∙27541∙79273 757∙32257∙2219869 757∙33013∙59221 757∙184843∙633151 757∙627481∙6506893
Carmichael number:  761∙2129∙31769 761∙2281∙3041 761∙3041∙771401 761∙6841∙19001 761∙8969∙1137569 761∙13681∙101081 761∙19001∙1032841 761∙41801∙497041 761∙230281∙1184081 761∙251941∙339341 761∙314641∙497801
Carmichael number:  769∙6529∙9601 769∙41729∙697601
Carmichael number:  773∙22003∙122363 773∙44777∙47093
Carmichael number:  787∙3931∙9433 787∙5503∙45589 787∙106373∙3348623
Carmichael number:  797∙2389∙476009 797∙3583∙16319 797∙5573∙11941 797∙21493∙428249 797∙58109∙7718813 797∙148853∙859681
Carmichael number:  809∙5657∙9697 809∙78781∙176549 809∙82013∙22116173 809∙176549∙2197357 809∙453289∙1171601
Carmichael number:  811∙1621∙438211 811∙4051∙19441 811∙4591∙744661 811∙6481∙17011 811∙19441∙3153331 811∙77761∙1189891 811∙86131∙478441
Carmichael number:  821∙1231∙6971 821∙15581∙42641 821∙137597∙6275953
Carmichael number:  823∙2467∙4111 823∙4111∙23017 823∙4933∙9043 823∙27127∙637873 823∙341953∙31269703
Carmichael number:  827∙2243∙2833 827∙4957∙5783 827∙24781∙476603 827∙101009∙2880499 827∙691363∙57175721
Carmichael number:  829∙1657∙17389 829∙9109∙15733 829∙10949∙2269181 829∙24841∙1872109 829∙140761∙5556709
Carmichael number:  839∙5867∙223747
Carmichael number:  853∙2557∙4261 853∙7669∙594697 853∙12781∙5451097 853∙17041∙309277 853∙19597∙185737
Carmichael number:  857∙6421∙127973 857∙10273∙160073 857∙95873∙115561 857∙796937∙9229393
Carmichael number:  859∙2861∙3719 859∙8581∙9439 859∙9439∙150151 859∙27457∙66067 859∙321751∙1039039
Carmichael number:  863∙24137∙38791 863∙28447∙153437 863∙38791∙62927 863∙56893∙68099
Carmichael number:  877∙1753∙56941 877∙3067∙30223 877∙6133∙8761 877∙24091∙7042603 877∙36793∙6453493 877∙263677∙8894029
Carmichael number:  881∙2861∙840181 881∙22441∙57641 881∙130241∙16391761
Carmichael number:  883∙2647∙44101 883∙8191∙267877 883∙11467∙35281 883∙15877∙824671 883∙16633∙358219 883∙21757∙3842287 883∙30871∙134947 883∙42337∙216091 883∙126127∙161407 883∙260191∙114874327 883∙403957∙10808911 883∙507151∙531847
Carmichael number:  887∙14177∙50503
Carmichael number:  907∙7853∙16007 907∙137713∙24981139
Carmichael number:  911∙2003∙912367 911∙9283∙1208117 911∙9311∙55441 911∙11831∙898171 911∙16381∙28211 911∙30941∙4026751 911∙55511∙12642631 911∙167441∙204751 911∙175631∙2962961 911∙185641∙1551551 911∙227501∙2328691
Carmichael number:  919∙8263∙949213 919∙15607∙170749 919∙60589∙11136259 919∙129439∙569161 919∙156979∙321301 919∙311203∙2918323 919∙877609∙21797911
Carmichael number:  929∙5569∙23201 929∙6961∙35729 929∙42689∙1071841 929∙139201∙307169
Carmichael number:  937∙1873∙70201 937∙6553∙7489 937∙7489∙1002457 937∙21529∙3362113 937∙38377∙5993209 937∙177841∙820873
Carmichael number:  941∙5171∙23971 941∙6581∙8461 941∙8461∙361901 941∙28201∙102461 941∙44651∙4668511 941∙209621∙1133641 941∙322891∙701711 941∙355321∙1732421
Carmichael number:  947∙29327∙1983763 947∙47129∙299539 947∙307451∙10398433
Carmichael number:  953∙2857∙9521 953∙5881∙18257 953∙17137∙69497 953∙52361∙159937 953∙159937∙2771273
Carmichael number:  967∙1289∙25439 967∙1933∙4831 967∙4831∙11593 967∙26083∙5044453 967∙62791∙7589863 967∙88873∙1909783 967∙156493∙30265747
Carmichael number:  971∙3881∙753691 971∙8731∙44621 971∙12611∙3061321 971∙110581∙635351 971∙142591∙2387171 971∙169751∙648931 971∙1324051∙3263081
Carmichael number:  977∙2441∙794953 977∙5857∙12689 977∙6833∙39041 977∙17569∙41969 977∙478241∙155747153
Carmichael number:  983∙3929∙8839 983∙8839∙1241249 983∙970217∙190744663
Carmichael number:  991∙4951∙58411 991∙10111∙501001 991∙16831∙26731 991∙56431∙607861 991∙99991∙5215321 991∙118801∙206911 991∙138403∙336997 991∙167311∙312841 991∙338581∙890011 991∙658351∙1924561
Carmichael number:  997∙1993∙56773 997∙8467∙367027 997∙12451∙4137883 997∙17929∙130477 997∙29383∙450691 997∙167329∙15166093 997∙1002973∙99996409

────────  1038  Carmichael numbers found.

Ruby

Works with: Ruby version 1.9

<lang ruby># Generate Charmichael Numbers

require 'prime'

Prime.each(61) do |p|

 (2...p).each do |h3|
   g = h3 + p
   (1...g).each do |d|
     next if (g*(p-1)) % d != 0 or (-p*p) % h3 != d % h3
     q = 1 + ((p - 1) * g / d)
     next unless q.prime?
     r = 1 + (p * q / h3)
     next unless r.prime? and (q * r) % (p - 1) == 1
     puts "#{p} x #{q} x #{r}" 
   end
 end
 puts

end</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

Rust

<lang rust> fn is_prime(n: i64) -> bool {

   if n > 1 {
       (2..((n / 2) + 1)).all(|x| n % x != 0)
   } else {
       false
   }

}

// The modulo operator actually calculates the remainder. fn modulo(n: i64, m: i64) -> i64 {

   ((n % m) + m) % m

}

fn carmichael(p1: i64) -> Vec<(i64, i64, i64)> {

   let mut results = Vec::new();
   if !is_prime(p1) {
       return results;
   }

   for h3 in 2..p1 {
       for d in 1..(h3 + p1) {
           if (h3 + p1) * (p1 - 1) % d != 0 || modulo(-p1 * p1, h3) != d % h3 {
               continue;
           }

           let p2 = 1 + ((p1 - 1) * (h3 + p1) / d);
           if !is_prime(p2) {
               continue;
           }

           let p3 = 1 + (p1 * p2 / h3);
           if !is_prime(p3) || ((p2 * p3) % (p1 - 1) != 1) {
               continue;
           }

           results.push((p1, p2, p3));
       }
   }

   results

}

fn main() {

   (1..62)
       .filter(|&x| is_prime(x))
       .map(carmichael)
       .filter(|x| !x.is_empty())
       .flat_map(|x| x)
       .inspect(|x| println!("{:?}", x))
       .count(); // Evaluate entire iterator

} </lang>

Output:

(3, 11, 17)
(5, 29, 73)
(5, 17, 29)
(5, 13, 17)
.
.
.
(61, 661, 2521)
(61, 271, 571)
(61, 241, 421)
(61, 3361, 4021)

Seed7

The function isPrime below is borrowed from the Seed7 algorithm collection.

<lang seed7>$ include "seed7_05.s7i";

const func boolean: isPrime (in integer: number) is func

 result
   var boolean: prime is FALSE;
 local
   var integer: upTo is 0;
   var integer: testNum is 3;
 begin
   if number = 2 then
     prime := TRUE;
   elsif odd(number) and number > 2 then
     upTo := sqrt(number);
     while number rem testNum <> 0 and testNum <= upTo do
       testNum +:= 2;
     end while;
     prime := testNum > upTo;
   end if;
 end func;

const proc: main is func

 local
   var integer: p1 is 0;
   var integer: h3 is 0;
   var integer: g is 0;
   var integer: d is 0;
   var integer: p2 is 0;
   var integer: p3 is 0;
 begin
   for p1 range 2 to 61 do
     if isPrime(p1) then
       for h3 range 2 to p1 do
         g := h3 + p1;
         for d range 1 to pred(g) do
           if (g * pred(p1)) mod d = 0 and -p1 ** 2 mod h3 = d mod h3 then
             p2 := 1 + pred(p1) * g div d;
             if isPrime(p2) then
               p3 := 1 + p1 * p2 div h3;
               if isPrime(p3) and (p2 * p3) mod pred(p1) = 1 then
                 writeln(p1 <& " * " <& p2 <& " * " <& p3 <& " = " <& p1*p2*p3);
               end if;
             end if;
           end if;
         end for;
       end for;
     end if;
   end for;
 end func;</lang>

Output:

3 * 11 * 17 = 561
5 * 29 * 73 = 10585
5 * 17 * 29 = 2465
5 * 13 * 17 = 1105
7 * 19 * 67 = 8911
7 * 31 * 73 = 15841
7 * 13 * 31 = 2821
7 * 23 * 41 = 6601
7 * 73 * 103 = 52633
7 * 13 * 19 = 1729
13 * 61 * 397 = 314821
13 * 37 * 241 = 115921
13 * 97 * 421 = 530881
13 * 37 * 97 = 46657
13 * 37 * 61 = 29341
17 * 41 * 233 = 162401
17 * 353 * 1201 = 7207201
19 * 43 * 409 = 334153
19 * 199 * 271 = 1024651
23 * 199 * 353 = 1615681
29 * 113 * 1093 = 3581761
29 * 197 * 953 = 5444489
31 * 991 * 15361 = 471905281
31 * 61 * 631 = 1193221
31 * 151 * 1171 = 5481451
31 * 61 * 271 = 512461
31 * 61 * 211 = 399001
31 * 271 * 601 = 5049001
31 * 181 * 331 = 1857241
37 * 109 * 2017 = 8134561
37 * 73 * 541 = 1461241
37 * 613 * 1621 = 36765901
37 * 73 * 181 = 488881
37 * 73 * 109 = 294409
41 * 1721 * 35281 = 2489462641
41 * 881 * 12041 = 434932961                                                                                                                                                 
41 * 101 * 461 = 1909001                                                                                                                                                     
41 * 241 * 761 = 7519441                                                                                                                                                     
41 * 241 * 521 = 5148001                                                                                                                                                     
41 * 73 * 137 = 410041                                                                                                                                                       
41 * 61 * 101 = 252601                                                                                                                                                       
43 * 631 * 13567 = 368113411                                                                                                                                                 
43 * 271 * 5827 = 67902031                                                                                                                                                   
43 * 127 * 2731 = 14913991                                                                                                                                                   
43 * 127 * 1093 = 5968873                                                                                                                                                    
43 * 211 * 757 = 6868261                                                                                                                                                     
43 * 631 * 1597 = 43331401                                                                                                                                                   
43 * 127 * 211 = 1152271
43 * 211 * 337 = 3057601
43 * 433 * 643 = 11972017
43 * 547 * 673 = 15829633
43 * 3361 * 3907 = 564651361
47 * 3359 * 6073 = 958762729
47 * 1151 * 1933 = 104569501
47 * 3727 * 5153 = 902645857
53 * 157 * 2081 = 17316001
53 * 79 * 599 = 2508013
53 * 157 * 521 = 4335241
59 * 1451 * 2089 = 178837201
61 * 421 * 12841 = 329769721
61 * 181 * 5521 = 60957361
61 * 1301 * 19841 = 1574601601
61 * 277 * 2113 = 35703361
61 * 181 * 1381 = 15247621
61 * 541 * 3001 = 99036001
61 * 661 * 2521 = 101649241
61 * 271 * 571 = 9439201
61 * 241 * 421 = 6189121
61 * 3361 * 4021 = 824389441

Sidef

Translation of: Perl

<lang ruby>func forprimes(a, b, callback) {

   for (a = (a-1 -> next_prime); a <= b; a.next_prime!) {
       callback(a)
   }

}

forprimes(3, 61, func(p) {

  for h3 in (2 ..^ p) {
     var ph3 = (p + h3)
     for d in (1 ..^ ph3) {
        ((-p * p) % h3) != (d % h3) && next
        ((p-1) * ph3) % d && next
        var q = 1+((p-1) * ph3 / d)
        q.is_prime || next
        var r = 1+((p*q - 1)/h3)
        r.is_prime || next
        (q*r) % (p-1) == 1 || next
        printf("%2d x %5d x %5d = %s\n",p,q,r, p*q*r)
     }
  }

})</lang>

Output:

 3 x    11 x    17 = 561
 5 x    29 x    73 = 10585
 5 x    17 x    29 = 2465
 5 x    13 x    17 = 1105
 ... full output is 69 lines ...
61 x   661 x  2521 = 101649241
61 x   271 x   571 = 9439201
61 x   241 x   421 = 6189121
61 x  3361 x  4021 = 824389441

Tcl

Using the primality tester from the Miller-Rabin task... <lang tcl>proc carmichael {limit {rounds 10}} {

   set carmichaels {}
   for {set p1 2} {$p1 <= $limit} {incr p1} {

if {![miller_rabin $p1 $rounds]} continue for {set h3 2} {$h3 < $p1} {incr h3} { set g [expr {$h3 + $p1}] for {set d 1} {$d < $h3+$p1} {incr d} { if {(($h3+$p1)*($p1-1))%$d != 0} continue if {(-($p1**2))%$h3 != $d%$h3} continue

set p2 [expr {1 + ($p1-1)*$g/$d}] if {![miller_rabin $p2 $rounds]} continue

set p3 [expr {1 + $p1*$p2/$h3}] if {![miller_rabin $p3 $rounds]} continue

if {($p2*$p3)%($p1-1) != 1} continue lappend carmichaels $p1 $p2 $p3 [expr {$p1*$p2*$p3}] } }

   }
   return $carmichaels

}</lang> Demonstrating: <lang tcl>set results [carmichael 61 2] puts "[expr {[llength $results]/4}] Carmichael numbers found" foreach {p1 p2 p3 c} $results {

   puts "$p1 x $p2 x $p3 = $c"

}</lang>

Output:

69 Carmichael numbers found
3 x 11 x 17 = 561
5 x 29 x 73 = 10585
5 x 17 x 29 = 2465
5 x 13 x 17 = 1105
7 x 19 x 67 = 8911
7 x 31 x 73 = 15841
7 x 13 x 31 = 2821
7 x 23 x 41 = 6601
7 x 73 x 103 = 52633
7 x 13 x 19 = 1729
13 x 61 x 397 = 314821
13 x 37 x 241 = 115921
13 x 97 x 421 = 530881
13 x 37 x 97 = 46657
13 x 37 x 61 = 29341
17 x 41 x 233 = 162401
17 x 353 x 1201 = 7207201
19 x 43 x 409 = 334153
19 x 199 x 271 = 1024651
23 x 199 x 353 = 1615681
29 x 113 x 1093 = 3581761
29 x 197 x 953 = 5444489
31 x 991 x 15361 = 471905281
31 x 61 x 631 = 1193221
31 x 151 x 1171 = 5481451
31 x 61 x 271 = 512461
31 x 61 x 211 = 399001
31 x 271 x 601 = 5049001
31 x 181 x 331 = 1857241
37 x 109 x 2017 = 8134561
37 x 73 x 541 = 1461241
37 x 613 x 1621 = 36765901
37 x 73 x 181 = 488881
37 x 73 x 109 = 294409
41 x 1721 x 35281 = 2489462641
41 x 881 x 12041 = 434932961
41 x 101 x 461 = 1909001
41 x 241 x 761 = 7519441
41 x 241 x 521 = 5148001
41 x 73 x 137 = 410041
41 x 61 x 101 = 252601
43 x 631 x 13567 = 368113411
43 x 271 x 5827 = 67902031
43 x 127 x 2731 = 14913991
43 x 127 x 1093 = 5968873
43 x 211 x 757 = 6868261
43 x 631 x 1597 = 43331401
43 x 127 x 211 = 1152271
43 x 211 x 337 = 3057601
43 x 433 x 643 = 11972017
43 x 547 x 673 = 15829633
43 x 3361 x 3907 = 564651361
47 x 3359 x 6073 = 958762729
47 x 1151 x 1933 = 104569501
47 x 3727 x 5153 = 902645857
53 x 157 x 2081 = 17316001
53 x 79 x 599 = 2508013
53 x 157 x 521 = 4335241
59 x 1451 x 2089 = 178837201
61 x 421 x 12841 = 329769721
61 x 181 x 5521 = 60957361
61 x 1301 x 19841 = 1574601601
61 x 277 x 2113 = 35703361
61 x 181 x 1381 = 15247621
61 x 541 x 3001 = 99036001
61 x 661 x 2521 = 101649241
61 x 271 x 571 = 9439201
61 x 241 x 421 = 6189121
61 x 3361 x 4021 = 824389441

zkl

Using the Miller-Rabin primality test in lib GMP. <lang zkl>var BN=Import("zklBigNum"), bi=BN(0); // gonna recycle bi primes:=T(2,3,5,7,11,13,17,19,23,29,31,37,41,43,47,53,59,61); var p2,p3; cs:=[[(p1,h3,d); primes; { [2..p1 - 1] }; // list comprehension

     { [1..h3 + p1 - 1] },

{ ((h3 + p1)*(p1 - 1)%d == 0 and ((-p1*p1):mod(_,h3) == d%h3)) },//guard { (p2=1 + (p1 - 1)*(h3 + p1)/d):bi.set(_).probablyPrime() },//guard { (p3=1 + (p1*p2/h3)):bi.set(_).probablyPrime() }, //guard { 1==(p2*p3)%(p1 - 1) }; //guard

  { T(p1,p2,p3) }  // return list of three primes in Carmichael number

]]; fcn mod(a,b) { m:=a%b; if(m<0) m+b else m }</lang> <lang>cs.len().println(" Carmichael numbers found:"); cs.pump(Console.println,fcn([(p1,p2,p3)]){

  "%2d * %4d * %5d = %d".fmt(p1,p2,p3,p1*p2*p3) });</lang>

Output:

69 Carmichael numbers found:
 3 *   11 *    17 = 561
 5 *   29 *    73 = 10585
 5 *   17 *    29 = 2465
 5 *   13 *    17 = 1105
 7 *   19 *    67 = 8911
...
61 *  181 *  1381 = 15247621
61 *  541 *  3001 = 99036001
61 *  661 *  2521 = 101649241
61 *  271 *   571 = 9439201
61 *  241 *   421 = 6189121
61 * 3361 *  4021 = 824389441

ZX Spectrum Basic

Translation of: C

<lang zxbasic>10 FOR p=2 TO 61 20 LET n=p: GO SUB 1000 30 IF NOT n THEN GO TO 200 40 FOR h=1 TO p-1 50 FOR d=1 TO h-1+p 60 IF NOT (FN m((h+p)*(p-1),d)=0 AND FN w(-p*p,h)=FN m(d,h)) THEN GO TO 180 70 LET q=INT (1+((p-1)*(h+p)/d)) 80 LET n=q: GO SUB 1000 90 IF NOT n THEN GO TO 180 100 LET r=INT (1+(p*q/h)) 110 LET n=r: GO SUB 1000 120 IF (NOT n) OR ((FN m((q*r),(p-1))<>1)) THEN GO TO 180 130 PRINT p;" ";q;" ";r 180 NEXT d 190 NEXT h 200 NEXT p 210 STOP 1000 IF n<4 THEN LET n=(n>1): RETURN 1010 IF (NOT FN m(n,2)) OR (NOT FN m(n,3)) THEN LET n=0: RETURN 1020 LET i=5 1030 IF NOT ((i*i)<=n) THEN LET n=1: RETURN 1040 IF (NOT FN m(n,i)) OR NOT FN m(n,(i+2)) THEN LET n=0: RETURN 1050 LET i=i+6 1060 GO TO 1030 2000 DEF FN m(a,b)=a-(INT (a/b)*b): REM Mod function 2010 DEF FN w(a,b)=FN m(FN m(a,b)+b,b): REM Mod function modified </lang>

Carmichael 3 strong pseudoprimes

Contents

Ada

ALGOL 68

C

Clojure

D

EchoLisp

Fortran

Plan

Source

Output

FreeBASIC

Go

Haskell

Icon and Unicon

J

Java

Julia

Kotlin

Mathematica / Wolfram Language

PARI/GP

Perl

Perl 6

PicoLisp

PL/I

Python

Racket

REXX

slightly optimized

more optimized

Ruby

Rust

Seed7

Sidef

Tcl

zkl

ZX Spectrum Basic