Carmichael 3 strong pseudoprimes

Carmichael 3 strong pseudoprimes
You are encouraged to solve this task according to the task description, using any language you may know.

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.

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   ${\displaystyle Prime_{1}}$

```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
```

Uses the Miller_Rabin package from Miller-Rabin primality test#ordinary integers.

`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;`
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```

ALGOL 68

Uses the Sieve of Eratosthenes code from the Smith Numbers task with an increased upper-bound (included here for convenience).

`# 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    FIOD`
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;} `
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

` (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)))) `
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

`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);      }    }  }}`
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

` ;; charmichaël numbers up to N-th prime ; 61 is 18-th prime(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))) `
Output:
` (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);; skipped ....💥    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 `

F#

This task uses Extensible Prime Generator (F#)

` // Carmichael Number . Nigel Galloway: November 19th., 2017let fN n = Seq.collect ((fun g->(Seq.map(fun e->(n,1+(n-1)*(n+g)/e,g,e))){1..(n+g-1)})){2..(n-1)}let fG (P1,P2,h3,d) =  let mod' n g = (n%g+g)%g  let fN P3 = if isPrime P3 && (P2*P3)%(P1-1)=1 then Some (P1,P2,P3) else None  if isPrime P2 && ((h3+P1)*(P1-1))%d=0 && mod' (-P1*P1) h3=d%h3 then fN (1+P1*P2/h3) else Nonelet carms g = primes|>Seq.takeWhile(fun n->n<=g)|>Seq.collect fN|>Seq.choose fGcarms 61 |> Seq.iter (fun (P1,P2,P3)->printfn "%2d x %4d x %5d = %10d" P1 P2 P3 ((uint64 P3)*(uint64 (P1*P2)))) `
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
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
```

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...
`      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`

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

`' version 17-10-2016' compile with: fbc -s console ' using a sieve for finding primes #Define max_sieve 10000000 ' 10^7ReDim 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 h3End Sub  ' ------=< MAIN >=------ Dim As UInteger i, j 'set up sieveFor i = 3 To max_sieve Step 2  isprime(i) = 1Next i isprime(2) = 1For 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 IfNext i For i = 2 To 61  carmichael3(i)Next i ' empty keyboard bufferWhile InKey <> "" : WendPrint : Print "hit any key to end program"SleepEnd`
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

`package main import "fmt" // Use this rather than % for negative integersfunc 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) }    }}`
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
```

Translation of: Ruby
Library: primes
Works with: GHC version 7.4.1
Works with: primes version 0.2.1.0
`#!/usr/bin/runhaskell import Data.Numbers.Primesimport 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`
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.

`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)%dend procedure format(p1,p2,p3)    return left(p1||" * "||p2||" * "||p3,20)||" = "||(p1*p2*p3)end`

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

` q =: (,"0 1~ >:@[email protected]<:@+/"1)&.>@(,&.>"0 1~ >:@i.)&.>@[email protected](1&p:@i.)@>:f1 =: (0: = {. | <:@{: * 1&{ + {:) *. ((1&{ | [email protected]*:@{:) = 1&{ | {.)f2 =: 1: = <:@{. | ({: * 1&{)p2 =: 0:`((* 1&p:)@(<[email protected](1: + <:@{: * {. %~ 1&{ + {:)))@.f1p3 =: 3:\$0:`((* 1&p:)@({: , {. , (<[email protected]>:@(1&{ %~ {. * {:))))@.(*@{.)@(p2 , }.)(-. 3:\$0:)@(((*"0 f2)@p3"1)@;@;@q) 61 `

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
`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);                }            }        }    }}`
```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

`using Primes function carmichael(pmax::Integer)    if pmax ≤ 0 throw(DomainError("pmax must be strictly positive")) end    car = Vector{typeof(pmax)}(0)    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)                if m % Δ != 0 || Δ % h₃ != pmh continue end                q = m ÷ Δ + 1                if !isprime(q) continue end                r = (p * q - 1) ÷ h₃ + 1                if !isprime(r) || mod(q * r, p - 1) == 1 continue end                append!(car, [p, q, r])            end        end    end    return reshape(car, 3, length(car) ÷ 3)end`

Main

`hi = 61car = carmichael(hi) curp = tcnt = 0print("Carmichael 3 (p×q×r) 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        @printf("\n\np = %d\n  ", p)        tcnt = 0    end    if tcnt == 4        print("\n  ")        tcnt = 1    else        tcnt += 1    end    @printf("p× %d × %d = %d  ", q, r, c)endprintln("\n\n", size(car)[2], " results in total.")`
Output:
```Carmichael 3 (p×q×r) pseudoprimes, up to p = 61:

p = 11
p× 29 × 107 = 34133  p× 37 × 59 = 24013

p = 17
p× 23 × 79 = 30889  p× 53 × 101 = 91001

p = 19
p× 59 × 113 = 126673  p× 139 × 661 = 1745701  p× 193 × 283 = 1037761

p = 23
p× 43 × 53 = 52417  p× 59 × 227 = 308039  p× 71 × 137 = 223721  p× 83 × 107 = 204263

p = 29
p× 41 × 109 = 129601  p× 89 × 173 = 446513  p× 97 × 149 = 419137  p× 149 × 541 = 2337661

p = 31
p× 67 × 1039 = 2158003  p× 73 × 79 = 178777  p× 79 × 307 = 751843  p× 223 × 1153 = 7970689
p× 313 × 463 = 4492489

p = 41
p× 89 × 1217 = 4440833  p× 97 × 569 = 2262913

p = 43
p× 67 × 241 = 694321  p× 107 × 461 = 2121061  p× 131 × 257 = 1447681  p× 139 × 1993 = 11912161
p× 157 × 751 = 5070001  p× 199 × 373 = 3191761

p = 47
p× 53 × 499 = 1243009  p× 89 × 103 = 430849  p× 101 × 1583 = 7514501  p× 107 × 839 = 4219331
p× 157 × 239 = 1763581

p = 53
p× 113 × 1997 = 11960033  p× 197 × 233 = 2432753  p× 281 × 877 = 13061161

p = 59
p× 131 × 1289 = 9962681  p× 139 × 821 = 6733021  p× 149 × 587 = 5160317  p× 173 × 379 = 3868453
p× 179 × 353 = 3728033

p = 61
p× 1009 × 2677 = 164766673

42 results in total.```

Kotlin

Translation of: D
`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")                            }                        }                    }                }            }        }    }}`
Output:

See D output.

Mathematica / Wolfram Language

`Cases[Cases[  Cases[Table[{p1, h3, d}, {p1, Array[Prime, [email protected]]}, {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]]`

PARI/GP

`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]", ")))`
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
`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;`
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.)

`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}";            }        }    }}`
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

`(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)`

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;`

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

`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)))`
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)))))) `

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 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) `

REXX

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.

`/*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*/saysay '──────── '     #     " 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`

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.
```

Ring

` # Project : Carmichael 3 strong pseudoprimes see "The following are Carmichael munbers for p1 <= 61:" + nlsee "p1     p2      p3     product" + nl for p = 2 to 61    carmichael3(p)next func carmichael3(p1)        if isprime(p1) = 0  return ok       for h3 = 1 to p1 -1            t1 = (h3 + p1) * (p1 -1)            t2 = (-p1 * p1) % h3            if t2 < 0               t2 = t2 + h3            ok            for d = 1 to h3 + p1 -1                 if t1 % d = 0 and t2 = (d % h3)                    p2 = 1 + (t1 / d)                   if isprime(p2) = 0                      loop                   ok                   p3 = 1 + floor((p1 * p2 / h3))                   if isprime(p3) = 0 or ((p2 * p3) % (p1 -1)) != 1                       loop                   ok                   see "" + p1 + "       " + p2 + "      " + p3 + "    " + p1*p2*p3 + nl                ok            next      next  func isprime(num)       if (num <= 1) return 0 ok       if (num % 2 = 0) and num != 2          return 0       ok       for i = 3 to floor(num / 2) -1 step 2           if (num % i = 0)               return 0           ok       next       return 1 `

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
```

Ruby

Works with: Ruby version 1.9
`# 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  putsend`
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

` 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} `
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.

`\$ 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;`
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
`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)      }   }})`
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...

`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}`

Demonstrating:

`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"}`
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.

`var BN=Import("zklBigNum"), bi=BN(0); // gonna recycle biprimes:=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 }`
`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) });`
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
`10 FOR p=2 TO 6120 LET n=p: GO SUB 100030 IF NOT n THEN GO TO 20040 FOR h=1 TO p-150 FOR d=1 TO h-1+p60 IF NOT (FN m((h+p)*(p-1),d)=0 AND FN w(-p*p,h)=FN m(d,h)) THEN GO TO 18070 LET q=INT (1+((p-1)*(h+p)/d))80 LET n=q: GO SUB 100090 IF NOT n THEN GO TO 180100 LET r=INT (1+(p*q/h))110 LET n=r: GO SUB 1000120 IF (NOT n) OR ((FN m((q*r),(p-1))<>1)) THEN GO TO 180130 PRINT p;" ";q;" ";r180 NEXT d190 NEXT h200 NEXT p210 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=51030 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+61060 GO TO 10302000 DEF FN m(a,b)=a-(INT (a/b)*b): REM Mod function2010 DEF FN w(a,b)=FN m(FN m(a,b)+b,b): REM Mod function modified `