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.
- 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
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
- related task
11l
<lang 11l>F mod_(n, m)
R ((n % m) + m) % m
F is_prime(n)
I n C (2, 3)
R 1B
E I n < 2 | n % 2 == 0 | n % 3 == 0
R 0B
V div = 5
V inc = 2
L div ^ 2 <= n
I n % div == 0
R 0B
div += inc
inc = 6 - inc
R 1B
L(p) 2 .< 62
I !is_prime(p)
L.continue
L(h3) 2 .< p
V g = h3 + p
L(d) 1 .< g
I (g * (p - 1)) % d != 0 | mod_(-p * p, h3) != d % h3
L.continue;
V q = 1 + (p - 1) * g I/ d;
I !is_prime(q)
L.continue
V r = 1 + (p * q I/ h3)
I !is_prime(r) | (q * r) % (p - 1) != 1
L.continue
print(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 ... 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
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
ALGOL 68
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
AWK
<lang AWK>
- syntax: GAWK -f CARMICHAEL_3_STRONG_PSEUDOPRIMES.AWK
- converted from C
BEGIN {
printf("%5s%8s%8s%13s\n","P1","P2","P3","PRODUCT")
for (p1=2; p1<62; p1++) {
if (!is_prime(p1)) { continue }
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 = int(1+((p1-1)*(h3+p1)/d))
if (!is_prime(p2)) { continue }
p3 = int(1+(p1*p2/h3))
if (!is_prime(p3) || (p2*p3)%(p1-1) != 1) { continue }
printf("%5d x %5d x %5d = %10d\n",p1,p2,p3,p1*p2*p3)
count++
}
}
}
}
printf("%d numbers\n",count)
exit(0)
} function is_prime(n, i) {
if (n <= 3) {
return(n > 1)
}
else if (!(n%2) || !(n%3)) {
return(0)
}
else {
for (i=5; i*i<=n; i+=6) {
if (!(n%i) || !(n%(i+2))) {
return(0)
}
}
return(1)
}
} function mod(n,m) {
- the % operator actually calculates the remainder of a / b so we need a small adjustment so it works as expected for negative values
return(((n%m)+m)%m)
} </lang>
- Output:
P1 P2 P3 PRODUCT
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
69 numbers
C
<lang 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
<lang scheme>
- 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)))
</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)
- 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 </lang>
F#
This task uses Extensible Prime Generator (F#) <lang fsharp> // Carmichael Number . Nigel Galloway: November 19th., 2017 let 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 None
let carms g = primes|>Seq.takeWhile(fun n->n<=g)|>Seq.collect fN|>Seq.choose fG carms 61 |> Seq.iter (fun (P1,P2,P3)->printfn "%2d x %4d x %5d = %10d" P1 P2 P3 ((uint64 P3)*(uint64 (P1*P2)))) </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 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
Factor
Note the use of rem instead of mod when the remainder should always be positive.
<lang factor>USING: formatting kernel locals math math.primes math.ranges
sequences ;
IN: rosetta-code.carmichael
- carmichael ( p1 -- )
1 p1 (a,b) [| h3 |
h3 p1 + [1,b) [| d |
h3 p1 + p1 1 - * d mod zero?
p1 neg p1 * h3 rem d h3 mod = and
[
p1 1 - h3 p1 + * d /i 1 + :> p2
p1 p2 * h3 /i 1 + :> p3
p2 p3 [ prime? ] both?
p2 p3 * p1 1 - mod 1 = and
[ p1 p2 p3 "%d %d %d\n" printf ] when
] when
] each
] each
- carmichael-demo ( -- ) 61 primes-upto [ carmichael ] each ;
MAIN: carmichael-demo</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
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
<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
<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>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</lang>
Main <lang julia>hi = 61 car = carmichael(hi)
curp = tcnt = 0 print("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)
end println("\n\n", size(car)[2], " results in total.")</lang>
- 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
<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
<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
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
Phix
Uses is_prime() from Extensible_prime_generator <lang Phix>integer count = 0 for p1=1 to 61 do
if is_prime(p1) then
for h3=1 to p1 do
atom h3p1 = h3 + p1
for d=1 to h3p1-1 do
if mod(h3p1*(p1-1),d)=0
and mod(-(p1*p1),h3) = mod(d,h3) then
atom p2 := 1 + floor(((p1-1)*h3p1)/d),
p3 := 1 +floor(p1*p2/h3)
if is_prime(p2)
and is_prime(p3)
and mod(p2*p3,p1-1)=1 then
if count<5 or count>55 then
printf(1,"%d * %d * %d = %d\n",{p1,p2,p3,p1*p2*p3})
elsif count=35 then puts(1,"...\n") end if
count += 1
end if
end if
end for
end for
end if
end for printf(1,"%d Carmichael numbers found\n",count)</lang>
- Output:
3 * 11 * 17 = 561 5 * 29 * 73 = 10585 5 * 17 * 29 = 2465 5 * 13 * 17 = 1105 7 * 19 * 67 = 8911 ... 61 * 271 * 571 = 9439201 61 * 241 * 421 = 6189121 61 * 3361 * 4021 = 824389441 69 Carmichael numbers found
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>
- 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
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== | 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.*/
c.= 0 /*the list of Carmichael #'s (so far).*/
do h3=2 for pm-1; g= h3 + p /*get Carmichael numbers for this prime*/
npsH3= ((nps // h3) + h3) // h3 /*define a couple of shortcuts for pgm.*/
gPM= g * pm /*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; c.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*/
$= /*build list of some Carmichael numbers*/
if tell then do j=bot to top by 2; if c.j\==0 then $= $ p"∙"j'∙'c.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 /*is 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
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
do k=29 by 6 until k*k>x; if x//k ==0 then return 0
if x//(k+2) ==0 then return 0
end /*k*/
@.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.
Ring
<lang ring>
- Project : Carmichael 3 strong pseudoprimes
see "The following are Carmichael munbers for p1 <= 61:" + nl see "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
</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
Ruby
<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
<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
Vala
<lang vala>long mod(long n, long m) {
return ((n % m) + m) % m;
}
bool is_prime(long n) {
if (n == 2 || n == 3)
return true;
else if (n < 2 || n % 2 == 0 || n % 3 == 0)
return false;
for (long div = 5, inc = 2; div * div <= n;
div += inc, inc = 6 - inc)
if (n % div == 0)
return false;
return true;
}
void main() {
for (long p = 2; p <= 63; p++) {
if (!is_prime(p)) continue;
for (long h3 = 2; h3 <= p; h3++) {
var g = h3 + p;
for (long d = 1; d <= g; d++) {
if ((g * (p - 1)) % d != 0 || mod(-p * p, h3) != d % h3)
continue;
var q = 1 + (p - 1) * g / d;
if (!is_prime(q)) continue;
var r = 1 + (p * q / h3);
if (!is_prime(r) || (q * r) % (p - 1) != 1) continue;
stdout.printf("%5ld × %5ld × %5ld = %10ld\n", p, q, r, p * q * r);
}
}
}
}</lang>
- Output:
3 × 11 × 17 = 561
3 × 3 × 5 = 45
5 × 29 × 73 = 10585
5 × 5 × 13 = 325
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 × 7 × 13 = 637
7 × 73 × 103 = 52633
7 × 13 × 19 = 1729
11 × 11 × 61 = 7381
11 × 11 × 41 = 4961
11 × 11 × 31 = 3751
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 × 17 × 97 = 28033
17 × 353 × 1201 = 7207201
19 × 43 × 409 = 334153
19 × 19 × 181 = 65341
19 × 19 × 73 = 26353
19 × 19 × 37 = 13357
19 × 199 × 271 = 1024651
23 × 23 × 89 = 47081
23 × 23 × 67 = 35443
23 × 199 × 353 = 1615681
29 × 29 × 421 = 354061
29 × 113 × 1093 = 3581761
29 × 29 × 281 = 236321
29 × 197 × 953 = 5444489
31 × 991 × 15361 = 471905281
31 × 61 × 631 = 1193221
31 × 151 × 1171 = 5481451
31 × 31 × 241 = 231601
31 × 61 × 271 = 512461
31 × 61 × 211 = 399001
31 × 271 × 601 = 5049001
31 × 31 × 61 = 58621
31 × 181 × 331 = 1857241
37 × 109 × 2017 = 8134561
37 × 73 × 541 = 1461241
37 × 613 × 1621 = 36765901
37 × 73 × 181 = 488881
37 × 37 × 73 = 99937
37 × 73 × 109 = 294409
41 × 1721 × 35281 = 2489462641
41 × 881 × 12041 = 434932961
41 × 41 × 281 = 472361
41 × 41 × 241 = 405121
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 × 43 × 463 = 856087
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 × 47 × 277 = 611893
47 × 47 × 139 = 307051
47 × 3359 × 6073 = 958762729
47 × 1151 × 1933 = 104569501
47 × 3727 × 5153 = 902645857
53 × 53 × 937 = 2632033
53 × 157 × 2081 = 17316001
53 × 79 × 599 = 2508013
53 × 53 × 313 = 879217
53 × 157 × 521 = 4335241
53 × 53 × 157 = 441013
59 × 59 × 1741 = 6060421
59 × 59 × 349 = 1214869
59 × 59 × 233 = 811073
59 × 1451 × 2089 = 178837201
61 × 421 × 12841 = 329769721
61 × 181 × 5521 = 60957361
61 × 61 × 1861 = 6924781
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
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
<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>