Strange unique prime triplets
Strange unique prime triplets is a draft programming task. It is not yet considered ready to be promoted as a complete task, for reasons that should be found in its talk page.
Primes n, m, and p are strange unique primes if n, m, and p are unique and their sum n + m + p is also prime. Assume n < m < p.
- Task
-
- Find all triplets of strange unique primes in which n, m, and p are all less than 30.
- (stretch goal) Show the count (only) of all the triplets of strange unique primes in which n, m, and p are all less than 1,000.
11l
F primes_upto(limit)
V is_prime = [0B] * 2 [+] [1B] * (limit - 1)
L(n) 0 .< Int(limit ^ 0.5 + 1.5)
I is_prime[n]
L(i) (n * n .< limit + 1).step(n)
is_prime[i] = 0B
R enumerate(is_prime).filter((i, prime) -> prime).map((i, prime) -> i)
F strange_triplets(Int mx = 30)
[(Int, Int, Int)] r
V primes = Array(primes_upto(mx))
V primes3 = Set(primes_upto(3 * mx))
L(n) primes
V i = L.index
L(m) primes[i + 1 ..]
V j = L.index + i + 1
L(p) primes[j + 1 ..]
I n + m + p C primes3
r.append((n, m, p))
R r
L(n, m, p) strange_triplets()
print(‘#2: #2+#2+#2 = #.’.format(L.index + 1, n, m, p, n + m + p))
V mx = 1'000
print("\nIf n, m, p < #. finds #.".format(mx, strange_triplets(mx).len))- Output:
1: 3+ 5+11 = 19 2: 3+ 5+23 = 31 3: 3+ 5+29 = 37 4: 3+ 7+13 = 23 5: 3+ 7+19 = 29 6: 3+11+17 = 31 7: 3+11+23 = 37 8: 3+11+29 = 43 9: 3+17+23 = 43 10: 5+ 7+11 = 23 11: 5+ 7+17 = 29 12: 5+ 7+19 = 31 13: 5+ 7+29 = 41 14: 5+11+13 = 29 15: 5+13+19 = 37 16: 5+13+23 = 41 17: 5+13+29 = 47 18: 5+17+19 = 41 19: 5+19+23 = 47 20: 5+19+29 = 53 21: 7+11+13 = 31 22: 7+11+19 = 37 23: 7+11+23 = 41 24: 7+11+29 = 47 25: 7+13+17 = 37 26: 7+13+23 = 43 27: 7+17+19 = 43 28: 7+17+23 = 47 29: 7+17+29 = 53 30: 7+23+29 = 59 31: 11+13+17 = 41 32: 11+13+19 = 43 33: 11+13+23 = 47 34: 11+13+29 = 53 35: 11+17+19 = 47 36: 11+19+23 = 53 37: 11+19+29 = 59 38: 13+17+23 = 53 39: 13+17+29 = 59 40: 13+19+29 = 61 41: 17+19+23 = 59 42: 19+23+29 = 71 If n, m, p < 1000 finds 241580
Action!
INCLUDE "H6:SIEVE.ACT"
PROC Main()
DEFINE MAXPRIME="29"
DEFINE MAX="99"
BYTE ARRAY primes(MAX+1)
BYTE n,m,p,c
INT count=[0]
Put(125) PutE() ;clear the screen
Sieve(primes,MAX+1)
c=0
FOR n=2 TO MAXPRIME-2
DO
IF primes(n) THEN
FOR m=n+1 TO MAXPRIME-1
DO
IF primes(m) THEN
FOR p=m+1 TO MAXPRIME
DO
IF primes(p)=1 AND primes(n+m+p)=1 THEN
PrintF("%I+%I+%I=%I ",n,m,p,n+m+p)
count==+1 c==+1
IF c=3 THEN
c=0 PutE()
FI
FI
OD
FI
OD
FI
OD
PrintF("%EThere are %I prime triplets",count)
RETURN- Output:
Screenshot from Atari 8-bit computer
3+5+11=19 3+5+23=31 3+5+29=37 3+7+13=23 3+7+19=29 3+11+17=31 3+11+23=37 3+11+29=43 3+17+23=43 5+7+11=23 5+7+17=29 5+7+19=31 5+7+29=41 5+11+13=29 5+13+19=37 5+13+23=41 5+13+29=47 5+17+19=41 5+19+23=47 5+19+29=53 7+11+13=31 7+11+19=37 7+11+23=41 7+11+29=47 7+13+17=37 7+13+23=43 7+17+19=43 7+17+23=47 7+17+29=53 7+23+29=59 11+13+17=41 11+13+19=43 11+13+23=47 11+13+29=53 11+17+19=47 11+19+23=53 11+19+29=59 13+17+23=53 13+17+29=59 13+19+29=61 17+19+23=59 19+23+29=71 There are 42 prime triplets
ALGOL 68
which is based on
BEGIN # find some strange unique primes - triplets of primes n, m, p #
# where n + m + p is also prime and n =/= m =/= p #
# we need to find the strange unique prime triplets below 1000 #
# so the maximum triplet sum could be roughly 3000 #
INT max number = 1000;
# sieve the primes to the maximum reuired prime #
PR read "primes.incl.a68" PR
[]BOOL prime = PRIMESIEVE ( max number * 3 );
# we need to find the strange unique prime triplets below 1000 #
INT s count := 0, c30 := 0;
# 2 cannot be one of the primes as the sum would be even otherwise #
FOR n FROM 3 BY 2 TO max number - 5 DO
IF prime[ n ] THEN
FOR m FROM n + 2 BY 2 TO max number- 3 DO
IF prime[ m ] THEN
FOR p FROM m + 2 BY 2 TO max number DO
IF prime[ p ] THEN
IF INT s = n + m + p;
prime[ s ]
THEN
# have 3 unique primes whose sum is prime #
s count +:= 1;
IF p <= 30 AND m <= 30 AND n <= 30 THEN
c30 +:= 1;
print( ( whole( c30, -3 ), ": "
, whole( n, -3 ), " + "
, whole( m, -3 ), " + "
, whole( p, -3 ), " = "
, whole( s, -3 ), newline
)
)
FI
FI
FI
OD # p #
FI
OD # m #
FI
OD # n # ;
print( ( "Found ", whole( c30, -6 ), " strange unique prime triplets up to 30", newline ) );
print( ( "Found ", whole( s count, -6 ), " strange unique prime triplets up to 1000", newline ) )
END- Output:
1: 3 + 5 + 11 = 19 2: 3 + 5 + 23 = 31 3: 3 + 5 + 29 = 37 4: 3 + 7 + 13 = 23 5: 3 + 7 + 19 = 29 6: 3 + 11 + 17 = 31 7: 3 + 11 + 23 = 37 8: 3 + 11 + 29 = 43 9: 3 + 17 + 23 = 43 10: 5 + 7 + 11 = 23 11: 5 + 7 + 17 = 29 12: 5 + 7 + 19 = 31 13: 5 + 7 + 29 = 41 14: 5 + 11 + 13 = 29 15: 5 + 13 + 19 = 37 16: 5 + 13 + 23 = 41 17: 5 + 13 + 29 = 47 18: 5 + 17 + 19 = 41 19: 5 + 19 + 23 = 47 20: 5 + 19 + 29 = 53 21: 7 + 11 + 13 = 31 22: 7 + 11 + 19 = 37 23: 7 + 11 + 23 = 41 24: 7 + 11 + 29 = 47 25: 7 + 13 + 17 = 37 26: 7 + 13 + 23 = 43 27: 7 + 17 + 19 = 43 28: 7 + 17 + 23 = 47 29: 7 + 17 + 29 = 53 30: 7 + 23 + 29 = 59 31: 11 + 13 + 17 = 41 32: 11 + 13 + 19 = 43 33: 11 + 13 + 23 = 47 34: 11 + 13 + 29 = 53 35: 11 + 17 + 19 = 47 36: 11 + 19 + 23 = 53 37: 11 + 19 + 29 = 59 38: 13 + 17 + 23 = 53 39: 13 + 17 + 29 = 59 40: 13 + 19 + 29 = 61 41: 17 + 19 + 23 = 59 42: 19 + 23 + 29 = 71 Found 42 strange unique prime triplets up to 30 Found 241580 strange unique prime triplets up to 1000
ALGOL W
Based on
begin % find some strange unique primes - triplets of primes n, m, p %
% where n + m + p is also prime and n =/= m =/= p %
% sets p( 1 :: n ) to a sieve of primes up to n %
procedure Eratosthenes ( logical array p( * ) ; integer value n ) ;
begin
p( 1 ) := false; p( 2 ) := true;
for i := 3 step 2 until n do p( i ) := true;
for i := 4 step 2 until n do p( i ) := false;
for i := 3 step 2 until truncate( sqrt( n ) ) do begin
integer ii; ii := i + i;
if p( i ) then for pr := i * i step ii until n do p( pr ) := false
end for_i ;
end Eratosthenes ;
% we need to find the strange unique prime triplets below 1000 %
integer MAX_PRIME;
MAX_PRIME := 1000;
begin
% the sum of the triplets could be (roughly) 3 x the largest prime %
logical array p ( 1 :: MAX_PRIME * 3 );
integer sCount, c30;
% construct a sieve of primes up to MAX_PRIME * 3 %
Eratosthenes( p, MAX_PRIME * 3 );
% count the strange prime triplets whose members are < 1000 and %
% whose sum is prime %
sCount := c30 := 0;
% 2 cannot be one of the primes as the sum would be even otherwise %
for n := 3 step 2 until MAX_PRIME - 5 do begin
if p( n ) then begin
for m := n + 2 step 2 until MAX_PRIME - 3 do begin
if p( m ) then begin
for l := m + 2 step 2 until MAX_PRIME do begin
if p( l ) then begin
integer s;
s := n + m + l;
if p( s ) then begin
sCount := sCount + 1;
if l <= 30 and m <= 30 and n <= 30 then begin
c30 := c30 + 1;
write( i_w := 3, s_w := 0, c30, ": ", n, " + ", m, " + ", l, " = ", s )
end if_l_m_n_le_30
end if_p_s
end if_p_l
end for_l
end if_p_m
end for_m
end if_p_n
end for_n ;
write( i_w := 3, s_w := 0, "Found ", c30, " strange unique prime triplets up to 30" );
write( i_w := 3, s_w := 0, "Found ", sCount, " strange unique prime triplets up to 1000" );
end
end.- Output:
1: 3 + 5 + 11 = 19 2: 3 + 5 + 23 = 31 3: 3 + 5 + 29 = 37 4: 3 + 7 + 13 = 23 5: 3 + 7 + 19 = 29 6: 3 + 11 + 17 = 31 7: 3 + 11 + 23 = 37 8: 3 + 11 + 29 = 43 9: 3 + 17 + 23 = 43 10: 5 + 7 + 11 = 23 11: 5 + 7 + 17 = 29 12: 5 + 7 + 19 = 31 13: 5 + 7 + 29 = 41 14: 5 + 11 + 13 = 29 15: 5 + 13 + 19 = 37 16: 5 + 13 + 23 = 41 17: 5 + 13 + 29 = 47 18: 5 + 17 + 19 = 41 19: 5 + 19 + 23 = 47 20: 5 + 19 + 29 = 53 21: 7 + 11 + 13 = 31 22: 7 + 11 + 19 = 37 23: 7 + 11 + 23 = 41 24: 7 + 11 + 29 = 47 25: 7 + 13 + 17 = 37 26: 7 + 13 + 23 = 43 27: 7 + 17 + 19 = 43 28: 7 + 17 + 23 = 47 29: 7 + 17 + 29 = 53 30: 7 + 23 + 29 = 59 31: 11 + 13 + 17 = 41 32: 11 + 13 + 19 = 43 33: 11 + 13 + 23 = 47 34: 11 + 13 + 29 = 53 35: 11 + 17 + 19 = 47 36: 11 + 19 + 23 = 53 37: 11 + 19 + 29 = 59 38: 13 + 17 + 23 = 53 39: 13 + 17 + 29 = 59 40: 13 + 19 + 29 = 61 41: 17 + 19 + 23 = 59 42: 19 + 23 + 29 = 71 Found 42 strange unique prime triplets up to 30 Found 241580 strange unique prime triplets up to 1000
Arturo
findTriplets: function [upTo][
results: []
loop select 2..upTo => prime? 'n ->
loop select n..upTo => prime? 'm ->
loop select m..upTo => prime? 'p ->
if all? @[
3 = size unique @[n m p]
prime? n+m+p
]-> 'results ++ @[@[n,m,p]]
return results
]
loop.with:'i findTriplets 30 'res ->
print [i+1 "->" join.with:" + " to [:string] res "=" sum res]
print ""
print ["If n, m, p < 1000 -> total number of triplets:" size findTriplets 1000]
- Output:
1 -> 3 + 5 + 11 = 19 2 -> 3 + 5 + 23 = 31 3 -> 3 + 5 + 29 = 37 4 -> 3 + 7 + 13 = 23 5 -> 3 + 7 + 19 = 29 6 -> 3 + 11 + 17 = 31 7 -> 3 + 11 + 23 = 37 8 -> 3 + 11 + 29 = 43 9 -> 3 + 17 + 23 = 43 10 -> 5 + 7 + 11 = 23 11 -> 5 + 7 + 17 = 29 12 -> 5 + 7 + 19 = 31 13 -> 5 + 7 + 29 = 41 14 -> 5 + 11 + 13 = 29 15 -> 5 + 13 + 19 = 37 16 -> 5 + 13 + 23 = 41 17 -> 5 + 13 + 29 = 47 18 -> 5 + 17 + 19 = 41 19 -> 5 + 19 + 23 = 47 20 -> 5 + 19 + 29 = 53 21 -> 7 + 11 + 13 = 31 22 -> 7 + 11 + 19 = 37 23 -> 7 + 11 + 23 = 41 24 -> 7 + 11 + 29 = 47 25 -> 7 + 13 + 17 = 37 26 -> 7 + 13 + 23 = 43 27 -> 7 + 17 + 19 = 43 28 -> 7 + 17 + 23 = 47 29 -> 7 + 17 + 29 = 53 30 -> 7 + 23 + 29 = 59 31 -> 11 + 13 + 17 = 41 32 -> 11 + 13 + 19 = 43 33 -> 11 + 13 + 23 = 47 34 -> 11 + 13 + 29 = 53 35 -> 11 + 17 + 19 = 47 36 -> 11 + 19 + 23 = 53 37 -> 11 + 19 + 29 = 59 38 -> 13 + 17 + 23 = 53 39 -> 13 + 17 + 29 = 59 40 -> 13 + 19 + 29 = 61 41 -> 17 + 19 + 23 = 59 42 -> 19 + 23 + 29 = 71 If n, m, p < 1000 -> total number of triplets: 241580
AWK
# syntax: GAWK -f STRANGE_UNIQUE_PRIME_TRIPLETS.AWK
# converted from Go
BEGIN {
main(29,1)
main(999,0)
exit(0)
}
function main(n,show, count,i,j,k,s) {
for (i=3; i<=n-4; i+=2) {
if (is_prime(i)) {
for (j=i+2; j<=n-2; j+=2) {
if (is_prime(j)) {
for (k=j+2; k<=n; k+=2) {
if (is_prime(k)) {
s = i + j + k
if (is_prime(s)) {
count++
if (show == 1) {
printf("%2d + %2d + %2d = %d\n",i,j,k,s)
}
}
}
}
}
}
}
}
printf("Unique prime triples 2-%d which sum to a prime: %'d\n\n",n,count)
}
function is_prime(x, i) {
if (x <= 1) {
return(0)
}
for (i=2; i<=int(sqrt(x)); i++) {
if (x % i == 0) {
return(0)
}
}
return(1)
}
- Output:
3 + 5 + 11 = 19 3 + 5 + 23 = 31 3 + 5 + 29 = 37 3 + 7 + 13 = 23 3 + 7 + 19 = 29 3 + 11 + 17 = 31 3 + 11 + 23 = 37 3 + 11 + 29 = 43 3 + 17 + 23 = 43 5 + 7 + 11 = 23 5 + 7 + 17 = 29 5 + 7 + 19 = 31 5 + 7 + 29 = 41 5 + 11 + 13 = 29 5 + 13 + 19 = 37 5 + 13 + 23 = 41 5 + 13 + 29 = 47 5 + 17 + 19 = 41 5 + 19 + 23 = 47 5 + 19 + 29 = 53 7 + 11 + 13 = 31 7 + 11 + 19 = 37 7 + 11 + 23 = 41 7 + 11 + 29 = 47 7 + 13 + 17 = 37 7 + 13 + 23 = 43 7 + 17 + 19 = 43 7 + 17 + 23 = 47 7 + 17 + 29 = 53 7 + 23 + 29 = 59 11 + 13 + 17 = 41 11 + 13 + 19 = 43 11 + 13 + 23 = 47 11 + 13 + 29 = 53 11 + 17 + 19 = 47 11 + 19 + 23 = 53 11 + 19 + 29 = 59 13 + 17 + 23 = 53 13 + 17 + 29 = 59 13 + 19 + 29 = 61 17 + 19 + 23 = 59 19 + 23 + 29 = 71 Unique prime triples 2-29 which sum to a prime: 42 Unique prime triples 2-999 which sum to a prime: 241,580
C
#include <stdbool.h>
#include <stdio.h>
#include <string.h>
#define LIMIT 3000
void init_sieve(unsigned char sieve[], int limit) {
int i, j;
for (i = 0; i < limit; i++) {
sieve[i] = 1;
}
sieve[0] = 0;
sieve[1] = 0;
for (i = 2; i < limit; i++) {
if (sieve[i]) {
for (j = i + i; j < limit; j += i) {
sieve[j] = 0;
}
}
}
}
void strange_unique_prime_triplets(unsigned char sieve[], int limit, bool verbose) {
int count = 0, sum;
int i, j, k, n, p;
int pi, pj, pk;
n = 0;
for (i = 0; i < limit; i++) {
if (sieve[i]) {
n++;
}
}
if (verbose) {
printf("Strange unique prime triplets < %d:\n", limit);
}
for (i = 0; i + 2 < n; i++) {
pi = 2;
p = i;
while (p > 0) {
pi++;
if (sieve[pi]) {
p--;
}
}
for (j = i + 1; j + 1 < n; j++) {
pj = pi;
p = j - i;
while (p > 0) {
pj++;
if (sieve[pj]) {
p--;
}
}
for (k = j + 1; k < n; k++) {
pk = pj;
p = k - j;
while (p > 0) {
pk++;
if (sieve[pk]) {
p--;
}
}
sum = pi + pj + pk;
if (sum < LIMIT && sieve[sum]) {
count++;
if (verbose) {
printf("%2d + %2d + %2d = %d\n", pi, pj, pk, sum);
}
}
}
}
}
printf("Count of strange unique prime triplets < %d is %d.\n\n", limit, count);
}
int main() {
unsigned char sieve[LIMIT];
init_sieve(sieve, LIMIT);
strange_unique_prime_triplets(sieve, 30, true);
strange_unique_prime_triplets(sieve, 1000, false);
return 0;
}
- Output:
Strange unique prime triplets < 30: 3 + 5 + 11 = 19 3 + 5 + 23 = 31 3 + 5 + 29 = 37 3 + 7 + 13 = 23 3 + 7 + 19 = 29 3 + 11 + 17 = 31 3 + 11 + 23 = 37 3 + 11 + 29 = 43 3 + 17 + 23 = 43 5 + 7 + 11 = 23 5 + 7 + 17 = 29 5 + 7 + 19 = 31 5 + 7 + 29 = 41 5 + 11 + 13 = 29 5 + 13 + 19 = 37 5 + 13 + 23 = 41 5 + 13 + 29 = 47 5 + 17 + 19 = 41 5 + 19 + 23 = 47 5 + 19 + 29 = 53 7 + 11 + 13 = 31 7 + 11 + 19 = 37 7 + 11 + 23 = 41 7 + 11 + 29 = 47 7 + 13 + 17 = 37 7 + 13 + 23 = 43 7 + 17 + 19 = 43 7 + 17 + 23 = 47 7 + 17 + 29 = 53 7 + 23 + 29 = 59 11 + 13 + 17 = 41 11 + 13 + 19 = 43 11 + 13 + 23 = 47 11 + 13 + 29 = 53 11 + 17 + 19 = 47 11 + 19 + 23 = 53 11 + 19 + 29 = 59 13 + 17 + 23 = 53 13 + 17 + 29 = 59 13 + 19 + 29 = 61 17 + 19 + 23 = 59 19 + 23 + 29 = 71 Count of strange unique prime triplets < 30 is 42. Count of strange unique prime triplets < 1000 is 241580.
C#
Just for fun, <30 sorted by sum, instead of order generated. One might think one should include the sieve generation time, but it is orders of magnitude smaller than the permute/sum time for these relatively low numbers.
using System; using System.Collections.Generic; using static System.Console; using System.Linq; using DT = System.DateTime;
class Program { static void Main(string[] args) { string s;
foreach (int lmt in new int[]{ 90, 300, 3000, 30000, 111000 }) {
var pr = PG.Primes(lmt).Skip(1).ToList(); DT st = DT.Now;
int d, f = 0; var r = new List<string>();
int i = -1, m, h = (m = lmt / 3), j, k, pra, prab;
while (i < 0) i = pr.IndexOf(h--); k = (j = i - 1) - 1;
for (int a = 0; a <= k; a++) { pra = pr[a];
for (int b = a + 1; b <= j; b++) { prab = pra + pr[b];
for (int c = b + 1; c <= i; c++) {
if (PG.flags[d = prab + pr[c]]) continue; f++;
if (lmt < 100) r.Add(string.Format("{3,5} = {0,2} + {1,2} + {2,2}", pra, pr[b], pr[c], d)); } } }
s = "s.u.p.t.s under "; r.Sort(); if (r.Count > 0) WriteLine("{0}{1}:\n{2}", s, m, string.Join("\n", r));
if (lmt > 100) WriteLine("Count of {0}{1,6:n0}: {2,13:n0} {3} sec", s, m, f, (DT.Now - st).ToString().Substring(6)); } } }
class PG { public static bool[] flags;
public static IEnumerable<int> Primes(int lim) {
flags = new bool[lim + 1]; int j = 2;
for (int d = 3, sq = 4; sq <= lim; j++, sq += d += 2)
if (!flags[j]) { yield return j;
for (int k = sq; k <= lim; k += j) flags[k] = true; }
for (; j <= lim; j++) if (!flags[j]) yield return j; } }
- Output:
Timings from tio.run
s.u.p.t.s under 30: 19 = 3 + 5 + 11 23 = 3 + 7 + 13 23 = 5 + 7 + 11 29 = 3 + 7 + 19 29 = 5 + 7 + 17 29 = 5 + 11 + 13 31 = 3 + 5 + 23 31 = 3 + 11 + 17 31 = 5 + 7 + 19 31 = 7 + 11 + 13 37 = 3 + 5 + 29 37 = 3 + 11 + 23 37 = 5 + 13 + 19 37 = 7 + 11 + 19 37 = 7 + 13 + 17 41 = 5 + 7 + 29 41 = 5 + 13 + 23 41 = 5 + 17 + 19 41 = 7 + 11 + 23 41 = 11 + 13 + 17 43 = 3 + 11 + 29 43 = 3 + 17 + 23 43 = 7 + 13 + 23 43 = 7 + 17 + 19 43 = 11 + 13 + 19 47 = 5 + 13 + 29 47 = 5 + 19 + 23 47 = 7 + 11 + 29 47 = 7 + 17 + 23 47 = 11 + 13 + 23 47 = 11 + 17 + 19 53 = 5 + 19 + 29 53 = 7 + 17 + 29 53 = 11 + 13 + 29 53 = 11 + 19 + 23 53 = 13 + 17 + 23 59 = 7 + 23 + 29 59 = 11 + 19 + 29 59 = 13 + 17 + 29 59 = 17 + 19 + 23 61 = 13 + 19 + 29 71 = 19 + 23 + 29 Count of s.u.p.t.s under 100: 891 00.0000243 sec Count of s.u.p.t.s under 1,000: 241,580 00.0054753 sec Count of s.u.p.t.s under 10,000: 74,588,542 01.8159964 sec Count of s.u.p.t.s under 37,000: 2,141,379,201 55.0369689 sec
C++
#include <iomanip>
#include <iostream>
#include <vector>
std::vector<bool> prime_sieve(size_t limit) {
std::vector<bool> sieve(limit, true);
if (limit > 0)
sieve[0] = false;
if (limit > 1)
sieve[1] = false;
for (size_t i = 4; i < limit; i += 2)
sieve[i] = false;
for (size_t p = 3; ; p += 2) {
size_t q = p * p;
if (q >= limit)
break;
if (sieve[p]) {
size_t inc = 2 * p;
for (; q < limit; q += inc)
sieve[q] = false;
}
}
return sieve;
}
void strange_unique_prime_triplets(int limit, bool verbose) {
std::vector<bool> sieve = prime_sieve(limit * 3);
std::vector<int> primes;
for (int p = 3; p < limit; p += 2) {
if (sieve[p])
primes.push_back(p);
}
size_t n = primes.size();
size_t count = 0;
if (verbose)
std::cout << "Strange unique prime triplets < " << limit << ":\n";
for (size_t i = 0; i + 2 < n; ++i) {
for (size_t j = i + 1; j + 1 < n; ++j) {
for (size_t k = j + 1; k < n; ++k) {
int sum = primes[i] + primes[j] + primes[k];
if (sieve[sum]) {
++count;
if (verbose) {
std::cout << std::setw(2) << primes[i] << " + "
<< std::setw(2) << primes[j] << " + "
<< std::setw(2) << primes[k] << " = " << sum
<< '\n';
}
}
}
}
}
std::cout << "\nCount of strange unique prime triplets < " << limit
<< " is " << count << ".\n";
}
int main() {
strange_unique_prime_triplets(30, true);
strange_unique_prime_triplets(1000, false);
return 0;
}
- Output:
Strange unique prime triplets < 30: 3 + 5 + 11 = 19 3 + 5 + 23 = 31 3 + 5 + 29 = 37 3 + 7 + 13 = 23 3 + 7 + 19 = 29 3 + 11 + 17 = 31 3 + 11 + 23 = 37 3 + 11 + 29 = 43 3 + 17 + 23 = 43 5 + 7 + 11 = 23 5 + 7 + 17 = 29 5 + 7 + 19 = 31 5 + 7 + 29 = 41 5 + 11 + 13 = 29 5 + 13 + 19 = 37 5 + 13 + 23 = 41 5 + 13 + 29 = 47 5 + 17 + 19 = 41 5 + 19 + 23 = 47 5 + 19 + 29 = 53 7 + 11 + 13 = 31 7 + 11 + 19 = 37 7 + 11 + 23 = 41 7 + 11 + 29 = 47 7 + 13 + 17 = 37 7 + 13 + 23 = 43 7 + 17 + 19 = 43 7 + 17 + 23 = 47 7 + 17 + 29 = 53 7 + 23 + 29 = 59 11 + 13 + 17 = 41 11 + 13 + 19 = 43 11 + 13 + 23 = 47 11 + 13 + 29 = 53 11 + 17 + 19 = 47 11 + 19 + 23 = 53 11 + 19 + 29 = 59 13 + 17 + 23 = 53 13 + 17 + 29 = 59 13 + 19 + 29 = 61 17 + 19 + 23 = 59 19 + 23 + 29 = 71 Count of strange unique prime triplets < 30 is 42. Count of strange unique prime triplets < 1000 is 241580.
Delphi
program Strange_primes;
{$APPTYPE CONSOLE}
uses
System.SysUtils;
function IsPrime(n: Integer): Boolean;
begin
if n < 2 then
exit(false);
if n mod 2 = 0 then
exit(n = 2);
if n mod 3 = 0 then
exit(n = 3);
var d := 5;
while d * d <= n do
begin
if n mod d = 0 then
exit(false);
inc(d, 2);
if n mod d = 0 then
exit(false);
inc(d, 4);
end;
Result := true;
end;
function Commatize(value: Integer): string;
begin
Result := FloatToStrF(value, ffNumber, 10, 0);
end;
function StrangePrimes(n: Integer; countOnly: Boolean): Integer;
begin
var c := 0;
var f := '%2d: %2d + %2d + %2d = %2d'#10;
var s: Integer := 0;
var i := 3;
while i <= n - 4 do
begin
if IsPrime(i) then
begin
var j := i + 2;
while j <= n - 2 do
begin
if IsPrime(j) then
begin
var k := j + 2;
while k <= n do
begin
if IsPrime(k) then
begin
s := i + j + k;
if IsPrime(s) then
begin
inc(c);
if not countOnly then
write(format(f, [c, i, j, k, s]));
end;
end;
inc(k, 2);
end;
end;
inc(j, 2);
end;
end;
inc(i, 2);
end;
Result := c;
end;
begin
Writeln('Unique prime triples under 30 which sum to a prime:');
strangePrimes(29, false);
var cs := commatize(strangePrimes(999, true));
writeln('There are ', cs, ' unique prime triples under 1,000 which sum to a prime.');
readln;
end.
EasyLang
fastfunc isprim num .
if num < 2
return 0
.
i = 2
while i <= sqrt num
if num mod i = 0
return 0
.
i += 1
.
return 1
.
lim = 29
for n = 1 to lim
for m = n + 1 to lim
for p = m + 1 to lim
sum = n + m + p
if isprim sum = 1 and isprim n = 1 and isprim m = 1 and isprim p = 1
write "(" & n & " " & m & " " & p & ") "
.
.
.
.
- Output:
(3 5 11) (3 5 23) (3 5 29) (3 7 13) (3 7 19) (3 11 17) (3 11 23) (3 11 29) (3 17 23) (5 7 11) (5 7 17) (5 7 19) (5 7 29) (5 11 13) (5 13 19) (5 13 23) (5 13 29) (5 17 19) (5 19 23) (5 19 29) (7 11 13) (7 11 19) (7 11 23) (7 11 29) (7 13 17) (7 13 23) (7 17 19) (7 17 23) (7 17 29) (7 23 29) (11 13 17) (11 13 19) (11 13 23) (11 13 29) (11 17 19) (11 19 23) (11 19 29) (13 17 23) (13 17 29) (13 19 29) (17 19 23) (19 23 29)
F#
This task uses Extensible Prime Generator (F#).
// Strange unique prime triplets. Nigel Galloway: March 12th., 2021
let sP n=let N=primes32()|>Seq.takeWhile((>)n)|>Array.ofSeq
seq{for n in 0..N.Length-1 do for i in n+1..N.Length-1 do for g in i+1..N.Length-1->(N.[n],N.[i],N.[g])}|>Seq.filter(fun(n,i,g)->isPrime(n+i+g))
sP 30|>Seq.iteri(fun n(i,g,l)->printfn "%2d: %2d+%2d+%2d=%2d")
printfn "%d" (Seq.length(sP 1000))
printfn "%d" (Seq.length(sP 10000))
- Output:
241580 74588542
Factor
USING: formatting io kernel math math.combinatorics math.primes
sequences tools.memory.private ;
: .triplet ( seq -- ) "%2d+%2d+%2d = %d\n" vprintf ;
: strange ( n -- )
primes-upto 3
[ dup sum dup prime? [ suffix .triplet ] [ 2drop ] if ]
each-combination ;
: count-strange ( n -- count )
0 swap primes-upto 3
[ sum prime? [ 1 + ] when ] each-combination ;
30 strange
1,000 count-strange commas nl
"Found %s strange prime triplets with n, m, p < 1,000.\n" printf
- Output:
3+ 5+11 = 19 3+ 5+23 = 31 3+ 5+29 = 37 3+ 7+13 = 23 3+ 7+19 = 29 3+11+17 = 31 3+11+23 = 37 3+11+29 = 43 3+17+23 = 43 5+ 7+11 = 23 5+ 7+17 = 29 5+ 7+19 = 31 5+ 7+29 = 41 5+11+13 = 29 5+13+19 = 37 5+13+23 = 41 5+13+29 = 47 5+17+19 = 41 5+19+23 = 47 5+19+29 = 53 7+11+13 = 31 7+11+19 = 37 7+11+23 = 41 7+11+29 = 47 7+13+17 = 37 7+13+23 = 43 7+17+19 = 43 7+17+23 = 47 7+17+29 = 53 7+23+29 = 59 11+13+17 = 41 11+13+19 = 43 11+13+23 = 47 11+13+29 = 53 11+17+19 = 47 11+19+23 = 53 11+19+29 = 59 13+17+23 = 53 13+17+29 = 59 13+19+29 = 61 17+19+23 = 59 19+23+29 = 71 Found 241,580 strange prime triplets with n, m, p < 1,000.
Fermat
Function IsSUPT(n,m,p) =
if Isprime(n) and Isprime(m) and Isprime(p) and Isprime(n+m+p) then 1 else 0 fi.
for n=3 to 19 do
for m=n+2 to 23 do
for p=m+2 to 29 do
if IsSUPT(n,m,p) then !!(n,m,p) fi;
od;
od;
odI'll leave the stretch goal for someone else.
FreeBASIC
Use the function at Primality by trial division#FreeBASIC as an include; I can't be bothered reproducing it here.
#include"isprime.bas"
dim as uinteger c = 0
for p as uinteger = 3 to 997
if not isprime(p) then continue for
for m as uinteger = p + 1 to 998
if not isprime(m) then continue for
for n as uinteger = m + 1 to 999
if not isprime(n) then continue for
if isprime(p + n + m) then
c = c + 1
if n < 30 then print p;" + ";m;" + ";n;" = "; p + m + n
end if
next n
next m
next p
print "There are ";c;" triples below 1000."
- Output:
3 + 5 + 11 = 193 + 5 + 23 = 31 3 + 5 + 29 = 37 3 + 7 + 13 = 23 3 + 7 + 19 = 29 3 + 11 + 17 = 31 3 + 11 + 23 = 37 3 + 11 + 29 = 43 3 + 17 + 23 = 43 5 + 7 + 11 = 23 5 + 7 + 17 = 29 5 + 7 + 19 = 31 5 + 7 + 29 = 41 5 + 11 + 13 = 29 5 + 13 + 19 = 37 5 + 13 + 23 = 41 5 + 13 + 29 = 47 5 + 17 + 19 = 41 5 + 19 + 23 = 47 5 + 19 + 29 = 53 7 + 11 + 13 = 31 7 + 11 + 19 = 37 7 + 11 + 23 = 41 7 + 11 + 29 = 47 7 + 13 + 17 = 37 7 + 13 + 23 = 43 7 + 17 + 19 = 43 7 + 17 + 23 = 47 7 + 17 + 29 = 53 7 + 23 + 29 = 59 11 + 13 + 17 = 41 11 + 13 + 19 = 43 11 + 13 + 23 = 47 11 + 13 + 29 = 53 11 + 17 + 19 = 47 11 + 19 + 23 = 53 11 + 19 + 29 = 59 13 + 17 + 23 = 53 13 + 17 + 29 = 59 13 + 19 + 29 = 61 17 + 19 + 23 = 59 19 + 23 + 29 = 71
There are 241580 triples below 1000.
Forth
: prime? ( n -- ? ) here + c@ 0= ;
: notprime! ( n -- ) here + 1 swap c! ;
: prime_sieve ( n -- )
here over erase
0 notprime!
1 notprime!
dup 4 > if
dup 4 do i notprime! 2 +loop
then
3
begin
2dup dup * >
while
dup prime? if
2dup dup * do
i notprime!
dup 2* +loop
then
2 +
repeat
2drop ;
: print_strange_unique_prime_triplets ( n -- )
dup 8 < if drop exit then
dup 3 * prime_sieve
dup 4 - 3 do
i prime? if
dup 2 - i 2 + do
i prime? if
dup i 2 + do
i prime? if
i j k + + dup prime? if
k 2 .r ." + " j 2 .r ." + " i 2 .r ." = " 2 .r cr
else
drop
then
then
2 +loop
then
2 +loop
then
2 +loop drop ;
: count_strange_unique_prime_triplets ( n -- n )
dup 8 < if drop 0 exit then
dup 3 * prime_sieve
0 swap
dup 4 - 3 do
i prime? if
dup 2 - i 2 + do
i prime? if
dup i 2 + do
i prime? if
i j k + + prime? if
swap 1+ swap
then
then
2 +loop
then
2 +loop
then
2 +loop drop ;
." Strange unique prime triplets < 30:" cr
30 print_strange_unique_prime_triplets
." Count of strange unique prime triplets < 1000: "
1000 count_strange_unique_prime_triplets . cr
bye
- Output:
Strange unique prime triplets < 30: 3 + 5 + 11 = 19 3 + 5 + 23 = 31 3 + 5 + 29 = 37 3 + 7 + 13 = 23 3 + 7 + 19 = 29 3 + 11 + 17 = 31 3 + 11 + 23 = 37 3 + 11 + 29 = 43 3 + 17 + 23 = 43 5 + 7 + 11 = 23 5 + 7 + 17 = 29 5 + 7 + 19 = 31 5 + 7 + 29 = 41 5 + 11 + 13 = 29 5 + 13 + 19 = 37 5 + 13 + 23 = 41 5 + 13 + 29 = 47 5 + 17 + 19 = 41 5 + 19 + 23 = 47 5 + 19 + 29 = 53 7 + 11 + 13 = 31 7 + 11 + 19 = 37 7 + 11 + 23 = 41 7 + 11 + 29 = 47 7 + 13 + 17 = 37 7 + 13 + 23 = 43 7 + 17 + 19 = 43 7 + 17 + 23 = 47 7 + 17 + 29 = 53 7 + 23 + 29 = 59 11 + 13 + 17 = 41 11 + 13 + 19 = 43 11 + 13 + 23 = 47 11 + 13 + 29 = 53 11 + 17 + 19 = 47 11 + 19 + 23 = 53 11 + 19 + 29 = 59 13 + 17 + 23 = 53 13 + 17 + 29 = 59 13 + 19 + 29 = 61 17 + 19 + 23 = 59 19 + 23 + 29 = 71 Count of strange unique prime triplets < 1000: 241580
Fōrmulæ
Fōrmulæ programs are not textual, visualization/edition of programs is done showing/manipulating structures but not text. Moreover, there can be multiple visual representations of the same program. Even though it is possible to have textual representation —i.e. XML, JSON— they are intended for storage and transfer purposes more than visualization and edition.
Programs in Fōrmulæ are created/edited online in its website.
In this page you can see and run the program(s) related to this task and their results. You can also change either the programs or the parameters they are called with, for experimentation, but remember that these programs were created with the main purpose of showing a clear solution of the task, and they generally lack any kind of validation.
Solution
Definitions:
Test case 1. Find all triplets of strange unique primes in which n, m, and p are all less than 30
Test case 2. (Stretch goal). Show the count (only) of all the triplets of strange unique primes in which n, m, and p are all less than 1,000
Go
Basic
package main
import "fmt"
func isPrime(n int) bool {
switch {
case n < 2:
return false
case n%2 == 0:
return n == 2
case n%3 == 0:
return n == 3
default:
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 commatize(n int) string {
s := fmt.Sprintf("%d", n)
if n < 0 {
s = s[1:]
}
le := len(s)
for i := le - 3; i >= 1; i -= 3 {
s = s[0:i] + "," + s[i:]
}
if n >= 0 {
return s
}
return "-" + s
}
func strangePrimes(n int, countOnly bool) int {
c := 0
f := "%2d: %2d + %2d + %2d = %2d\n"
var s int
for i := 3; i <= n-4; i += 2 {
if isPrime(i) {
for j := i + 2; j <= n-2; j += 2 {
if isPrime(j) {
for k := j + 2; k <= n; k += 2 {
if isPrime(k) {
s = i + j + k
if isPrime(s) {
c++
if !countOnly {
fmt.Printf(f, c, i, j, k, s)
}
}
}
}
}
}
}
}
return c
}
func main() {
fmt.Println("Unique prime triples under 30 which sum to a prime:")
strangePrimes(29, false)
cs := commatize(strangePrimes(999, true))
fmt.Printf("\nThere are %s unique prime triples under 1,000 which sum to a prime.\n", cs)
}
- Output:
Unique prime triples under 30 which sum to a prime: 1: 3 + 5 + 11 = 19 2: 3 + 5 + 23 = 31 3: 3 + 5 + 29 = 37 4: 3 + 7 + 13 = 23 5: 3 + 7 + 19 = 29 6: 3 + 11 + 17 = 31 7: 3 + 11 + 23 = 37 8: 3 + 11 + 29 = 43 9: 3 + 17 + 23 = 43 10: 5 + 7 + 11 = 23 11: 5 + 7 + 17 = 29 12: 5 + 7 + 19 = 31 13: 5 + 7 + 29 = 41 14: 5 + 11 + 13 = 29 15: 5 + 13 + 19 = 37 16: 5 + 13 + 23 = 41 17: 5 + 13 + 29 = 47 18: 5 + 17 + 19 = 41 19: 5 + 19 + 23 = 47 20: 5 + 19 + 29 = 53 21: 7 + 11 + 13 = 31 22: 7 + 11 + 19 = 37 23: 7 + 11 + 23 = 41 24: 7 + 11 + 29 = 47 25: 7 + 13 + 17 = 37 26: 7 + 13 + 23 = 43 27: 7 + 17 + 19 = 43 28: 7 + 17 + 23 = 47 29: 7 + 17 + 29 = 53 30: 7 + 23 + 29 = 59 31: 11 + 13 + 17 = 41 32: 11 + 13 + 19 = 43 33: 11 + 13 + 23 = 47 34: 11 + 13 + 29 = 53 35: 11 + 17 + 19 = 47 36: 11 + 19 + 23 = 53 37: 11 + 19 + 29 = 59 38: 13 + 17 + 23 = 53 39: 13 + 17 + 29 = 59 40: 13 + 19 + 29 = 61 41: 17 + 19 + 23 = 59 42: 19 + 23 + 29 = 71 There are 241,580 unique prime triples under 1,000 which sum to a prime.
Faster
package main
import "fmt"
var sieved []bool
var p = []int{2}
func sieve(limit int) []bool {
limit++
// True denotes composite, false denotes prime.
c := make([]bool, limit) // all false by default
c[0] = true
c[1] = true
// no need to bother with even numbers over 2 for this task
p := 3 // Start from 3.
for {
p2 := p * p
if p2 >= limit {
break
}
for i := p2; i < limit; i += 2 * p {
c[i] = true
}
for {
p += 2
if !c[p] {
break
}
}
}
return c
}
func commatize(n int) string {
s := fmt.Sprintf("%d", n)
if n < 0 {
s = s[1:]
}
le := len(s)
for i := le - 3; i >= 1; i -= 3 {
s = s[0:i] + "," + s[i:]
}
if n >= 0 {
return s
}
return "-" + s
}
func strangePrimes(n int, countOnly bool) int {
c := 0
f := "%2d: %2d + %2d + %2d = %2d\n"
var r, s int
m := 0
for ; m < len(p) && p[m] <= n; m++ {
}
for i := 1; i < m-2; i++ {
for j := i + 1; j < m-1; j++ {
r = p[i] + p[j]
for k := j + 1; k < m; k++ {
s = r + p[k]
if !sieved[s] {
c++
if !countOnly {
fmt.Printf(f, c, p[i], p[j], p[k], s)
}
}
}
}
}
return c
}
func main() {
const max = 1000
sieved = sieve(3*max)
for i := 3; i <= max; i += 2 {
if !sieved[i] {
p = append(p, i)
}
}
fmt.Println("Unique prime triples under 30 which sum to a prime:")
strangePrimes(29, false)
cs := commatize(strangePrimes(999, true))
fmt.Printf("\nThere are %s unique prime triples under 1,000 which sum to a prime.\n", cs)
}
- Output:
Same as 'basic' version.
J
cb=. ;@:({. <@,. @\.)}.
comb3=. ]cb cb
triplets=. (#~ 1 p: +/"1)@comb3@(i.&.(p:inv)"0)
- Output:
triplets 30 3 5 11 3 5 23 3 5 29 3 7 13 3 7 19 3 11 17 3 11 23 3 11 29 3 17 23 5 7 11 5 7 17 5 7 19 5 7 29 5 11 13 5 13 19 5 13 23 5 13 29 5 17 19 5 19 23 5 19 29 7 11 13 7 11 19 7 11 23 7 11 29 7 13 17 7 13 23 7 17 19 7 17 23 7 17 29 7 23 29 11 13 17 11 13 19 11 13 23 11 13 29 11 17 19 11 19 23 11 19 29 13 17 23 13 17 29 13 19 29 17 19 23 19 23 29 #@triplets 30 1000 42 241580
Java
import java.util.*;
public class StrangeUniquePrimeTriplets {
public static void main(String[] args) {
strangeUniquePrimeTriplets(30, true);
strangeUniquePrimeTriplets(1000, false);
}
private static void strangeUniquePrimeTriplets(int limit, boolean verbose) {
boolean[] sieve = primeSieve(limit * 3);
List<Integer> primeList = new ArrayList<>();
for (int p = 3; p < limit; p += 2) {
if (sieve[p])
primeList.add(p);
}
int n = primeList.size();
// Convert object list to primitive array for performance
int[] primes = new int[n];
for (int i = 0; i < n; ++i)
primes[i] = primeList.get(i);
int count = 0;
if (verbose)
System.out.printf("Strange unique prime triplets < %d:\n", limit);
for (int i = 0; i + 2 < n; ++i) {
for (int j = i + 1; j + 1 < n; ++j) {
int s = primes[i] + primes[j];
for (int k = j + 1; k < n; ++k) {
int sum = s + primes[k];
if (sieve[sum]) {
++count;
if (verbose)
System.out.printf("%2d + %2d + %2d = %2d\n", primes[i], primes[j], primes[k], sum);
}
}
}
}
System.out.printf("\nCount of strange unique prime triplets < %d is %d.\n", limit, count);
}
private static boolean[] primeSieve(int limit) {
boolean[] sieve = new boolean[limit];
Arrays.fill(sieve, true);
if (limit > 0)
sieve[0] = false;
if (limit > 1)
sieve[1] = false;
for (int i = 4; i < limit; i += 2)
sieve[i] = false;
for (int p = 3; ; p += 2) {
int q = p * p;
if (q >= limit)
break;
if (sieve[p]) {
int inc = 2 * p;
for (; q < limit; q += inc)
sieve[q] = false;
}
}
return sieve;
}
}
- Output:
Strange unique prime triplets < 30: 3 + 5 + 11 = 19 3 + 5 + 23 = 31 3 + 5 + 29 = 37 3 + 7 + 13 = 23 3 + 7 + 19 = 29 3 + 11 + 17 = 31 3 + 11 + 23 = 37 3 + 11 + 29 = 43 3 + 17 + 23 = 43 5 + 7 + 11 = 23 5 + 7 + 17 = 29 5 + 7 + 19 = 31 5 + 7 + 29 = 41 5 + 11 + 13 = 29 5 + 13 + 19 = 37 5 + 13 + 23 = 41 5 + 13 + 29 = 47 5 + 17 + 19 = 41 5 + 19 + 23 = 47 5 + 19 + 29 = 53 7 + 11 + 13 = 31 7 + 11 + 19 = 37 7 + 11 + 23 = 41 7 + 11 + 29 = 47 7 + 13 + 17 = 37 7 + 13 + 23 = 43 7 + 17 + 19 = 43 7 + 17 + 23 = 47 7 + 17 + 29 = 53 7 + 23 + 29 = 59 11 + 13 + 17 = 41 11 + 13 + 19 = 43 11 + 13 + 23 = 47 11 + 13 + 29 = 53 11 + 17 + 19 = 47 11 + 19 + 23 = 53 11 + 19 + 29 = 59 13 + 17 + 23 = 53 13 + 17 + 29 = 59 13 + 19 + 29 = 61 17 + 19 + 23 = 59 19 + 23 + 29 = 71 Count of strange unique prime triplets < 30 is 42. Count of strange unique prime triplets < 1000 is 241580.
jq
Works with gojq, the Go implementation of jq
See e.g. Erdős-primes#jq for a suitable implementation of `is_prime`.
def count(s): reduce s as $x (null; .+1);
def task($n):
[2, (range(3;$n;2)|select(is_prime))]
| . as $p
| range(0; length) as $i
| range($i+1; length) as $j
| range($j+1; length) as $k
| [.[$i], .[$j], .[$k]]
| select( add| is_prime) ;
task(30),
"\nStretch goal: \(count(task(1000)))"- Output:
[3,5,11] [3,5,23] [3,5,29] [3,7,13] [3,7,19] [3,11,17] [3,11,23] [3,11,29] [3,17,23] [5,7,11] [5,7,17] [5,7,19] [5,7,29] [5,11,13] [5,13,19] [5,13,23] [5,13,29] [5,17,19] [5,19,23] [5,19,29] [7,11,13] [7,11,19] [7,11,23] [7,11,29] [7,13,17] [7,13,23] [7,17,19] [7,17,23] [7,17,29] [7,23,29] [11,13,17] [11,13,19] [11,13,23] [11,13,29] [11,17,19] [11,19,23] [11,19,29] [13,17,23] [13,17,29] [13,19,29] [17,19,23] [19,23,29] Stretch goal: 241580
Julia
using Primes
function prime_sum_prime_triplets_to(N, verbose=false)
a = primes(3, N)
prime_sieve_set = primesmask(1, N * 3)
len, triplets, n = length(a), Dict{Tuple{Int64,Int64,Int64}, Int}(), 0
for i in eachindex(a), j in i+1:len, k in j+1:len
if prime_sieve_set[a[i] + a[j] + a[k]]
verbose && (triplets[(a[i], a[j], a[k])] = 1)
n += 1
end
end
if verbose
len = (length(string(N)) + 2) * 3
println("\n", rpad("Triplet", len), "Sum\n", "-"^(len+3))
for k in sort(collect(keys(triplets)), lt = (x, y) -> collect(x) < collect(y))
println(rpad(k, len), sum(k))
end
end
println("\n\n$n unique triplets of 3 primes between 2 and $N sum to a prime.")
return triplets
end
prime_sum_prime_triplets_to(30, true)
prime_sum_prime_triplets_to(1000)
@time prime_sum_prime_triplets_to(10000)
@time prime_sum_prime_triplets_to(100000)
- Output:
Triplet Sum --------------- (3, 5, 11) 19 (3, 5, 23) 31 (3, 5, 29) 37 (3, 7, 13) 23 (3, 7, 19) 29 (3, 11, 17) 31 (3, 11, 23) 37 (3, 11, 29) 43 (3, 17, 23) 43 (5, 7, 11) 23 (5, 7, 17) 29 (5, 7, 19) 31 (5, 7, 29) 41 (5, 11, 13) 29 (5, 13, 19) 37 (5, 13, 23) 41 (5, 13, 29) 47 (5, 17, 19) 41 (5, 19, 23) 47 (5, 19, 29) 53 (7, 11, 13) 31 (7, 11, 19) 37 (7, 11, 23) 41 (7, 11, 29) 47 (7, 13, 17) 37 (7, 13, 23) 43 (7, 17, 19) 43 (7, 17, 23) 47 (7, 17, 29) 53 (7, 23, 29) 59 (11, 13, 17)41 (11, 13, 19)43 (11, 13, 23)47 (11, 13, 29)53 (11, 17, 19)47 (11, 19, 23)53 (11, 19, 29)59 (13, 17, 23)53 (13, 17, 29)59 (13, 19, 29)61 (17, 19, 23)59 (19, 23, 29)71 42 unique triplets of 3 primes between 2 and 30 sum to a prime. 241580 unique triplets of 3 primes between 2 and 1000 sum to a prime. 74588542 unique triplets of 3 primes between 2 and 10000 sum to a prime. 0.509732 seconds (31 allocations: 25.938 KiB) 28694800655 unique triplets of 3 primes between 2 and 100000 sum to a prime. 224.940756 seconds (35 allocations: 218.156 KiB)
Mathematica /Wolfram Language
p = Prime[Range@PrimePi[30]];
Select[Subsets[p, {3}], Total/*PrimeQ]
p = Prime[Range@PrimePi[1000]];
Length[Select[Subsets[p, {3}], Total/*PrimeQ]]
- Output:
{{3,5,11},{3,5,23},{3,5,29},{3,7,13},{3,7,19},{3,11,17},{3,11,23},{3,11,29},{3,17,23},{5,7,11},{5,7,17},{5,7,19},{5,7,29},{5,11,13},{5,13,19},{5,13,23},{5,13,29},{5,17,19},{5,19,23},{5,19,29},{7,11,13},{7,11,19},{7,11,23},{7,11,29},{7,13,17},{7,13,23},{7,17,19},{7,17,23},{7,17,29},{7,23,29},{11,13,17},{11,13,19},{11,13,23},{11,13,29},{11,17,19},{11,19,23},{11,19,29},{13,17,23},{13,17,29},{13,19,29},{17,19,23},{19,23,29}}
241580
Nim
import strformat, strutils, sugar
func isPrime(n: Positive): bool =
if n < 2: return false
if n mod 2 == 0: return n == 2
if n mod 3 == 0: return n == 3
var d = 5
while d * d <= n:
if n mod d == 0: return false
inc d, 2
if n mod d == 0: return false
inc d, 4
result = true
iterator triplets(primes: openArray[int]): (int, int, int) =
## Yield the triplets.
for i in 0..primes.high-2:
let n = primes[i]
for j in (i+1)..primes.high-1:
let m = primes[j]
for k in (j+1)..primes.high:
let p = primes[k]
if (n + m + p).isPrime:
yield (n, m, p)
const Primes30 = [2, 3, 5, 7, 11, 13, 17, 19, 23, 29]
echo "List of strange unique prime triplets for n < m < p < 30:"
for (n, m, p) in Primes30.triplets():
echo &"{n:2} + {m:2} + {p:2} = {n+m+p}"
echo()
const Primes1000 = collect(newSeq):
for n in 2..999:
if n.isPrime: n
var count = 0
for _ in Primes1000.triplets(): inc count
echo "Count of strange unique prime triplets for n < m < p < 1000: ", ($count).insertSep()
- Output:
List of strange unique prime triplets for n < m < p < 30: 3 + 5 + 11 = 19 3 + 5 + 23 = 31 3 + 5 + 29 = 37 3 + 7 + 13 = 23 3 + 7 + 19 = 29 3 + 11 + 17 = 31 3 + 11 + 23 = 37 3 + 11 + 29 = 43 3 + 17 + 23 = 43 5 + 7 + 11 = 23 5 + 7 + 17 = 29 5 + 7 + 19 = 31 5 + 7 + 29 = 41 5 + 11 + 13 = 29 5 + 13 + 19 = 37 5 + 13 + 23 = 41 5 + 13 + 29 = 47 5 + 17 + 19 = 41 5 + 19 + 23 = 47 5 + 19 + 29 = 53 7 + 11 + 13 = 31 7 + 11 + 19 = 37 7 + 11 + 23 = 41 7 + 11 + 29 = 47 7 + 13 + 17 = 37 7 + 13 + 23 = 43 7 + 17 + 19 = 43 7 + 17 + 23 = 47 7 + 17 + 29 = 53 7 + 23 + 29 = 59 11 + 13 + 17 = 41 11 + 13 + 19 = 43 11 + 13 + 23 = 47 11 + 13 + 29 = 53 11 + 17 + 19 = 47 11 + 19 + 23 = 53 11 + 19 + 29 = 59 13 + 17 + 23 = 53 13 + 17 + 29 = 59 13 + 19 + 29 = 61 17 + 19 + 23 = 59 19 + 23 + 29 = 71 Count of strange unique prime triplets for n < m < p < 1000: 241_580
Pascal
program PrimeTriplets;
//Free Pascal Compiler version 3.2.1 [2020/11/03] for x86_64fpc 3.2.1
{$IFDEF FPC}
{$MODE DELPHI}
{$Optimization ON,ALL}
{$ELSE}
{$APPTYPE CONSOLE}
{$ENDIF}
const
MAXZAHL = 100000;// > 3
MAXSUM = 3*MAXZAHL;
CountOfPrimes = trunc(MAXZAHL/(ln(MAXZAHL)-1.08))+100;
type
tChkprimes = array[0..MAXSUM] of byte;//prime == 1 , nonprime == 0
var
Chkprimes:tChkprimes;
primes : array[0..CountOfPrimes]of Uint32;//here starting with 3
count,primeCount:NativeInt;
procedure InitPrimes;
//sieve of eratothenes
var
i,j : NativeInt;
begin
fillchar(Chkprimes,SizeOf(tChkprimes),#1);
i := 2;
j := 2*2;
if j> MAXSUM then
EXIT;
repeat
Chkprimes[j]:= 0;
inc(j,i);
until j> Maxsum;
For i := 3 to MAXSUM do
Begin
if Chkprimes[i] <>0 then
Begin
j := i*i;
if j> MAXSUM then
Break;
repeat
Chkprimes[j]:= 0;
inc(j,2*i);
until j> Maxsum;
end;
end;
j := 0;
For i := 3 to MAXZAHL do
IF Chkprimes[i]<>0 then
Begin
primes[j] := i;
inc(j);
end;
primeCount := j-1;
j :=CountOfPrimes -primeCount;
IF j <0 then
begin
writeln(' Need more space for primes ', -j);
HALT(-243);
end;
end;
function GetMaxPrimeIdx(lmt:NativeInt):NativeInt;
begin
if lmt >= Maxzahl then
Begin
result := primecount;
EXIT;
end;
result := 0;
while (result < primecount) AND (primes[result]<lmt) do
inc(result);
dec(result);
end;
procedure Out_Check(lmt:nativeInt);
//simplest version
var
i,j,k,s,pc: NativeInt;
Begin
pc:= GetMaxPrimeIdx(lmt);
count := 0;
For i := 0 to pc do
For j := i+1 to pc do
For k := j+1 to pc do
Begin
s := primes[i]+primes[j]+Primes[k];
//if takes the longest time
if ChkPrimes[s]<> 0 then
begin
inc(count);
writeln(count:3,': ',primes[i],'+',primes[j],'+',primes[k],' = ',s);
end;
end;
writeln;
end;
procedure Count_Check(pc:nativeInt);
// the power of many registers ( 64-Bit )
var
cnt : Uint64;
pPrimes : pUint32;
pChkPrimes : ^tChkprimes;
pi,pij,i,j,k: NativeInt;
Begin
cnt := 0;
pPrimes := @primes[0];
pChkPrimes := @Chkprimes[0];
For i := 0 to pc do
Begin
pi := pPrimes[i];
For j := i+1 to pc do
begin
pij := pi+pPrimes[j];
For k := j+1 to pc do
inc(cnt,pChkPrimes^[pij+pPrimes[k]]);
end;
end;
count := cnt;
end;
procedure Check_Limit(lmt:NativeInt);
Begin
If lmt>primes[primecount] then
lmt := MaxZahl;
write('Limit = ',lmt,' count: ');
Count_Check(GetMaxPrimeIdx(lmt));
writeln(count);
end;
BEGIN
InitPrimes;
Out_Check(30);
Check_Limit(100);
Check_Limit(1000);
Check_Limit(10000);
//Check_Limit(MAXZAHL);
END.
- Output:
1: 3+5+11 = 19 2: 3+5+23 = 31 3: 3+5+29 = 37 4: 3+7+13 = 23 5: 3+7+19 = 29 6: 3+11+17 = 31 7: 3+11+23 = 37 8: 3+11+29 = 43 9: 3+17+23 = 43 10: 5+7+11 = 23 11: 5+7+17 = 29 12: 5+7+19 = 31 13: 5+7+29 = 41 14: 5+11+13 = 29 15: 5+13+19 = 37 16: 5+13+23 = 41 17: 5+13+29 = 47 18: 5+17+19 = 41 19: 5+19+23 = 47 20: 5+19+29 = 53 21: 7+11+13 = 31 22: 7+11+19 = 37 23: 7+11+23 = 41 24: 7+11+29 = 47 25: 7+13+17 = 37 26: 7+13+23 = 43 27: 7+17+19 = 43 28: 7+17+23 = 47 29: 7+17+29 = 53 30: 7+23+29 = 59 31: 11+13+17 = 41 32: 11+13+19 = 43 33: 11+13+23 = 47 34: 11+13+29 = 53 35: 11+17+19 = 47 36: 11+19+23 = 53 37: 11+19+29 = 59 38: 13+17+23 = 53 39: 13+17+29 = 59 40: 13+19+29 = 61 41: 17+19+23 = 59 42: 19+23+29 = 71 Limit = 100 count: 891 Limit = 1000 count: 241580 Limit = 10000 count: 74588542 //real 0m0,142s Limit = 100000 count: 28694800655 real 1m5,378s
Perl
use strict;
use warnings;
use List::Util 'sum';
use ntheory <primes is_prime>;
use Algorithm::Combinatorics 'combinations';
for my $n (30, 1000) {
printf "Found %d strange unique prime triplets up to $n.\n",
scalar grep { is_prime(sum @$_) } combinations(primes($n), 3);
}
- Output:
Found 42 strange unique prime triplets up to 30. Found 241580 strange unique prime triplets up to 1000.
Phix
with javascript_semantics requires("0.8.4") function create_sieve(integer limit) sequence sieve = repeat(true,limit) sieve[1] = false for i=4 to limit by 2 do sieve[i] = false end for for p=3 to floor(sqrt(limit)) by 2 do integer p2 = p*p if sieve[p2] then for k=p2 to limit by p*2 do sieve[k] = false end for end if end for return sieve end function procedure strange_triplets(integer lim, bool bCountOnly=true) atom t0 = time(), t1 = t0+1 sequence primes = get_primes_le(lim), sieve = create_sieve(lim*3), res = {} atom count = 0 -- -- It is not worth involving 2, ie primes[1], -- since (2 + any other two primes) is even, -- also we may as well leave space for {j,k}, -- {k} in the two outer loops. -- Using a sieve on the inner test is over -- ten times faster than is_prime(), whereas -- using a separate table of primes for the -- two outer loops is about twice as fast as -- scanning the sieve skipping falsies. Also -- interestingly, using nm = n+m is twice as -- fast as nmp = n+m+p. -- for i=2 to length(primes)-2 do integer n = primes[i] for j=i+1 to length(primes)-1 do integer m = primes[j], nm = n+m for k=j+1 to length(primes) do integer p = primes[k], nmp = nm+p if sieve[nmp] then count += 1 if not bCountOnly then res = append(res,sprintf("%2d: %2d+%2d+%2d = %d", {count, n, m, p, nmp})) end if end if if platform()!=JS and time()>t1 then progress("Working... (%,d)\r",{count}) t1 = time()+1 end if end for end for end for if platform()!=JS then progress("") end if string r = iff(bCountOnly?sprintf(" (%s)",{elapsed(time()-t0)}) :sprintf(":\n%s",{join(shorten(res,"",3),"\n")})) printf(1,"%,d strange triplets < %,d found%s\n\n",{count,lim,r}) end procedure strange_triplets(30,false) strange_triplets(1000) strange_triplets(10000)
- Output:
42 strange triplets < 30 found: 1: 3+ 5+11 = 19 2: 3+ 5+23 = 31 3: 3+ 5+29 = 37 ... 40: 13+19+29 = 61 41: 17+19+23 = 59 42: 19+23+29 = 71 241,580 strange triplets < 1,000 found (0.0s) 74,588,542 strange triplets < 10,000 found (11.4s)
Prolog
primes(2, Limit):- 2 =< Limit.
primes(N, Limit):-
between(3, Limit, N),
N /\ 1 > 0, % odd
M is floor(sqrt(N)) - 1, % reverse 2*I+1
Max is M div 2,
forall(between(1, Max, I), N mod (2*I+1) > 0).
primeComb(N, List, Comb):-
comb(N, List, Comb),
sumlist(Comb, Sum),
primes(Sum, inf).
comb(0, _, []).
comb(N, [X|T], [X|Comb]):-
N > 0,
N1 is N - 1,
comb(N1, T, Comb).
comb(N, [_|T], Comb):-
N > 0,
comb(N, T, Comb).
tripletList(Limit, List, Len):-
findall(N, primes(N, Limit), PrimeList),
findall(Comb, primeComb(3, PrimeList, Comb), List),
length(List, Len).
showList([]).
showList([[I, J, K]|TList]):-
Sum is I + J + K,
writef('%3r +%3r +%3r =%3r\n', [I, J, K, Sum]),
showList(TList).
run([]).
run([Limit|TLimits]):-
tripletList(Limit, List, Len),
( Limit < 50
-> List1 = List
; List1 = []
),
showList(List1),
writef('number of prime Triplets up to%5r is%7r\n', [Limit, Len]),
run(TLimits).
do:- run([30, 1000]).
- Output:
?- do.
3 + 5 + 11 = 19
3 + 5 + 23 = 31
3 + 5 + 29 = 37
...
13 + 19 + 29 = 61
17 + 19 + 23 = 59
19 + 23 + 29 = 71
number of prime Triplets up to 30 is 42
number of prime Triplets up to 1000 is 241580
true.
Python
Using sympy.primerange.
from sympy import primerange
def strange_triplets(mx: int = 30) -> None:
primes = list(primerange(0, mx))
primes3 = set(primerange(0, 3 * mx))
for i, n in enumerate(primes):
for j, m in enumerate(primes[i + 1:], i + 1):
for p in primes[j + 1:]:
if n + m + p in primes3:
yield n, m, p
for c, (n, m, p) in enumerate(strange_triplets(), 1):
print(f"{c:2}: {n:2}+{m:2}+{p:2} = {n + m + p}")
mx = 1_000
print(f"\nIf n, m, p < {mx:_} finds {sum(1 for _ in strange_triplets(mx)):_}")
- Output:
1: 3+ 5+11 = 19 2: 3+ 5+23 = 31 3: 3+ 5+29 = 37 4: 3+ 7+13 = 23 5: 3+ 7+19 = 29 6: 3+11+17 = 31 7: 3+11+23 = 37 8: 3+11+29 = 43 9: 3+17+23 = 43 10: 5+ 7+11 = 23 11: 5+ 7+17 = 29 12: 5+ 7+19 = 31 13: 5+ 7+29 = 41 14: 5+11+13 = 29 15: 5+13+19 = 37 16: 5+13+23 = 41 17: 5+13+29 = 47 18: 5+17+19 = 41 19: 5+19+23 = 47 20: 5+19+29 = 53 21: 7+11+13 = 31 22: 7+11+19 = 37 23: 7+11+23 = 41 24: 7+11+29 = 47 25: 7+13+17 = 37 26: 7+13+23 = 43 27: 7+17+19 = 43 28: 7+17+23 = 47 29: 7+17+29 = 53 30: 7+23+29 = 59 31: 11+13+17 = 41 32: 11+13+19 = 43 33: 11+13+23 = 47 34: 11+13+29 = 53 35: 11+17+19 = 47 36: 11+19+23 = 53 37: 11+19+29 = 59 38: 13+17+23 = 53 39: 13+17+29 = 59 40: 13+19+29 = 61 41: 17+19+23 = 59 42: 19+23+29 = 71 If n, m, p < 1_000 finds 241_580
Quackery
isprime is defined at Primality by trial division#Quackery.
comb is defined at Combinations#Quackery.
[ dup size dip
[ witheach
[ over swap peek swap ] ]
nip pack ] is arrange ( [ [ --> [ )
[ 0 swap witheach + ] is sum ( [ --> n )
[] 30 times
[ i^ isprime if
[ i^ join ] ]
behead drop
3 over size comb
[] unrot
witheach
[ over swap arrange
dup sum
isprime not iff
drop done
nested swap dip join ]
drop
sortwith [ sum dip sum > ]
dup size echo
say " strange unique prime triplets found:"
cr cr
witheach
[ dup witheach
[ echo
i if say "+" ]
say " = " sum echo
cr ]- Output:
42 strange unique prime triplets found: 3+5+11 = 19 3+7+13 = 23 5+7+11 = 23 3+7+19 = 29 5+7+17 = 29 5+11+13 = 29 3+5+23 = 31 3+11+17 = 31 5+7+19 = 31 7+11+13 = 31 3+5+29 = 37 3+11+23 = 37 5+13+19 = 37 7+11+19 = 37 7+13+17 = 37 5+7+29 = 41 5+13+23 = 41 5+17+19 = 41 7+11+23 = 41 11+13+17 = 41 3+11+29 = 43 3+17+23 = 43 7+13+23 = 43 7+17+19 = 43 11+13+19 = 43 5+13+29 = 47 5+19+23 = 47 7+11+29 = 47 7+17+23 = 47 11+13+23 = 47 11+17+19 = 47 5+19+29 = 53 7+17+29 = 53 11+13+29 = 53 11+19+23 = 53 13+17+23 = 53 7+23+29 = 59 11+19+29 = 59 13+17+29 = 59 17+19+23 = 59 13+19+29 = 61 19+23+29 = 71
Raku
(formerly Perl 6)
# 20210312 Raku programming solution
for 30, 1000 -> \k {
given (2..k).grep(*.is-prime).combinations(3).grep(*.sum.is-prime) {
say "Found ", +$_, " strange unique prime triplets up to ", k
}
}
- Output:
Found 42 strange unique prime triplets up to 30 Found 241580 strange unique prime triplets up to 1000
REXX
/*REXX program finds/lists triplet strange primes (<HI) where the triplets' sum is prime*/
parse arg hi . /*obtain optional argument from the CL.*/
if hi=='' | hi=="," then hi= 30 /*Not specified? Then use the default.*/
tell= hi>0; hi= abs(hi); hi= hi - 1 /*use absolute value of HI for limit. */
if tell>0 then say 'list of unique triplet strange primes whose sum is a prime.:'
call genP /*build array of semaphores for primes.*/
finds= 0 /*# of triplet strange primes (so far).*/
say
do m=2+1 by 2 to hi; if \!.m then iterate /*just use the odd primes. */
do n=m+2 by 2 to hi; if \!.n then iterate /* " " " " " */
mn= m + n /*partial sum (deep loops).*/
do p=n+2 by 2 to hi; if \!.p then iterate /*just use the odd primes. */
sum= mn + p /*compute sum of 3 primes. */
if \!.sum then iterate /*Is the sum prime? No, then skip it.*/
finds= finds + 1 /*bump # of triplet "strange" primes.*/
if tell then say right(m, w+9) right(n, w) right(p, w) ' sum to:' right(sum, w+2)
end /*p*/
end /*n*/
end /*m*/
say
say 'Found ' commas(finds) " unique triplet strange primes < " commas(hi+1) ,
" which sum to a prime."
exit 0 /*stick a fork in it, we're all done. */
/*──────────────────────────────────────────────────────────────────────────────────────*/
commas: parse arg ?; do jc=length(?)-3 to 1 by -3; ?=insert(',', ?, jc); end; return ?
/*──────────────────────────────────────────────────────────────────────────────────────*/
genP: !.= 0; w= length(hi) /*semaphores for primes; width of #'s.*/
@.1=2; @.2=3; @.3=5; @.4=7; @.5=11 /*define some low primes. */
!.2=1; !.3=1; !.5=1; !.7=1; !.11=1 /* " " " " semaphores. */
#=5; s.#= @.# **2 /*number of primes so far; prime². */
/* [↓] generate more primes ≤ high.*/
do j=@.#+2 by 2 for hi*3%2 /*find odd primes from here on. */
parse var j '' -1 _; if _==5 then iterate /*J divisible by 5? (right dig)*/
if j// 3==0 then iterate /*" " " 3? */
if j// 7==0 then iterate /*" " " 7? */
/* [↑] the above five lines saves time*/
do k=5 while s.k<=j /* [↓] divide by the known odd primes.*/
if j // @.k == 0 then iterate j /*Is J ÷ X? Then not prime. ___ */
end /*k*/ /* [↑] only process numbers ≤ √ J */
#= #+1; @.#= j; s.#= j*j; !.j= 1 /*bump # of Ps; assign next P; P²; P# */
end /*j*/; return
- output when using the default input:
list of unique triplet strange primes that sum to a prime:
prime generation took 0.02 seconds.
3 5 11 sum to: 19
3 5 23 sum to: 31
3 5 29 sum to: 37
3 7 13 sum to: 23
3 7 19 sum to: 29
3 11 17 sum to: 31
3 11 23 sum to: 37
3 11 29 sum to: 43
3 17 23 sum to: 43
5 7 11 sum to: 23
5 7 17 sum to: 29
5 7 19 sum to: 31
5 7 29 sum to: 41
5 11 13 sum to: 29
5 13 19 sum to: 37
5 13 23 sum to: 41
5 13 29 sum to: 47
5 17 19 sum to: 41
5 19 23 sum to: 47
5 19 29 sum to: 53
7 11 13 sum to: 31
7 11 19 sum to: 37
7 11 23 sum to: 41
7 11 29 sum to: 47
7 13 17 sum to: 37
7 13 23 sum to: 43
7 17 19 sum to: 43
7 17 23 sum to: 47
7 17 29 sum to: 53
7 23 29 sum to: 59
11 13 17 sum to: 41
11 13 19 sum to: 43
11 13 23 sum to: 47
11 13 29 sum to: 53
11 17 19 sum to: 47
11 19 23 sum to: 53
11 19 29 sum to: 59
13 17 23 sum to: 53
13 17 29 sum to: 59
13 19 29 sum to: 61
17 19 23 sum to: 59
19 23 29 sum to: 71
Found 42 unique triplet strange primes < 30 which sum to a prime.
- output when using the input of: -1000
Found 241,580 unique triplet strange primes < 1,000 which sum to a prime.
Ring
load "stdlib.ring"
num = 0
limit = 30
see "working..." + nl
see "the strange primes are:" + nl
for n = 1 to limit
for m = n+1 to limit
for p = m+1 to limit
sum = n+m+p
if isprime(sum) and isprime(n) and isprime(m) and isprime(p)
num = num + 1
see "" + num + ": " + n + "+" + m + "+" + p + " = " + sum + nl
ok
next
next
next
see "done..." + nl- Output:
working... the strange primes are: 1: 3+5+11 = 19 2: 3+5+23 = 31 3: 3+5+29 = 37 4: 3+7+13 = 23 5: 3+7+19 = 29 6: 3+11+17 = 31 7: 3+11+23 = 37 8: 3+11+29 = 43 9: 3+17+23 = 43 10: 5+7+11 = 23 11: 5+7+17 = 29 12: 5+7+19 = 31 13: 5+7+29 = 41 14: 5+11+13 = 29 15: 5+13+19 = 37 16: 5+13+23 = 41 17: 5+13+29 = 47 18: 5+17+19 = 41 19: 5+19+23 = 47 20: 5+19+29 = 53 21: 7+11+13 = 31 22: 7+11+19 = 37 23: 7+11+23 = 41 24: 7+11+29 = 47 25: 7+13+17 = 37 26: 7+13+23 = 43 27: 7+17+19 = 43 28: 7+17+23 = 47 29: 7+17+29 = 53 30: 7+23+29 = 59 31: 11+13+17 = 41 32: 11+13+19 = 43 33: 11+13+23 = 47 34: 11+13+29 = 53 35: 11+17+19 = 47 36: 11+19+23 = 53 37: 11+19+29 = 59 38: 13+17+23 = 53 39: 13+17+29 = 59 40: 13+19+29 = 61 41: 17+19+23 = 59 42: 19+23+29 = 71 done...
RPL
« { }
DO SWAP OVER + SWAP NEXTPRIME
UNTIL DUP 30 > END
DROP DUP SIZE
→ primes size
« { }
1 size 2 - FOR m
m 1 + size 1 - FOR n
n 1 + size FOR p
primes m GET primes n GET primes p GET
IF 3 DUPN + + ISPRIME? THEN
ROT "+" + ROT + "+" + SWAP + +
ELSE 3 DROPN END
NEXT NEXT NEXT
DUP SIZE
» » 'TASK' STO
- Output:
2: { "3+5+11" "3+5+23" "3+5+29" "3+7+13" "3+7+19" "3+11+17" "3+11+23" "3+11+29" "3+17+23" "5+7+11" "5+7+17" "5+7+19" "5+7+29" "5+11+13" "5+13+19" "5+13+23" "5+13+29" "5+17+19" "5+19+23" "5+19+29" "7+11+13" "7+11+19" "7+11+23" "7+11+29" "7+13+17" "7+13+23" "7+17+19" "7+17+23" "7+17+29" "7+23+29" "11+13+17" "11+13+19" "11+13+23" "11+13+29" "11+17+19" "11+19+23" "11+19+29" "13+17+23" "13+17+29" "13+19+29" "17+19+23" "19+23+29" }
1: 42.
Ruby
require 'prime'
Prime.each(30).to_a.combination(3).select{|trio| trio.sum.prime? }.each do |a,b,c|
puts "#{a} + #{b} + #{c} = #{a+b+c}"
end
m = 1000
count = Prime.each(m).to_a.combination(3).count{|trio| trio.sum.prime? }
puts "Count of strange unique prime triplets < #{m} is #{count}."
- Output:
3 + 5 + 11 = 19 3 + 5 + 23 = 31 3 + 5 + 29 = 37 3 + 7 + 13 = 23 3 + 7 + 19 = 29 3 + 11 + 17 = 31 3 + 11 + 23 = 37 3 + 11 + 29 = 43 3 + 17 + 23 = 43 5 + 7 + 11 = 23 5 + 7 + 17 = 29 5 + 7 + 19 = 31 5 + 7 + 29 = 41 5 + 11 + 13 = 29 5 + 13 + 19 = 37 5 + 13 + 23 = 41 5 + 13 + 29 = 47 5 + 17 + 19 = 41 5 + 19 + 23 = 47 5 + 19 + 29 = 53 7 + 11 + 13 = 31 7 + 11 + 19 = 37 7 + 11 + 23 = 41 7 + 11 + 29 = 47 7 + 13 + 17 = 37 7 + 13 + 23 = 43 7 + 17 + 19 = 43 7 + 17 + 23 = 47 7 + 17 + 29 = 53 7 + 23 + 29 = 59 11 + 13 + 17 = 41 11 + 13 + 19 = 43 11 + 13 + 23 = 47 11 + 13 + 29 = 53 11 + 17 + 19 = 47 11 + 19 + 23 = 53 11 + 19 + 29 = 59 13 + 17 + 23 = 53 13 + 17 + 29 = 59 13 + 19 + 29 = 61 17 + 19 + 23 = 59 19 + 23 + 29 = 71 Count of strange unique prime triplets < 1000 is 241580.
Rust
fn prime_sieve(limit: usize) -> Vec<bool> {
let mut sieve = vec![true; limit];
if limit > 0 {
sieve[0] = false;
}
if limit > 1 {
sieve[1] = false;
}
for i in (4..limit).step_by(2) {
sieve[i] = false;
}
let mut p = 3;
loop {
let mut q = p * p;
if q >= limit {
break;
}
if sieve[p] {
let inc = 2 * p;
while q < limit {
sieve[q] = false;
q += inc;
}
}
p += 2;
}
sieve
}
fn strange_unique_prime_triplets(limit: usize, verbose: bool) {
if limit < 6 {
return;
}
let mut primes = Vec::new();
let sieve = prime_sieve(limit * 3);
for p in (3..limit).step_by(2) {
if sieve[p] {
primes.push(p);
}
}
if verbose {
println!("Strange unique prime triplets < {}:", limit);
}
let mut count = 0;
let n = primes.len();
for i in 0..n - 2 {
for j in i + 1..n - 1 {
for k in j + 1..n {
let sum = primes[i] + primes[j] + primes[k];
if sieve[sum] {
count += 1;
if verbose {
println!(
"{:2} + {:2} + {:2} = {:2}",
primes[i], primes[j], primes[k], sum
);
}
}
}
}
}
println!(
"Count of strange unique prime triplets < {} is {}.",
limit, count
);
}
fn main() {
strange_unique_prime_triplets(30, true);
strange_unique_prime_triplets(1000, false);
}
- Output:
Strange unique prime triplets < 30: 3 + 5 + 11 = 19 3 + 5 + 23 = 31 3 + 5 + 29 = 37 3 + 7 + 13 = 23 3 + 7 + 19 = 29 3 + 11 + 17 = 31 3 + 11 + 23 = 37 3 + 11 + 29 = 43 3 + 17 + 23 = 43 5 + 7 + 11 = 23 5 + 7 + 17 = 29 5 + 7 + 19 = 31 5 + 7 + 29 = 41 5 + 11 + 13 = 29 5 + 13 + 19 = 37 5 + 13 + 23 = 41 5 + 13 + 29 = 47 5 + 17 + 19 = 41 5 + 19 + 23 = 47 5 + 19 + 29 = 53 7 + 11 + 13 = 31 7 + 11 + 19 = 37 7 + 11 + 23 = 41 7 + 11 + 29 = 47 7 + 13 + 17 = 37 7 + 13 + 23 = 43 7 + 17 + 19 = 43 7 + 17 + 23 = 47 7 + 17 + 29 = 53 7 + 23 + 29 = 59 11 + 13 + 17 = 41 11 + 13 + 19 = 43 11 + 13 + 23 = 47 11 + 13 + 29 = 53 11 + 17 + 19 = 47 11 + 19 + 23 = 53 11 + 19 + 29 = 59 13 + 17 + 23 = 53 13 + 17 + 29 = 59 13 + 19 + 29 = 61 17 + 19 + 23 = 59 19 + 23 + 29 = 71 Count of strange unique prime triplets < 30 is 42. Count of strange unique prime triplets < 1000 is 241580.
Scala
Scala3 ready
val primeStream5 = LazyList.from(5, 6)
.flatMap(n => Seq(n, n + 2))
.filter(p => (5 to math.sqrt(p).floor.toInt by 6).forall(a => p % a > 0 && p % (a + 2) > 0))
val primes = LazyList(2, 3) ++ primeStream5
def isPrime(n: Int): Boolean =
if (n < 5) (n | 1) == 3
else primes.takeWhile(_ <= math.sqrt(n)).forall(n % _ > 0)
def triplets(limit: Int): Iterator[Seq[Int]] =
primes.takeWhile(_ <= limit)
.combinations{3}
.filter(primeTriplet => isPrime(primeTriplet.sum))
@main def main: Unit = {
for (list <- triplets(30)) {
val Seq(k, l, m) = list
println(f"$k%2d + $l%2d + $m%2d = ${list.sum}%2d")
}
for (limit <- Seq(30, 1000)) {
val start = System.currentTimeMillis
val num = triplets(limit).length
val duration = System.currentTimeMillis - start
println(f"number of prime triplets up to $limit%4d is $num%6d [time(ms): $duration%4d]")
}
}
- Output:
3 + 5 + 11 = 19 3 + 5 + 23 = 31 3 + 5 + 29 = 37 3 + 7 + 13 = 23 3 + 7 + 19 = 29 3 + 11 + 17 = 31 3 + 11 + 23 = 37 3 + 11 + 29 = 43 3 + 17 + 23 = 43 5 + 7 + 11 = 23 5 + 7 + 17 = 29 5 + 7 + 19 = 31 5 + 7 + 29 = 41 5 + 11 + 13 = 29 5 + 13 + 19 = 37 5 + 13 + 23 = 41 5 + 13 + 29 = 47 5 + 17 + 19 = 41 5 + 19 + 23 = 47 5 + 19 + 29 = 53 7 + 11 + 13 = 31 7 + 11 + 19 = 37 7 + 11 + 23 = 41 7 + 11 + 29 = 47 7 + 13 + 17 = 37 7 + 13 + 23 = 43 7 + 17 + 19 = 43 7 + 17 + 23 = 47 7 + 17 + 29 = 53 7 + 23 + 29 = 59 11 + 13 + 17 = 41 11 + 13 + 19 = 43 11 + 13 + 23 = 47 11 + 13 + 29 = 53 11 + 17 + 19 = 47 11 + 19 + 23 = 53 11 + 19 + 29 = 59 13 + 17 + 23 = 53 13 + 17 + 29 = 59 13 + 19 + 29 = 61 17 + 19 + 23 = 59 19 + 23 + 29 = 71 number of prime triplets up to 30 is 42 [time(ms): 2] number of prime triplets up to 1000 is 241580 [time(ms): 1271]
Sidef
for n in (30, 1000) {
var triplets = []
combinations(n.primes, 3, {|*a|
triplets << a if a.sum.is_prime
})
if (n == 30) {
say "Unique prime triplets (p,q,r) <= #{n} such that p+q+r is prime:"
triplets.slices(6).each{.join(' ').say}
}
printf("Found %d strange unique prime triplets up to %s.\n", triplets.len, n)
}
- Output:
Unique prime triplets (p,q,r) <= 30 such that p+q+r is prime: [3, 5, 11] [3, 5, 23] [3, 5, 29] [3, 7, 13] [3, 7, 19] [3, 11, 17] [3, 11, 23] [3, 11, 29] [3, 17, 23] [5, 7, 11] [5, 7, 17] [5, 7, 19] [5, 7, 29] [5, 11, 13] [5, 13, 19] [5, 13, 23] [5, 13, 29] [5, 17, 19] [5, 19, 23] [5, 19, 29] [7, 11, 13] [7, 11, 19] [7, 11, 23] [7, 11, 29] [7, 13, 17] [7, 13, 23] [7, 17, 19] [7, 17, 23] [7, 17, 29] [7, 23, 29] [11, 13, 17] [11, 13, 19] [11, 13, 23] [11, 13, 29] [11, 17, 19] [11, 19, 23] [11, 19, 29] [13, 17, 23] [13, 17, 29] [13, 19, 29] [17, 19, 23] [19, 23, 29] Found 42 strange unique prime triplets up to 30. Found 241580 strange unique prime triplets up to 1000.
Swift
import Foundation
func primeSieve(limit: Int) -> [Bool] {
guard limit > 0 else {
return []
}
var sieve = Array(repeating: true, count: limit)
sieve[0] = false
if limit > 1 {
sieve[1] = false
}
if limit > 4 {
for i in stride(from: 4, to: limit, by: 2) {
sieve[i] = false
}
}
var p = 3
while true {
var q = p * p
if q >= limit {
break
}
if sieve[p] {
let inc = 2 * p
while q < limit {
sieve[q] = false
q += inc
}
}
p += 2
}
return sieve
}
func strangeUniquePrimeTriplets(limit: Int, verbose: Bool) {
guard limit > 5 else {
return;
}
let sieve = primeSieve(limit: 3 * limit)
var primes: [Int] = []
for p in stride(from: 3, to: limit, by: 2) {
if sieve[p] {
primes.append(p)
}
}
let n = primes.count
var count = 0
if verbose {
print("Strange unique prime triplets < \(limit):")
}
for i in (0..<n - 2) {
for j in (i + 1..<n - 1) {
for k in (j + 1..<n) {
let sum = primes[i] + primes[j] + primes[k]
if sieve[sum] {
count += 1
if verbose {
print(String(format: "%2d + %2d + %2d = %2d",
primes[i], primes[j], primes[k], sum))
}
}
}
}
}
print("\nCount of strange unique prime triplets < \(limit) is \(count).")
}
strangeUniquePrimeTriplets(limit: 30, verbose: true)
strangeUniquePrimeTriplets(limit: 1000, verbose: false)
- Output:
Strange unique prime triplets < 30: 3 + 5 + 11 = 19 3 + 5 + 23 = 31 3 + 5 + 29 = 37 3 + 7 + 13 = 23 3 + 7 + 19 = 29 3 + 11 + 17 = 31 3 + 11 + 23 = 37 3 + 11 + 29 = 43 3 + 17 + 23 = 43 5 + 7 + 11 = 23 5 + 7 + 17 = 29 5 + 7 + 19 = 31 5 + 7 + 29 = 41 5 + 11 + 13 = 29 5 + 13 + 19 = 37 5 + 13 + 23 = 41 5 + 13 + 29 = 47 5 + 17 + 19 = 41 5 + 19 + 23 = 47 5 + 19 + 29 = 53 7 + 11 + 13 = 31 7 + 11 + 19 = 37 7 + 11 + 23 = 41 7 + 11 + 29 = 47 7 + 13 + 17 = 37 7 + 13 + 23 = 43 7 + 17 + 19 = 43 7 + 17 + 23 = 47 7 + 17 + 29 = 53 7 + 23 + 29 = 59 11 + 13 + 17 = 41 11 + 13 + 19 = 43 11 + 13 + 23 = 47 11 + 13 + 29 = 53 11 + 17 + 19 = 47 11 + 19 + 23 = 53 11 + 19 + 29 = 59 13 + 17 + 23 = 53 13 + 17 + 29 = 59 13 + 19 + 29 = 61 17 + 19 + 23 = 59 19 + 23 + 29 = 71 Count of strange unique prime triplets < 30 is 42. Count of strange unique prime triplets < 1000 is 241580.
Visual Basic .NET
Imports DT = System.DateTime
Module Module1
Iterator Function Primes(lim As Integer) As IEnumerable(Of Integer)
Dim flags(lim) As Boolean
Dim j = 2
Dim d = 3
Dim sq = 4
While sq <= lim
If Not flags(j) Then
Yield j
For k = sq To lim Step j
flags(k) = True
Next
End If
j += 1
d += 2
sq += d
End While
While j <= lim
If Not flags(j) Then
Yield j
End If
j += 1
End While
End Function
Sub Main()
For Each lmt In {90, 300, 3000, 30000, 111000}
Dim pr = Primes(lmt).Skip(1).ToList()
Dim st = DT.Now
Dim f = 0
Dim r As New List(Of String)
Dim i = -1
Dim m = lmt \ 3
Dim h = m
While i < 0
i = pr.IndexOf(h)
h -= 1
End While
Dim j = i - 1
Dim k = j - 1
For a = 0 To k
Dim pra = pr(a)
For b = a + 1 To j
Dim prab = pra + pr(b)
For c = b + 1 To i
Dim d = prab + pr(c)
If Not pr.Contains(d) Then
Continue For
End If
f += 1
If lmt < 100 Then
r.Add(String.Format("{3,5} = {0,2} + {1,2} + {2,2}", pra, pr(b), pr(c), d))
End If
Next
Next
Next
Dim s = "s.u.p.t.s under "
r.Sort()
If r.Count > 0 Then
Console.WriteLine("{0}{1}:" + vbNewLine + "{2}", s, m, String.Join(vbNewLine, r))
End If
If lmt > 100 Then
Console.WriteLine("Count of {0}{1,6:n0}: {2,13:n0} {3} sec", s, m, f, (DT.Now - st).ToString().Substring(6))
End If
Next
End Sub
End Module
- Output:
Same as C#
Wren
Basic
import "./math" for Int
import "./iterate" for Stepped
import "./fmt" for Fmt
var strangePrimes = Fn.new { |n, countOnly|
var c = 0
var s
for (i in Stepped.new(3..n-4, 2)) {
if (Int.isPrime(i)) {
for (j in Stepped.new(i+2..n-2, 2)) {
if (Int.isPrime(j)) {
for (k in Stepped.new(j+2..n, 2)) {
if (Int.isPrime(k) && Int.isPrime(s = i + j + k)) {
c = c + 1
if (!countOnly) Fmt.print("$2d: $2d + $2d + $2d = $2d", c, i, j, k, s)
}
}
}
}
}
}
return c
}
System.print("Unique prime triples under 30 which sum to a prime:")
strangePrimes.call(29, false)
var c = strangePrimes.call(999, true)
Fmt.print("\nThere are $,d unique prime triples under 1,000 which sum to a prime.", c)
- Output:
Unique prime triples under 30 which sum to a prime: 1: 3 + 5 + 11 = 19 2: 3 + 5 + 23 = 31 3: 3 + 5 + 29 = 37 4: 3 + 7 + 13 = 23 5: 3 + 7 + 19 = 29 6: 3 + 11 + 17 = 31 7: 3 + 11 + 23 = 37 8: 3 + 11 + 29 = 43 9: 3 + 17 + 23 = 43 10: 5 + 7 + 11 = 23 11: 5 + 7 + 17 = 29 12: 5 + 7 + 19 = 31 13: 5 + 7 + 29 = 41 14: 5 + 11 + 13 = 29 15: 5 + 13 + 19 = 37 16: 5 + 13 + 23 = 41 17: 5 + 13 + 29 = 47 18: 5 + 17 + 19 = 41 19: 5 + 19 + 23 = 47 20: 5 + 19 + 29 = 53 21: 7 + 11 + 13 = 31 22: 7 + 11 + 19 = 37 23: 7 + 11 + 23 = 41 24: 7 + 11 + 29 = 47 25: 7 + 13 + 17 = 37 26: 7 + 13 + 23 = 43 27: 7 + 17 + 19 = 43 28: 7 + 17 + 23 = 47 29: 7 + 17 + 29 = 53 30: 7 + 23 + 29 = 59 31: 11 + 13 + 17 = 41 32: 11 + 13 + 19 = 43 33: 11 + 13 + 23 = 47 34: 11 + 13 + 29 = 53 35: 11 + 17 + 19 = 47 36: 11 + 19 + 23 = 53 37: 11 + 19 + 29 = 59 38: 13 + 17 + 23 = 53 39: 13 + 17 + 29 = 59 40: 13 + 19 + 29 = 61 41: 17 + 19 + 23 = 59 42: 19 + 23 + 29 = 71 There are 241,580 unique prime triples under 1,000 which sum to a prime.
Faster
The following version uses a prime sieve and is about 17 times faster than the 'basic' version.
import "./math" for Int
import "./fmt" for Fmt
var max = 1000
var sieved = Int.primeSieve(3*max, false) // includes composites
var p = Int.primeSieve(max, true) // primes only
var strangePrimes = Fn.new { |n, countOnly|
var c = 0
var m = 0
while (m < p.count && p[m] <= n) m = m + 1
var r
var s
for (i in 1...m-2) {
for (j in i+1...m-1) {
r = p[i] + p[j]
for (k in j+1...m) {
if (!sieved[s = r + p[k]]) {
c = c + 1
if (!countOnly) Fmt.print("$2d: $2d + $2d + $2d = $2d", c, p[i], p[j], p[k], s)
}
}
}
}
return c
}
System.print("Unique prime triples under 30 which sum to a prime:")
strangePrimes.call(29, false)
var c = strangePrimes.call(999, true)
Fmt.print("\nThere are $,d unique prime triples under 1,000 which sum to a prime.", c)
- Output:
Same as 'basic' version.
XPL0
func IsPrime(N); \Return 'true' if N is prime
int N, I;
[if N <= 2 then return N = 2;
if (N&1) = 0 then \even >2\ return false;
for I:= 3 to sqrt(N) do
[if rem(N/I) = 0 then return false;
I:= I+1;
];
return true;
]; \IsPrime
int Primes, Cnt, P, M, N, S;
[Primes:= [2, 3, 5, 7, 11, 13, 17, 19, 23, 29];
Format(2, 0);
Cnt:= 0;
for P:= 2 to 9 do
for M:= 1 to P-1 do
for N:= 0 to M-1 do
[S:= Primes(N) + Primes(M) + Primes(P);
if IsPrime(S) then
[Cnt:= Cnt+1;
RlOut(0, float(Cnt));
Text(0, ": ");
RlOut(0, float(Primes(N)));
Text(0, " + ");
RlOut(0, float(Primes(M)));
Text(0, " + ");
RlOut(0, float(Primes(P)));
Text(0, " = ");
RlOut(0, float(S));
CrLf(0);
];
];
]- Output:
1: 3 + 5 + 11 = 19 2: 5 + 7 + 11 = 23 3: 3 + 7 + 13 = 23 4: 5 + 11 + 13 = 29 5: 7 + 11 + 13 = 31 6: 5 + 7 + 17 = 29 7: 3 + 11 + 17 = 31 8: 7 + 13 + 17 = 37 9: 11 + 13 + 17 = 41 10: 3 + 7 + 19 = 29 11: 5 + 7 + 19 = 31 12: 7 + 11 + 19 = 37 13: 5 + 13 + 19 = 37 14: 11 + 13 + 19 = 43 15: 5 + 17 + 19 = 41 16: 7 + 17 + 19 = 43 17: 11 + 17 + 19 = 47 18: 3 + 5 + 23 = 31 19: 3 + 11 + 23 = 37 20: 7 + 11 + 23 = 41 21: 5 + 13 + 23 = 41 22: 7 + 13 + 23 = 43 23: 11 + 13 + 23 = 47 24: 3 + 17 + 23 = 43 25: 7 + 17 + 23 = 47 26: 13 + 17 + 23 = 53 27: 5 + 19 + 23 = 47 28: 11 + 19 + 23 = 53 29: 17 + 19 + 23 = 59 30: 3 + 5 + 29 = 37 31: 5 + 7 + 29 = 41 32: 3 + 11 + 29 = 43 33: 7 + 11 + 29 = 47 34: 5 + 13 + 29 = 47 35: 11 + 13 + 29 = 53 36: 7 + 17 + 29 = 53 37: 13 + 17 + 29 = 59 38: 5 + 19 + 29 = 53 39: 11 + 19 + 29 = 59 40: 13 + 19 + 29 = 61 41: 7 + 23 + 29 = 59 42: 19 + 23 + 29 = 71
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