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Concatenate two primes is also prime

From Rosetta Code
Concatenate two primes is also prime 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.
Task

Find and show here when the concatenation of two primes   (p1,  p2)   shown in base ten is also prime,   where   p1,   p2  <  100.

ALGOL 68[edit]

BEGIN  # find primes whose decimal representation is the concatenation of 2 primes #
INT max low prime = 99; # for the task, only need component primes up to 99 #
INT max prime = max low prime * max low prime;
# sieve the primes to max prime #
PR read "primes.incl.a68" PR
[]BOOL prime = PRIMESIEVE max prime;
# construct a list of the primes up to the maximum low prime to consider #
[]INT low prime = EXTRACTPRIMESUPTO max low prime FROMPRIMESIEVE prime;
# find the primes that can be concatenated to form another prime #
[ 1 : max prime ]BOOL concat prime; FOR i TO UPB concat prime DO concat prime[ i ] := FALSE OD;
# note that all possible concatenated primes have at least 2 digits, so the final digit can't be 2 #
# ( or 5 but we don't check that here ) #
FOR i TO UPB low prime WHILE low prime[ i ] <= max low prime DO
INT p1 = low prime[ i ];
FOR j FROM 2 TO UPB low prime WHILE low prime[ j ] <= max low prime DO
INT p2 = low prime[ j ];
INT pc = ( p1 * IF p2 < 10 THEN 10 ELSE 100 FI ) + p2;
concat prime[ pc ] := prime[ pc ]
OD
OD;
# show the concatenated primes #
INT c count := 0;
FOR n TO UPB concat prime DO
IF concat prime[ n ] THEN
print( ( whole( n, -5 ) ) );
IF ( c count +:= 1 ) MOD 10 = 0 THEN print( ( newline ) ) FI
FI
OD;
print( ( newline, newline, "Found ", whole( c count, 0 ), " concat primes", newline ) )
END
Output:
   23   37   53   73  113  137  173  193  197  211
  223  229  233  241  271  283  293  311  313  317
  331  337  347  353  359  367  373  379  383  389
  397  433  523  541  547  571  593  613  617  673
  677  719  733  743  761  773  797  977 1117 1123
 1129 1153 1171 1319 1361 1367 1373 1723 1741 1747
 1753 1759 1783 1789 1913 1931 1973 1979 1997 2311
 2341 2347 2371 2383 2389 2917 2953 2971 3119 3137
 3167 3719 3761 3767 3779 3797 4111 4129 4153 4159
 4337 4373 4397 4723 4729 4759 4783 4789 5323 5347
 5923 5953 6113 6131 6143 6173 6197 6719 6737 6761
 6779 7129 7159 7331 7919 7937 8311 8317 8329 8353
 8389 8923 8929 8941 8971 9719 9743 9767

Found 128 concat primes

ALGOL W[edit]

The Algol W for loop allows the loop counter values to be specified as a list - as there are only 25 primes below 100, this feature is used here to save looking through the sieve for the low primes.

begin % find primes whose decimal representation is the concatenation of 2 primes %
integer MAX_PRIME;
MAX_PRIME := 99 * 99;
begin
logical array isPrime ( 1 :: MAX_PRIME );
logical array concatPrime ( 1 :: MAX_PRIME );
integer cCount;
 % sieve the primes to MAX_PRIME  %
isPrime( 1 ) := false; isPrime( 2 ) := true;
for i := 3 step 2 until MAX_PRIME do isPrime( i ) := true;
for i := 4 step 2 until MAX_PRIME do isPrime( i ) := false;
for i := 3 step 2 until truncate( sqrt( MAX_PRIME ) ) do begin
integer ii; ii := i + i;
if isPrime( i ) then for pr := i * i step ii until MAX_PRIME do isPrime( pr ) := false
end for_i ;
 % find the concatenated primes, note their final digit can't be 2 or 5  %
for i := 1 until MAX_PRIME do concatPrime( i ) := false;
cCount := 0;
for p1 := 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37
, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97 do begin
for p2 := 3, 7, 11, 13, 17, 19, 23, 29, 31, 37
, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97 do begin
integer pc;
pc := ( p1 * ( if p2 < 10 then 10 else 100 ) ) + p2;
concatPrime( pc ) := isPrime( pc )
end for_p2
end for_p1;
 % print the concatenated primes  %
cCount := 0;
for i := 1 until MAX_PRIME do begin
if concatPrime( i ) then begin
writeon( i_w := 5, s_w := 0, i );
cCount := cCount + 1;
if cCount rem 10 = 0 then write()
end if_concatPrime_i
end for_i;
write();write( i_w := 1, s_w := 0, "Found ", cCount, " concat primes" )
end
end.
Output:
   23   37   53   73  113  137  173  193  197  211
  223  229  233  241  271  283  293  311  313  317
  331  337  347  353  359  367  373  379  383  389
  397  433  523  541  547  571  593  613  617  673
  677  719  733  743  761  773  797  977 1117 1123
 1129 1153 1171 1319 1361 1367 1373 1723 1741 1747
 1753 1759 1783 1789 1913 1931 1973 1979 1997 2311
 2341 2347 2371 2383 2389 2917 2953 2971 3119 3137
 3167 3719 3761 3767 3779 3797 4111 4129 4153 4159
 4337 4373 4397 4723 4729 4759 4783 4789 5323 5347
 5923 5953 6113 6131 6143 6173 6197 6719 6737 6761
 6779 7129 7159 7331 7919 7937 8311 8317 8329 8353
 8389 8923 8929 8941 8971 9719 9743 9767

Found 128 concat primes

AWK[edit]

 
# syntax: GAWK -f CONCATENATE_TWO_PRIMES_IS_ALSO_PRIME.AWK
#
# sorting:
# PROCINFO["sorted_in"] is used by GAWK
# SORTTYPE is used by Thompson Automation's TAWK
#
BEGIN {
start = 1
stop = 99
for (i=start; i<=stop; i++) {
if (is_prime(i)) {
for (j=start; j<=stop; j++) {
if (is_prime(j)) {
if (is_prime(i j)) {
arr[i j] = ""
}
}
}
}
}
PROCINFO["sorted_in"] = "@ind_num_asc" ; SORTTYPE = 1
for (i in arr) {
printf("%5d%1s",i,++count%10?"":"\n")
}
printf("\nConcatenate two primes is also prime %d-%d: %d\n",start,stop,count)
exit(0)
}
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:
   23    37    53    73   113   137   173   193   197   211
  223   229   233   241   271   283   293   311   313   317
  331   337   347   353   359   367   373   379   383   389
  397   433   523   541   547   571   593   613   617   673
  677   719   733   743   761   773   797   977  1117  1123
 1129  1153  1171  1319  1361  1367  1373  1723  1741  1747
 1753  1759  1783  1789  1913  1931  1973  1979  1997  2311
 2341  2347  2371  2383  2389  2917  2953  2971  3119  3137
 3167  3719  3761  3767  3779  3797  4111  4129  4153  4159
 4337  4373  4397  4723  4729  4759  4783  4789  5323  5347
 5923  5953  6113  6131  6143  6173  6197  6719  6737  6761
 6779  7129  7159  7331  7919  7937  8311  8317  8329  8353
 8389  8923  8929  8941  8971  9719  9743  9767
Concatenate two primes is also prime 1-99: 128

Factor[edit]

Works with: Factor version 0.99 2021-06-02
USING: formatting grouping io kernel math.parser math.primes
present prettyprint sequences sets sorting ;
 
"Concatenated-pair primes from primes < 100:" print nl
99 primes-upto [ present ] map dup [ append dec> ] cartesian-map
concat [ prime? ] filter members natural-sort [ length ] keep
8 group simple-table. "\nFound %d concatenated primes.\n" printf
Output:
Concatenated-pair primes from primes < 100:

23   37   53   73   113  137  173  193
197  211  223  229  233  241  271  283
293  311  313  317  331  337  347  353
359  367  373  379  383  389  397  433
523  541  547  571  593  613  617  673
677  719  733  743  761  773  797  977
1117 1123 1129 1153 1171 1319 1361 1367
1373 1723 1741 1747 1753 1759 1783 1789
1913 1931 1973 1979 1997 2311 2341 2347
2371 2383 2389 2917 2953 2971 3119 3137
3167 3719 3761 3767 3779 3797 4111 4129
4153 4159 4337 4373 4397 4723 4729 4759
4783 4789 5323 5347 5923 5953 6113 6131
6143 6173 6197 6719 6737 6761 6779 7129
7159 7331 7919 7937 8311 8317 8329 8353
8389 8923 8929 8941 8971 9719 9743 9767

Found 128 concatenated primes.

Go[edit]

Library: Go-rcu
package main
 
import (
"fmt"
"rcu"
"sort"
)
 
func main() {
const LIMIT = 99
primes := rcu.Primes(LIMIT)
rmap := make(map[int]bool)
for _, p := range primes {
for _, q := range primes {
var pq int
if q < 10 {
pq = p*10 + q
} else {
pq = p*100 + q
}
if rcu.IsPrime(pq) {
rmap[pq] = true
}
}
}
results := make([]int, len(rmap))
i := 0
for k := range rmap {
results[i] = k
i++
}
sort.Ints(results)
fmt.Println("Two primes under 100 concatenated together to form another prime:")
for i, p := range results {
fmt.Printf("%5s ", rcu.Commatize(p))
if (i+1)%10 == 0 {
fmt.Println()
}
}
fmt.Println("\n\nFound", len(results), "such concatenated primes.")
}
Output:
Two primes under 100 concatenated together to form another prime:
   23    37    53    73   113   137   173   193   197   211 
  223   229   233   241   271   283   293   311   313   317 
  331   337   347   353   359   367   373   379   383   389 
  397   433   523   541   547   571   593   613   617   673 
  677   719   733   743   761   773   797   977 1,117 1,123 
1,129 1,153 1,171 1,319 1,361 1,367 1,373 1,723 1,741 1,747 
1,753 1,759 1,783 1,789 1,913 1,931 1,973 1,979 1,997 2,311 
2,341 2,347 2,371 2,383 2,389 2,917 2,953 2,971 3,119 3,137 
3,167 3,719 3,761 3,767 3,779 3,797 4,111 4,129 4,153 4,159 
4,337 4,373 4,397 4,723 4,729 4,759 4,783 4,789 5,323 5,347 
5,923 5,953 6,113 6,131 6,143 6,173 6,197 6,719 6,737 6,761 
6,779 7,129 7,159 7,331 7,919 7,937 8,311 8,317 8,329 8,353 
8,389 8,923 8,929 8,941 8,971 9,719 9,743 9,767 

Found 128 such concatenated primes.

jq[edit]

Works with: jq

Works with gojq, the Go implementation of jq

Preliminaries

def is_prime:
. as $n
| if ($n < 2) then false
elif ($n % 2 == 0) then $n == 2
elif ($n % 3 == 0) then $n == 3
elif ($n % 5 == 0) then $n == 5
elif ($n % 7 == 0) then $n == 7
elif ($n % 11 == 0) then $n == 11
elif ($n % 13 == 0) then $n == 13
elif ($n % 17 == 0) then $n == 17
elif ($n % 19 == 0) then $n == 19
else {i:23}
| until( (.i * .i) > $n or ($n % .i == 0); .i += 2)
| .i * .i > $n
end;
 
# Emit an array of primes less than `.`
def primes:
if . < 2 then []
else
[2] + [range(3; .; 2) | select(is_prime)]
end;
 
# Pretty-printing
def nwise($n):
def n: if length <= $n then . else .[0:$n] , (.[$n:] | n) end;
n;
 
def lpad($len): tostring | ($len - length) as $l | (" " * $l)[:$l] + .;
 

The task

# Emit [p1,p2] where p1 < p2 < . and the concatenation is prime
def concatenative_primes:
primes
| range(0;length) as $i
| range($i+1;length) as $j
| [.[$i], .[$j]], [.[$j], .[$i]]
| select( map(tostring) | add | tonumber | is_prime);
 
[100 | concatenative_primes | join("||")]
| (nwise(10) | map(lpad(6)) | join(" "))
Output:
  2||3  2||11  2||23  2||29  2||41  2||71  2||83   5||3   3||7   7||3
 3||11  11||3  3||13  3||17  17||3  19||3  23||3  29||3  3||31  31||3
 3||37  37||3  43||3  3||47  3||53  3||59  59||3  61||3  3||67  67||3
 3||73  73||3  3||79  3||83  3||89  3||97  5||23  5||41  5||47  5||71
 13||7  7||19  19||7  31||7  7||43  7||61  61||7  67||7  7||73  79||7
 7||97  97||7 11||17 11||23 23||11 11||29 41||11 11||53 11||71 83||11
13||19 19||13 13||61 61||13 13||67 13||73 17||23 29||17 17||41 17||47
17||53 17||59 17||83 83||17 17||89 19||31 31||19 37||19 67||19 19||73
19||79 79||19 19||97 97||19 23||41 23||47 47||23 53||23 59||23 23||71
23||83 23||89 89||23 41||29 47||29 29||53 29||71 71||29 83||29 89||29
31||37 61||31 31||67 73||31 43||37 37||61 37||67 67||37 37||79 79||37
37||97 41||53 41||59 89||41 61||43 43||73 43||97 97||43 53||47 47||59
47||83 47||89 59||53 83||53 71||59 67||61 61||73 61||97 67||79 97||67

Julia[edit]

using Primes
 
function catprimes()
found, pri = Int[], primes(100)
for x in pri, y in pri
n = x + y * (x < 10 ? 10 : 100)
isprime(n) && push!(found, n)
end
return sort!(unique(found))
end
 
foreach(p -> print(lpad(last(p), 5), first(p) % 16 == 0 ? "\n" : ""),
catprimes() |> enumerate)
 
Output:
   23   37   53   73  113  137  173  193  197  211  223  229  233  241  271  283
  293  311  313  317  331  337  347  353  359  367  373  379  383  389  397  433
  523  541  547  571  593  613  617  673  677  719  733  743  761  773  797  977
 1117 1123 1129 1153 1171 1319 1361 1367 1373 1723 1741 1747 1753 1759 1783 1789
 1913 1931 1973 1979 1997 2311 2341 2347 2371 2383 2389 2917 2953 2971 3119 3137
 3167 3719 3761 3767 3779 3797 4111 4129 4153 4159 4337 4373 4397 4723 4729 4759
 4783 4789 5323 5347 5923 5953 6113 6131 6143 6173 6197 6719 6737 6761 6779 7129
 7159 7331 7919 7937 8311 8317 8329 8353 8389 8923 8929 8941 8971 9719 9743 9767

Mathematica/Wolfram Language[edit]

Select[Catenate /* FromDigits /@ Map[IntegerDigits, Tuples[Prime[Range[PrimePi[100]]], 2], {2}], PrimeQ] // Union
Output:
{23, 37, 53, 73, 113, 137, 173, 193, 197, 211, 223, 229, 233, 241, 271, 283, 293, 311, 313, 317, 331, 337, 347, 353, 359, 367, 373, 379, 383, 389, 397, 433, 523, 541, 547, 571, 593, 613, 617, 673, 677, 719, 733, 743, 761, 773, 797, 977, 1117, 1123, 1129, 1153, 1171, 1319, 1361, 1367, 1373, 1723, 1741, 1747, 1753, 1759, 1783, 1789, 1913, 1931, 1973, 1979, 1997, 2311, 2341, 2347, 2371, 2383, 2389, 2917, 2953, 2971, 3119, 3137, 3167, 3719, 3761, 3767, 3779, 3797, 4111, 4129, 4153, 4159, 4337, 4373, 4397, 4723, 4729, 4759, 4783, 4789, 5323, 5347, 5923, 5953, 6113, 6131, 6143, 6173, 6197, 6719, 6737, 6761, 6779, 7129, 7159, 7331, 7919, 7937, 8311, 8317, 8329, 8353, 8389, 8923, 8929, 8941, 8971, 9719, 9743, 9767}

Nim[edit]

import strutils, sugar
 
func isPrime(n: Positive): bool =
if n == 1: return false
if (n and 1) == 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
if n mod (d + 2) == 0: return false
inc d, 6
result = true
 
# List of primes.
const Primes = collect(newSeq, for n in 2..<100: (if n.isPrime: n))
 
var concatPrimes: set[0..9999] # Using a set to eliminate duplicates and avoid sort.
for p1 in Primes:
for p2 in Primes:
let n = p2 + p1 * (if p2 < 10: 10 else: 100)
if n.isPrime:
concatPrimes.incl n
 
echo "Found $# primes which are a concatenation of two primes below 100:".format(concatPrimes.len)
var i = 1
for n in concatPrimes:
stdout.write ($n).align(4), if i mod 16 == 0: '\n' else: ' '
inc i
Output:
Found 128 primes which are a concatenation of two primes below 100:
  23   37   53   73  113  137  173  193  197  211  223  229  233  241  271  283
 293  311  313  317  331  337  347  353  359  367  373  379  383  389  397  433
 523  541  547  571  593  613  617  673  677  719  733  743  761  773  797  977
1117 1123 1129 1153 1171 1319 1361 1367 1373 1723 1741 1747 1753 1759 1783 1789
1913 1931 1973 1979 1997 2311 2341 2347 2371 2383 2389 2917 2953 2971 3119 3137
3167 3719 3761 3767 3779 3797 4111 4129 4153 4159 4337 4373 4397 4723 4729 4759
4783 4789 5323 5347 5923 5953 6113 6131 6143 6173 6197 6719 6737 6761 6779 7129
7159 7331 7919 7937 8311 8317 8329 8353 8389 8923 8929 8941 8971 9719 9743 9767

Perl[edit]

#!/usr/bin/perl
 
use strict; # https://rosettacode.org/wiki/Concatenate_two_primes_is_also_prime
use warnings;
use ntheory qw( primes is_prime );
use List::Util qw( uniq );
 
my @primes = @{ primes(100) };
my @valid = uniq sort { $a <=> $b } grep is_prime($_),
map { my $prefix = $_; map "$prefix$_", @primes } @primes;
print @valid . " primes found\n\n@valid\n" =~ s/.{79}\K /\n/gr;
Output:
128 primes found

23 37 53 73 113 137 173 193 197 211 223 229 233 241 271 283 293 311 313 317 331
337 347 353 359 367 373 379 383 389 397 433 523 541 547 571 593 613 617 673 677
719 733 743 761 773 797 977 1117 1123 1129 1153 1171 1319 1361 1367 1373 1723 1741
1747 1753 1759 1783 1789 1913 1931 1973 1979 1997 2311 2341 2347 2371 2383 2389
2917 2953 2971 3119 3137 3167 3719 3761 3767 3779 3797 4111 4129 4153 4159 4337
4373 4397 4723 4729 4759 4783 4789 5323 5347 5923 5953 6113 6131 6143 6173 6197
6719 6737 6761 6779 7129 7159 7331 7919 7937 8311 8317 8329 8353 8389 8923 8929
8941 8971 9719 9743 9767

Phix[edit]

with javascript_semantics
sequence primes = get_primes_le(100),
         result = {}
for i=1 to length(primes) do
    integer p = primes[i]
    for j=1 to length(primes) do
        integer q = primes[j],
                pq = p*iff(q<10?10:100)+q
        if is_prime(pq) then result &= pq end if
    end for
end for
result = unique(deep_copy(result)) -- (see note)
printf(1,"Found %d such primes: %V\n",{length(result),shorten(result,"",5)})

Note: The deep_copy() [for pwa/p2js] on such calls to unique() would be unnecessary were result routine-local, due to automatic pbr, but here that variable is file-level.
While the (human-readable) error message (without it) is deep within sort(), the call stack makes it clear where that call should best go.

Output:
Found 128 such primes: {23,37,53,73,113,"...",8941,8971,9719,9743,9767}

Raku[edit]

Inefficient, but for a limit of 100, who cares?

my @p = ^1e2 .grep: *.is-prime;
 
say display ( @p X~ @p ).grep( *.is-prime ).unique.sort( +* );
 
sub display ($list, :$cols = 10, :$fmt = '%6d', :$title = "{+$list} matching:\n") {
cache $list;
$title ~ $list.batch($cols)».fmt($fmt).join: "\n"
}
Output:
128 matching:
    23     37     53     73    113    137    173    193    197    211
   223    229    233    241    271    283    293    311    313    317
   331    337    347    353    359    367    373    379    383    389
   397    433    523    541    547    571    593    613    617    673
   677    719    733    743    761    773    797    977   1117   1123
  1129   1153   1171   1319   1361   1367   1373   1723   1741   1747
  1753   1759   1783   1789   1913   1931   1973   1979   1997   2311
  2341   2347   2371   2383   2389   2917   2953   2971   3119   3137
  3167   3719   3761   3767   3779   3797   4111   4129   4153   4159
  4337   4373   4397   4723   4729   4759   4783   4789   5323   5347
  5923   5953   6113   6131   6143   6173   6197   6719   6737   6761
  6779   7129   7159   7331   7919   7937   8311   8317   8329   8353
  8389   8923   8929   8941   8971   9719   9743   9767

REXX[edit]

/*REXX pgm finds base ten neighbor primes P1 & P2,  when concatenated,  is also a prime.*/
parse arg hip cols . /*obtain optional arguments from the CL*/
if hip=='' | hip=="," then hip= 100 /*Not specified? Then use the default.*/
if cols=='' | cols=="," then cols= 10 /* " " " " " " */
call genP /*build array of semaphores for primes.*/
title= ' concatenation of two neighbor primes (P1, P2) in base ten that results in' ,
"a prime, where P1 & P2 are <" commas(hip)
w= 10 /*width of a number in any column. */
if cols>0 then say ' index │'center(title, 1 + cols*(w+1) )
if cols>0 then say '───────┼'center("" , 1 + cols*(w+1), '─')
a.= 0 /*array of "2cat" primes */
do j=1 for ## /*search through specified primes. */
do k=1 for ##; cat2= @.j || @.k /*create a concatenated prime (base 10)*/
if !.cat2 then a.cat2= 1 /*flag it as being a "2cat" prime. */
end /*k*/ /* [↑] forgoes the need for sorting. */
end /*j*/
found= 0; idx= 1 /*initialize # of primes found; IDX. */
$=; do n=1 by 2 for hip**2%2 /*search for odd "cat2" primes. */
if \a.n then iterate /*search through allowable primes. */
found= found + 1 /*bump the number of primes found. */
if cols<=0 then iterate /*Build the list (to be shown later)? */
c= commas(n) /*maybe add commas to the number. */
$= $ right(c, max(w, length(c) ) ) /*add a prime ──► $ list, allow big #*/
if found//cols\==0 then iterate /*have we populated a line of output? */
say center(idx, 7)'│' substr($, 2); $= /*display what we have so far (cols). */
idx= idx + cols /*bump the index count for the output*/
end /*n*/
 
if $\=='' then say center(idx, 7)"│" substr($, 2) /*possible display residual output.*/
if cols>0 then say '───────┴'center("" , 1 + cols*(w+1), '─')
say
say 'Found ' commas(found) title
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: @.1=2; @.2=3; @.3=5; @.4=7; @.5=11 /*define some low primes. */
 !.=0;  !.2=1; !.3=1; !.5=1; !.7=1;  !.11=1 /* " " " " semaphores. */
#= 5; sq.#= @.#**2 /*the square of the highest low prime. */
do [email protected].#+2 by 2 to hip**2-1 /*find odd primes from here on. */
parse var j '' -1 _; if _==5 then iterate /*J ÷ by 5? (right digit).*/
if j//3==0 then iterate; if j//7==0 then iterate /*" " " 3? J ÷ by 7? */
do k=5 while sq.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; sq.#= j*j;  !.j= 1 /*bump # of Ps; assign next P; P²; P# */
if j<hip then ##= # /*find a shortcut for the 1st DO loop. */
end /*j*/; return
output   when using the default inputs:
 index │   concatenation of two neighbor primes (P1, P2) in base ten that results in a prime, where P1 & P2 are < 100
───────┼───────────────────────────────────────────────────────────────────────────────────────────────────────────────
   1   │         23         37         53         73        113        137        173        193        197        211
  11   │        223        229        233        241        271        283        293        311        313        317
  21   │        331        337        347        353        359        367        373        379        383        389
  31   │        397        433        523        541        547        571        593        613        617        673
  41   │        677        719        733        743        761        773        797        977      1,117      1,123
  51   │      1,129      1,153      1,171      1,319      1,361      1,367      1,373      1,723      1,741      1,747
  61   │      1,753      1,759      1,783      1,789      1,913      1,931      1,973      1,979      1,997      2,311
  71   │      2,341      2,347      2,371      2,383      2,389      2,917      2,953      2,971      3,119      3,137
  81   │      3,167      3,719      3,761      3,767      3,779      3,797      4,111      4,129      4,153      4,159
  91   │      4,337      4,373      4,397      4,723      4,729      4,759      4,783      4,789      5,323      5,347
  101  │      5,923      5,953      6,113      6,131      6,143      6,173      6,197      6,719      6,737      6,761
  111  │      6,779      7,129      7,159      7,331      7,919      7,937      8,311      8,317      8,329      8,353
  121  │      8,389      8,923      8,929      8,941      8,971      9,719      9,743      9,767
───────┴───────────────────────────────────────────────────────────────────────────────────────────────────────────────

Found  128  concatenation of two neighbor primes (P1, P2) in base ten that results in a prime, where P1 & P2 are < 100

Ring[edit]

 
load "stdlib.ring"
see "working..." + nl
see "Concatenate two primes is also prime:" + nl
row = 0
prime = []
 
for n = 1 to 100
for m = 1 to 100
ps1 = string(n)
ps2 = string(m)
ps12 = ps1 + ps2
ps21 = ps2 + ps1
p1 = number(ps12)
p2 = number(ps21)
if isprime(n) and isprime(m)
if isprime(p1)
pf = find(prime,p1)
if pf < 1
add(prime,p1)
ok
ok
if isprime(p2)
pf = find(prime,p2)
if pf < 1
add(prime,p2)
ok
ok
ok
next
next
 
prime = sort(prime)
for n = 1 to len(prime)
row++
see "" + prime[n] + " "
if row%10 = 0
see nl
ok
next
 
see nl + "Found " + row + " prime numbers" + nl
see "done..." + nl
 
Output:
working...
Concatenate two primes is also prime:
23 37 53 73 113 137 173 193 197 211 
223 229 233 241 271 283 293 311 313 317 
331 337 347 353 359 367 373 379 383 389 
397 433 523 541 547 571 593 613 617 673 
677 719 733 743 761 773 797 977 1117 1123 
1129 1153 1171 1319 1361 1367 1373 1723 1741 1747 
1753 1759 1783 1789 1913 1931 1973 1979 1997 2311 
2341 2347 2371 2383 2389 2917 2953 2971 3119 3137 
3167 3719 3761 3767 3779 3797 4111 4129 4153 4159 
4337 4373 4397 4723 4729 4759 4783 4789 5323 5347 
5923 5953 6113 6131 6143 6173 6197 6719 6737 6761 
6779 7129 7159 7331 7919 7937 8311 8317 8329 8353 
8389 8923 8929 8941 8971 9719 9743 9767 
Found 128 prime numbers
done...

Ruby[edit]

require "prime"
 
concats = Prime.each(100).to_a.repeated_permutation(2).filter_map do |pair|
p1p2 = pair.map(&:to_s).join.to_i
p1p2 if p1p2.prime?
end
concats = concats.sort.uniq
 
concats.each_slice(10){|slice|puts slice.map{|el| el.to_s.ljust(6)}.join }
Output:
23    37    53    73    113   137   173   193   197   211   
223   229   233   241   271   283   293   311   313   317   
331   337   347   353   359   367   373   379   383   389   
397   433   523   541   547   571   593   613   617   673   
677   719   733   743   761   773   797   977   1117  1123  
1129  1153  1171  1319  1361  1367  1373  1723  1741  1747  
1753  1759  1783  1789  1913  1931  1973  1979  1997  2311  
2341  2347  2371  2383  2389  2917  2953  2971  3119  3137  
3167  3719  3761  3767  3779  3797  4111  4129  4153  4159  
4337  4373  4397  4723  4729  4759  4783  4789  5323  5347  
5923  5953  6113  6131  6143  6173  6197  6719  6737  6761  
6779  7129  7159  7331  7919  7937  8311  8317  8329  8353  
8389  8923  8929  8941  8971  9719  9743  9767  

Sidef[edit]

var upto = 100
var arr = upto.primes
var base = 10
 
arr = arr.map {|p|
var d = p.digits(base);
arr.map {|q| [q.digits(base)..., d...].digits2num(base) }.grep { .is_prime }
}.flat.uniq.sort
 
say "Concatenated primes from primes p,q <= #{upto}:"
 
arr.each_slice(10, {|*a|
say a.map { '%6s' % _ }.join(' ')
})
 
say "\nFound #{arr.len} such concatenated primes."
Output:
Concatenated primes from primes p,q <= 100:
    23     37     53     73    113    137    173    193    197    211
   223    229    233    241    271    283    293    311    313    317
   331    337    347    353    359    367    373    379    383    389
   397    433    523    541    547    571    593    613    617    673
   677    719    733    743    761    773    797    977   1117   1123
  1129   1153   1171   1319   1361   1367   1373   1723   1741   1747
  1753   1759   1783   1789   1913   1931   1973   1979   1997   2311
  2341   2347   2371   2383   2389   2917   2953   2971   3119   3137
  3167   3719   3761   3767   3779   3797   4111   4129   4153   4159
  4337   4373   4397   4723   4729   4759   4783   4789   5323   5347
  5923   5953   6113   6131   6143   6173   6197   6719   6737   6761
  6779   7129   7159   7331   7919   7937   8311   8317   8329   8353
  8389   8923   8929   8941   8971   9719   9743   9767

Found 128 such concatenated primes.

Wren[edit]

Library: Wren-math
Library: Wren-fmt
Library: Wren-seq
import "/math" for Int
import "/fmt" for Fmt
import "/seq" for Lst
 
var limit = 99
var primes = Int.primeSieve(limit)
var results = []
for (p in primes) {
for (q in primes) {
var pq = (q < 10) ? p*10 + q : p*100 + q
if (Int.isPrime(pq)) results.add(pq)
}
}
results = Lst.distinct(results)
results.sort()
System.print("Two primes under 100 concatenated together to form another prime:")
for (chunk in Lst.chunks(results, 10)) Fmt.print("$,6d", chunk)
System.print("\nFound %(results.count) such concatenated primes.")
Output:
Two primes under 100 concatenated together to form another prime:
    23     37     53     73    113    137    173    193    197    211
   223    229    233    241    271    283    293    311    313    317
   331    337    347    353    359    367    373    379    383    389
   397    433    523    541    547    571    593    613    617    673
   677    719    733    743    761    773    797    977  1,117  1,123
 1,129  1,153  1,171  1,319  1,361  1,367  1,373  1,723  1,741  1,747
 1,753  1,759  1,783  1,789  1,913  1,931  1,973  1,979  1,997  2,311
 2,341  2,347  2,371  2,383  2,389  2,917  2,953  2,971  3,119  3,137
 3,167  3,719  3,761  3,767  3,779  3,797  4,111  4,129  4,153  4,159
 4,337  4,373  4,397  4,723  4,729  4,759  4,783  4,789  5,323  5,347
 5,923  5,953  6,113  6,131  6,143  6,173  6,197  6,719  6,737  6,761
 6,779  7,129  7,159  7,331  7,919  7,937  8,311  8,317  8,329  8,353
 8,389  8,923  8,929  8,941  8,971  9,719  9,743  9,767

Found 128 such concatenated primes.

XPL0[edit]

func IsPrime(N);        \Return 'true' if N is a prime number
int N, I;
[if N <= 1 then return false;
for I:= 2 to sqrt(N) do
if rem(N/I) = 0 then return false;
return true;
];
 
char Set(9999+1);
int P1, P2, P, Count;
[for P:= 0 to 9999 do Set(P):= false;
for P1:= 0 to 99 do
if IsPrime(P1) then
for P2:= 0 to 99 do
if IsPrime(P2) then
[if P2 >= 10 then
P:= P1*100 + P2
else P:= P1*10 + P2;
if IsPrime(P) then Set(P):= true;
];
Count:= 0;
for P:= 0 to 9999 do
if Set(P) then
[IntOut(0, P);
Count:= Count+1;
if rem(Count/10) = 0 then CrLf(0) else ChOut(0, 9\tab\);
];
CrLf(0);
IntOut(0, Count);
Text(0, " such concatenated primes found.
");
]
Output:
23      37      53      73      113     137     173     193     197     211
223     229     233     241     271     283     293     311     313     317
331     337     347     353     359     367     373     379     383     389
397     433     523     541     547     571     593     613     617     673
677     719     733     743     761     773     797     977     1117    1123
1129    1153    1171    1319    1361    1367    1373    1723    1741    1747
1753    1759    1783    1789    1913    1931    1973    1979    1997    2311
2341    2347    2371    2383    2389    2917    2953    2971    3119    3137
3167    3719    3761    3767    3779    3797    4111    4129    4153    4159
4337    4373    4397    4723    4729    4759    4783    4789    5323    5347
5923    5953    6113    6131    6143    6173    6197    6719    6737    6761
6779    7129    7159    7331    7919    7937    8311    8317    8329    8353
8389    8923    8929    8941    8971    9719    9743    9767    
128 such concatenated primes found.