# Sum of two adjacent numbers are primes

Sum of two adjacent numbers are primes 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.

Show on this page the first 20 prime sums of two consecutive integers.

Extra credit

Show the ten millionth such prime sum.

## Action!

```;;; Find numbers n such that n + n+1 is prime
;;; Library: Action! Sieve of Eratosthenes
INCLUDE "H6:SIEVE.ACT"

PROC Main()
DEFINE MAX_PRIME = "255"

BYTE ARRAY primes(MAX_PRIME)
CARD n, n1, count

Sieve(primes,MAX_PRIME)

count = 0;
n1 = 1
WHILE count < 20 DO
n = n1
n1 ==+ 1
IF primes( n + n1 ) THEN
count ==+ 1
PrintF( "%2U: %2U + %2U = %2U%E", count, n, n1, n + n1 )
FI
OD

RETURN```
Output:
``` 1:  1 +  2 =  3
2:  2 +  3 =  5
3:  3 +  4 =  7
4:  5 +  6 = 11
5:  6 +  7 = 13
6:  8 +  9 = 17
7:  9 + 10 = 19
8: 11 + 12 = 23
9: 14 + 15 = 29
10: 15 + 16 = 31
11: 18 + 19 = 37
12: 20 + 21 = 41
13: 21 + 22 = 43
14: 23 + 24 = 47
15: 26 + 27 = 53
16: 29 + 30 = 59
17: 30 + 31 = 61
18: 33 + 34 = 67
19: 35 + 36 = 71
20: 36 + 37 = 73
```

## ALGOL 68

Using Pete Lomax's observation that all odd primes are n + n+1 for some n - see #Phix

```BEGIN # find the first 20 primes which are n + ( n + 1 ) for some n  #
PR read "primes.incl.a68" PR           # include prime utilities #
[]BOOL prime = PRIMESIEVE 200;         # should be enough primes #
INT p count := 0;
FOR p FROM 3 BY 2 WHILE p count < 20 DO
IF prime[ p ] THEN
INT n  = p OVER 2;
INT n1 = n + 1;
p count +:= 1;
print( ( whole( n, -2 ), " + ", whole( n1, -2 ), " = ", whole( p, -3 ), newline ) )
FI
OD
END```
Output:
``` 1 +  2 =   3
2 +  3 =   5
3 +  4 =   7
5 +  6 =  11
6 +  7 =  13
8 +  9 =  17
9 + 10 =  19
11 + 12 =  23
14 + 15 =  29
15 + 16 =  31
18 + 19 =  37
20 + 21 =  41
21 + 22 =  43
23 + 24 =  47
26 + 27 =  53
29 + 30 =  59
30 + 31 =  61
33 + 34 =  67
35 + 36 =  71
36 + 37 =  73
```

## ALGOL W

```begin % find numbers n such that n + n+1 is prime                            %

% sets s to a sieve of primes, using the sieve of Eratosthenes           %
procedure sieve( logical array s ( * ); integer value n ) ;
begin
for i := 1 until n do s( i ) := true;
% sieve out the non-primes                                           %
s( 1 ) := false;
for i := 2 until truncate( sqrt( n ) )
do begin
if s( i )
then begin
for p := i * i step i until n do s( p ) := false
end if_s_i
end for_i ;
end sieve ;

integer sieveMax;
sieveMax := 200;
begin
logical array prime ( 1 :: sieveMax );
integer count, n, n1;
i_w := 2; % set output integer field width                           %
s_w := 1; % and output separator width                               %

% find and display the numbers                                       %
sieve( prime, sieveMax );
count := 0;
n1    := 1;
while count < 20 do begin
n  := n1;
n1 := n1 + 1;
if prime( n + n1 ) then begin
count := count + 1;
write( count, ": ", n, " + ", n1, " = ", n + n1 )
end if_prime__n_plus_n1
end while_count_lt_20
end

end.```
Output:
``` 1 :  1  +  2  =  3
2 :  2  +  3  =  5
3 :  3  +  4  =  7
4 :  5  +  6  = 11
5 :  6  +  7  = 13
6 :  8  +  9  = 17
7 :  9  + 10  = 19
8 : 11  + 12  = 23
9 : 14  + 15  = 29
10 : 15  + 16  = 31
11 : 18  + 19  = 37
12 : 20  + 21  = 41
13 : 21  + 22  = 43
14 : 23  + 24  = 47
15 : 26  + 27  = 53
16 : 29  + 30  = 59
17 : 30  + 31  = 61
18 : 33  + 34  = 67
19 : 35  + 36  = 71
20 : 36  + 37  = 73
```

## AWK

```# syntax: GAWK -f SUM_OF_TWO_ADJACENT_NUMBERS_ARE_PRIMES.AWK
BEGIN {
n = 1
stop = 20
printf("The first %d pairs of numbers whose sum is prime:\n",stop)
while (count < stop) {
if (is_prime(2*n + 1)) {
printf("%2d + %2d = %2d\n",n,n+1,2*n+1)
count++
}
n++
}
exit(0)
}
function is_prime(n,  d) {
d = 5
if (n < 2) { return(0) }
if (n % 2 == 0) { return(n == 2) }
if (n % 3 == 0) { return(n == 3) }
while (d*d <= n) {
if (n % d == 0) { return(0) }
d += 2
if (n % d == 0) { return(0) }
d += 4
}
return(1)
}
```
Output:
```The first 20 pairs of numbers whose sum is prime:
1 +  2 =  3
2 +  3 =  5
3 +  4 =  7
5 +  6 = 11
6 +  7 = 13
8 +  9 = 17
9 + 10 = 19
11 + 12 = 23
14 + 15 = 29
15 + 16 = 31
18 + 19 = 37
20 + 21 = 41
21 + 22 = 43
23 + 24 = 47
26 + 27 = 53
29 + 30 = 59
30 + 31 = 61
33 + 34 = 67
35 + 36 = 71
36 + 37 = 73
```

## BASIC

### BASIC256

Translation of: FreeBASIC
```function isPrime(v)
if v < 2 then return False
if v mod 2 = 0 then return v = 2
if v mod 3 = 0 then return v = 3
d = 5
while d * d <= v
if v mod d = 0 then return False else d += 2
end while
return True
end function

n = 0
num = 0

print "The first 20 pairs of numbers whose sum is prime:"
while True
n += 1
suma = 2*n+1
if isPrime(suma) then
num += 1
if num < 21 then
print n; " + "; (n+1); " = "; suma
else
exit while
end if
end if
end while
end```
Output:
```Igual que la entrada de FreeBASIC.
```

### FreeBASIC

```Function isPrime(Byval ValorEval As Integer) As Boolean
If ValorEval <= 1 Then Return False
For i As Integer = 2 To Int(Sqr(ValorEval))
If ValorEval Mod i = 0 Then Return False
Next i
Return True
End Function

Dim As Integer n = 0, num = 0, suma
print "The first 20 pairs of numbers whose sum is prime:"
Do
n += 1
suma = 2*n+1
If isPrime(suma) Then
num += 1
If num < 21 Then
Print using "## + ## = ##"; n; (n+1); suma
Else
Exit Do
End If
End If
Loop
Sleep
```
Output:
```The first 20 pairs of numbers whose sum is prime:
1 +  2 =  3
2 +  3 =  5
3 +  4 =  7
5 +  6 = 11
6 +  7 = 13
8 +  9 = 17
9 + 10 = 19
11 + 12 = 23
14 + 15 = 29
15 + 16 = 31
18 + 19 = 37
20 + 21 = 41
21 + 22 = 43
23 + 24 = 47
26 + 27 = 53
29 + 30 = 59
30 + 31 = 61
33 + 34 = 67
35 + 36 = 71
36 + 37 = 73```

### PureBasic

```Procedure isPrime(v.i)
If     v <= 1    : ProcedureReturn #False
ElseIf v < 4     : ProcedureReturn #True
ElseIf v % 2 = 0 : ProcedureReturn #False
ElseIf v < 9     : ProcedureReturn #True
ElseIf v % 3 = 0 : ProcedureReturn #False
Else
Protected r = Round(Sqr(v), #PB_Round_Down)
Protected f = 5
While f <= r
If v % f = 0 Or v % (f + 2) = 0
ProcedureReturn #False
EndIf
f + 6
Wend
EndIf
ProcedureReturn #True
EndProcedure

If OpenConsole()
Define n.i = 0, num.i = 0, suma.i

PrintN("The first 20 pairs of numbers whose sum is prime:")
Repeat
n + 1
suma = 2*n+1
If isPrime(suma)
num + 1
If num < 21
PrintN(Str(n) + " + " + Str(n+1) + " = " + Str(suma))
Else
Break
EndIf
EndIf
ForEver
CloseConsole()
EndIf
```
Output:
```Igual que la entrada de FreeBASIC.
```

### QBasic

Works with: QBasic version 1.1
Works with: QuickBasic version 4.5
Translation of: FreeBASIC
```FUNCTION isPrime (ValorEval)
IF ValorEval <= 1 THEN isPrime = 0
FOR i = 2 TO INT(SQR(ValorEval))
IF ValorEval MOD i = 0 THEN isPrime = 0
NEXT i
isPrime = 1
END FUNCTION

n = 0
num = 0
PRINT "The first 20 pairs of numbers whose sum is prime:"
DO
n = n + 1
suma = 2 * n + 1
IF isPrime(suma) THEN
num = num + 1
IF num < 21 THEN
PRINT USING "## + ## = ##"; n; (n + 1); suma
ELSE
EXIT DO
END IF
END IF
LOOP
END
```
Output:
```Igual que la entrada de FreeBASIC.
```

### True BASIC

```FUNCTION isPrime (ValorEval)
IF ValorEval <= 1 THEN LET isPrime = 0
FOR i = 2 TO INT(SQR(ValorEval))
IF MOD(ValorEval, i) = 0 THEN LET isPrime = 0
NEXT i
LET isPrime = 1
END FUNCTION

LET n = 0
LET num = 0
PRINT "The first 20 pairs of numbers whose sum is prime:"
DO
LET n = n + 1
LET suma = 2 * n + 1
IF isPrime(suma) = 1 THEN
LET num = num + 1
IF num < 21 THEN
PRINT USING "##": n;
PRINT " + ";
PRINT USING "##": n+1;
PRINT " = ";
PRINT USING "##": suma
ELSE
EXIT DO
END IF
END IF
LOOP
END
```
Output:
```Igual que la entrada de FreeBASIC.
```

### Yabasic

```sub isPrime(v)
if v < 2 then return False : fi
if mod(v, 2) = 0 then return v = 2 : fi
if mod(v, 3) = 0 then return v = 3 : fi
d = 5
while d * d <= v
if mod(v, d) = 0 then return False else d = d + 2 : fi
wend
return True
end sub

n = 0
num = 0
print "The first 20 pairs of numbers whose sum is prime:"
do
n = n + 1
suma = 2*n+1
if isPrime(suma) then
num = num + 1
if num < 21 then
print n using "##", " + ", (n+1) using "##", " = ", suma using "##"
else
break
end if
end if
loop
end```
Output:
```Igual que la entrada de FreeBASIC.
```

## C

Translation of: Go
```#include <stdio.h>
#include <stdlib.h>
#include <math.h>

#define TRUE 1
#define FALSE 0

typedef int bool;

void primeSieve(int *c, int limit, bool processEven, bool primesOnly) {
int i, ix, p, p2;
limit++;
c[0] = TRUE;
c[1] = TRUE;
if (processEven) {
for (i = 4; i < limit; i +=2) c[i] = TRUE;
}
p = 3;
while (TRUE) {
p2 = p * p;
if (p2 >= limit) break;
for (i = p2; i < limit; i += 2*p) c[i] = TRUE;
while (TRUE) {
p += 2;
if (!c[p]) break;
}
}
if (primesOnly) {
/* move the actual primes to the front of the array */
c[0] = 2;
for (i = 3, ix = 1; i < limit; i += 2) {
if (!c[i]) c[ix++] = i;
}
}
}

int main() {
int i, p, hp, n = 10000000;
int limit = (int)(log(n) * (double)n * 1.2);  /* should be more than enough */
int *primes = (int *)calloc(limit, sizeof(int));
primeSieve(primes, limit-1, FALSE, TRUE);
printf("The first 20 pairs of natural numbers whose sum is prime are:\n");
for (i = 1; i <= 20; ++i) {
p = primes[i];
hp = p / 2;
printf("%2d + %2d = %2d\n", hp, hp+1, p);
}
printf("\nThe 10 millionth such pair is:\n");
p = primes[n];
hp = p / 2;
printf("%2d + %2d = %2d\n", hp, hp+1, p);
free(primes);
return 0;
}
```
Output:
```The first 20 pairs of natural numbers whose sum is prime are:
1 +  2 =  3
2 +  3 =  5
3 +  4 =  7
5 +  6 = 11
6 +  7 = 13
8 +  9 = 17
9 + 10 = 19
11 + 12 = 23
14 + 15 = 29
15 + 16 = 31
18 + 19 = 37
20 + 21 = 41
21 + 22 = 43
23 + 24 = 47
26 + 27 = 53
29 + 30 = 59
30 + 31 = 61
33 + 34 = 67
35 + 36 = 71
36 + 37 = 73

The 10 millionth such pair is:
89712345 + 89712346 = 179424691
```

## F#

This task uses Extensible Prime Generator (F#)

```// 2n+1 is prime. Nigel Galloway: Januuary 22nd., 2022
primes32()|>Seq.skip 1|>Seq.take 20|>Seq.map(fun n->n/2)|>Seq.iteri(fun n g->printfn "%2d: %2d + %2d=%d" (n+1) g (g+1) (g+g+1))
```
Output:
``` 1:  1 +  2=3
2:  2 +  3=5
3:  3 +  4=7
4:  5 +  6=11
5:  6 +  7=13
6:  8 +  9=17
7:  9 + 10=19
8: 11 + 12=23
9: 14 + 15=29
10: 15 + 16=31
11: 18 + 19=37
12: 20 + 21=41
13: 21 + 22=43
14: 23 + 24=47
15: 26 + 27=53
16: 29 + 30=59
17: 30 + 31=61
18: 33 + 34=67
19: 35 + 36=71
20: 36 + 37=73
```

## Factor

Works with: Factor version 0.99 2021-06-02
```USING: arrays formatting kernel lists lists.lazy math
math.primes.lists sequences ;

20 lprimes cdr [ 2/ dup 1 + 2array ] lmap-lazy ltake
[ dup sum suffix "%d + %d = %d\n" vprintf ] leach
```
Output:
```1 + 2 = 3
2 + 3 = 5
3 + 4 = 7
5 + 6 = 11
6 + 7 = 13
8 + 9 = 17
9 + 10 = 19
11 + 12 = 23
14 + 15 = 29
15 + 16 = 31
18 + 19 = 37
20 + 21 = 41
21 + 22 = 43
23 + 24 = 47
26 + 27 = 53
29 + 30 = 59
30 + 31 = 61
33 + 34 = 67
35 + 36 = 71
36 + 37 = 73
```

## Go

Library: Go-rcu
```package main

import (
"fmt"
"math"
"rcu"
)

func main() {
limit := int(math.Log(1e7) * 1e7 * 1.2) // should be more than enough
primes := rcu.Primes(limit)
fmt.Println("The first 20 pairs of natural numbers whose sum is prime are:")
for i := 1; i <= 20; i++ {
p := primes[i]
hp := p / 2
fmt.Printf("%2d + %2d = %2d\n", hp, hp+1, p)
}
fmt.Println("\nThe 10 millionth such pair is:")
p := primes[1e7]
hp := p / 2
fmt.Printf("%2d + %2d = %2d\n", hp, hp+1, p)
}
```
Output:
```The first 20 pairs of natural numbers whose sum is prime are:
1 +  2 =  3
2 +  3 =  5
3 +  4 =  7
5 +  6 = 11
6 +  7 = 13
8 +  9 = 17
9 + 10 = 19
11 + 12 = 23
14 + 15 = 29
15 + 16 = 31
18 + 19 = 37
20 + 21 = 41
21 + 22 = 43
23 + 24 = 47
26 + 27 = 53
29 + 30 = 59
30 + 31 = 61
33 + 34 = 67
35 + 36 = 71
36 + 37 = 73

The 10 millionth such pair is:
89712345 + 89712346 = 179424691
```

```import Data.Numbers.Primes

primeSumsOfConsecutiveNumbers :: Integral a => [(a, (a, a))]
primeSumsOfConsecutiveNumbers =
let go a b = [(n, (a, b)) | let n = a + b, isPrime n]
in concat \$ zipWith go [1 ..] [2 ..]

main :: IO ()
main = mapM_ print \$ take 20 primeSumsOfConsecutiveNumbers
```
Output:
```(3,(1,2))
(5,(2,3))
(7,(3,4))
(11,(5,6))
(13,(6,7))
(17,(8,9))
(19,(9,10))
(23,(11,12))
(29,(14,15))
(31,(15,16))
(37,(18,19))
(41,(20,21))
(43,(21,22))
(47,(23,24))
(53,(26,27))
(59,(29,30))
(61,(30,31))
(67,(33,34))
(71,(35,36))
(73,(36,37))```

## J

Here, we generate the first 20 odd primes, divide by 2, drop the fractional part and add 0 and 1 to that value. Then we format that pair of numbers with their sum and with decorations indicating the relevant arithmetic:

```   (+/,&":'=',{.,&":'+',&":{:)"#. 0 1+/~<.-: p:1+i.20
3=1+2
5=2+3
7=3+4
11=5+6
13=6+7
17=8+9
19=9+10
23=11+12
29=14+15
31=15+16
37=18+19
41=20+21
43=21+22
47=23+24
53=26+27
59=29+30
61=30+31
67=33+34
71=35+36
73=36+37
```

## jq

Works with: jq

Works with gojq, the Go implementation of jq

This entry uses a memory-less approach to computing `is_prime`, and so a naive approach to computing the first 20 numbers satisfying the condition is appropriate.

```def is_prime:
. as \$n
| if (\$n < 2)         then false
elif (\$n % 2 == 0)  then \$n == 2
elif (\$n % 3 == 0)  then \$n == 3
else 5
| until( . <= 0;
if .*. > \$n then -1
elif (\$n % . == 0) then 0
else . + 2
|  if (\$n % . == 0) then 0
else . + 4
end
end)
| . == -1
end;

def naive:
limit(20;
range(0; infinite) as \$i
| (2*\$i + 1) as \$q
| select(\$q | is_prime)
| "\(\$i) + \(\$i + 1) = \(\$q)" );

naive```
Output:
```1 + 2 = 3
2 + 3 = 5
3 + 4 = 7
5 + 6 = 11
6 + 7 = 13
8 + 9 = 17
9 + 10 = 19
11 + 12 = 23
14 + 15 = 29
15 + 16 = 31
18 + 19 = 37
20 + 21 = 41
21 + 22 = 43
23 + 24 = 47
26 + 27 = 53
29 + 30 = 59
30 + 31 = 61
33 + 34 = 67
35 + 36 = 71
36 + 37 = 73
```

## Julia

```using Lazy
using Primes

seq = @>> Lazy.range() filter(n -> isprime(2n + 1))
for n in take(20, seq)
println("\$n + \$(n + 1) = \$(n + n + 1)")
end

let
i, cnt = 0, 0
while cnt < 10_000_000
i += 1
if isprime(2i + 1)
cnt += 1
end
end
println("Ten millionth: \$i + \$(i + 1) = \$(i + i + 1)")
end
```
Output:
```1 + 2 = 3
2 + 3 = 5
3 + 4 = 7
5 + 6 = 11
6 + 7 = 13
8 + 9 = 17
9 + 10 = 19
11 + 12 = 23
14 + 15 = 29
15 + 16 = 31
18 + 19 = 37
20 + 21 = 41
21 + 22 = 43
23 + 24 = 47
26 + 27 = 53
29 + 30 = 59
30 + 31 = 61
33 + 34 = 67
35 + 36 = 71
36 + 37 = 73
Ten millionth: 89712345 + 89712346 = 179424691
```

## Mathematica/Wolfram Language

```Column[Row[{Floor[#/2], " + ", Ceiling[#/2], " = ", #}] & /@ Prime[1 + Range[20]]]
Row[{Floor[#/2], " + ", Ceiling[#/2], " = ", #}] &[Prime[10^7 + 1]]
```
Output:
```1 + 2 = 3
2 + 3 = 5
3 + 4 = 7
5 + 6 = 11
6 + 7 = 13
8 + 9 = 17
9 + 10 = 19
11 + 12 = 23
14 + 15 = 29
15 + 16 = 31
18 + 19 = 37
20 + 21 = 41
21 + 22 = 43
23 + 24 = 47
26 + 27 = 53
29 + 30 = 59
30 + 31 = 61
33 + 34 = 67
35 + 36 = 71
36 + 37 = 73
89712345 + 89712346 = 179424691```

## Perl

Library: ntheory
```use strict;
use warnings;
use ntheory 'is_prime';

my(\$n,\$c);
while () { is_prime(1 + 2*++\$n) and printf "%2d + %2d = %2d\n", \$n, \$n+1, 1+2*\$n and ++\$c == 20 and last }
```
Output:
``` 1 +  2 =  3
2 +  3 =  5
3 +  4 =  7
5 +  6 = 11
6 +  7 = 13
8 +  9 = 17
9 + 10 = 19
11 + 12 = 23
14 + 15 = 29
15 + 16 = 31
18 + 19 = 37
20 + 21 = 41
21 + 22 = 43
23 + 24 = 47
26 + 27 = 53
29 + 30 = 59
30 + 31 = 61
33 + 34 = 67
35 + 36 = 71
36 + 37 = 73```

## Phix

Every prime p greater than 2 is odd, hence p/2 is k.5 and the adjacent numbers needed are k and k+1. DOH.

```with javascript_semantics
procedure doh(integer p)
printf(1,"%d + %d = %d\n",{floor(p/2),ceil(p/2),p})
end procedure
papply(get_primes(-21)[2..\$],doh)
doh(get_prime(1e7+1))
```
Output:
```1 + 2 = 3
2 + 3 = 5
3 + 4 = 7
5 + 6 = 11
6 + 7 = 13
8 + 9 = 17
9 + 10 = 19
11 + 12 = 23
14 + 15 = 29
15 + 16 = 31
18 + 19 = 37
20 + 21 = 41
21 + 22 = 43
23 + 24 = 47
26 + 27 = 53
29 + 30 = 59
30 + 31 = 61
33 + 34 = 67
35 + 36 = 71
36 + 37 = 73
89712345 + 89712346 = 179424691
```

## PL/M

Works with: 8080 PL/M Compiler
... under CP/M (or an emulator)
```100H: /* FIND NUMBERS N SUCH THAT N + N + 1 IS PRIME                         */

/* CP/M SYSTEM CALL AND I/O ROUTINES                                      */
BDOS:      PROCEDURE( FN, ARG ); DECLARE FN BYTE, ARG ADDRESS; GOTO 5; END;
PR\$CHAR:   PROCEDURE( C ); DECLARE C BYTE;    CALL BDOS( 2, C );  END;
PR\$STRING: PROCEDURE( S ); DECLARE S ADDRESS; CALL BDOS( 9, S );  END;
PR\$NL:     PROCEDURE;   CALL PR\$CHAR( 0DH ); CALL PR\$CHAR( 0AH ); END;
PR\$NUMBER: PROCEDURE( N ); /* PRINTS A NUMBER IN THE MINIMUN FIELD WIDTH  */
DECLARE V ADDRESS, N\$STR ( 6 )BYTE, W BYTE;
V = N;
W = LAST( N\$STR );
N\$STR( W ) = '\$';
N\$STR( W := W - 1 ) = '0' + ( V MOD 10 );
DO WHILE( ( V := V / 10 ) > 0 );
N\$STR( W := W - 1 ) = '0' + ( V MOD 10 );
END;
CALL PR\$STRING( .N\$STR( W ) );
END PR\$NUMBER;

PR\$NUMBER2: PROCEDURE( N ); /* PRINT A NUMBER IN AT LEAST 2 POSITIONS     */
IF N < 10 THEN CALL PR\$CHAR( ' ' );
CALL PR\$NUMBER( N );
END PR\$NUMBER2;

/* ASSUMING THE FIRST 20 SUCH NUMBERS WILL BE < 255                       */

DECLARE MAX\$NUMBER LITERALLY '255'
, FALSE      LITERALLY '0'
, TRUE       LITERALLY '0FFH'
;

DECLARE PRIME ( MAX\$NUMBER )BYTE;   /* SIEVE THE PRIMES TO MAX\$NUMBER - 1 */
PRIME( 0 ), PRIME( 1 ) = FALSE;
PRIME( 2 ) = TRUE;
DO I = 3 TO LAST( PRIME ) BY 2; PRIME( I ) = TRUE;  END;
DO I = 4 TO LAST( PRIME ) BY 2; PRIME( I ) = FALSE; END;
DO I = 3 TO LAST( PRIME ) BY 2;
IF PRIME( I ) THEN DO;
DO S = I + I TO LAST( PRIME ) BY I;
PRIME( S ) = FALSE;
END;
END;
END;

/* FIND THE NUMBERS                                                       */
DECLARE ( N, N1, COUNT ) BYTE;
COUNT = 0;
N1    = 1;
DO WHILE COUNT < 20;
N  = N1;
N1 = N1 + 1;
IF PRIME( N + N1 ) THEN DO;
COUNT = COUNT + 1;
CALL PR\$NUMBER2( COUNT );
CALL PR\$STRING( .': \$' );
CALL PR\$NUMBER2( N );
CALL PR\$STRING( .' + \$' );
CALL PR\$NUMBER2( N1 );
CALL PR\$STRING( .' = \$' );
CALL PR\$NUMBER2( N + N1 );
CALL PR\$NL;
END;
END;

EOF```
Output:
``` 1:  1 +  2 =  3
2:  2 +  3 =  5
3:  3 +  4 =  7
4:  5 +  6 = 11
5:  6 +  7 = 13
6:  8 +  9 = 17
7:  9 + 10 = 19
8: 11 + 12 = 23
9: 14 + 15 = 29
10: 15 + 16 = 31
11: 18 + 19 = 37
12: 20 + 21 = 41
13: 21 + 22 = 43
14: 23 + 24 = 47
15: 26 + 27 = 53
16: 29 + 30 = 59
17: 30 + 31 = 61
18: 33 + 34 = 67
19: 35 + 36 = 71
20: 36 + 37 = 73
```

## Python

### Procedural

```#!/usr/bin/python

def isPrime(n):
for i in range(2, int(n**0.5) + 1):
if n % i == 0:
return False
return True

if __name__ == "__main__":
n = 0
num = 0

print('The first 20 pairs of numbers whose sum is prime:')
while True:
n += 1
suma = 2*n+1
if isPrime(suma):
num += 1
if num < 21:
print('{:2}'.format(n), "+", '{:2}'.format(n+1), "=", '{:2}'.format(suma))
else:
break
```
Output:
```The first 20 pairs of numbers whose sum is prime:
1 +  2 =  3
2 +  3 =  5
3 +  4 =  7
5 +  6 = 11
6 +  7 = 13
8 +  9 = 17
9 + 10 = 19
11 + 12 = 23
14 + 15 = 29
15 + 16 = 31
18 + 19 = 37
20 + 21 = 41
21 + 22 = 43
23 + 24 = 47
26 + 27 = 53
29 + 30 = 59
30 + 31 = 61
33 + 34 = 67
35 + 36 = 71
36 + 37 = 73```

### Functional

```'''Prime sums of two consecutive integers'''

from itertools import chain, count, islice

# primeSumsOfTwoConsecutiveIntegers :: [Int]
def primeSumsOfTwoConsecutiveIntegers():
'''Infinite series of prime sums of
two consecutive integers.
'''
def go(a, b):
n = a + b
return [(n, (a, b))] if isPrime(n) else []

return chain.from_iterable(
map(go, count(1), count(2))
)

# ------------------------- TEST -------------------------
# main :: IO ()
def main():
'''First 20 prime sums of two consecutive integers.'''
print(
'\n'.join([
f'{n} = {a} + {b}'  for n, (a, b) in islice(
primeSumsOfTwoConsecutiveIntegers(),
20
)
])
)

# ----------------------- GENERIC ------------------------

# isPrime :: Int -> Bool
def isPrime(n):
'''True if n is prime.'''
if n in (2, 3):
return True
if 2 > n or 0 == n % 2:
return False
if 9 > n:
return True
if 0 == n % 3:
return False

def p(x):
return 0 == n % x or 0 == n % (2 + x)

return not any(map(p, range(5, 1 + int(n ** 0.5), 6)))

# MAIN ---
if __name__ == '__main__':
main()
```

or as a list comprehension (the walrus operator requires Python 3.8+)

```'''Prime sums of two consecutive integers'''

from itertools import count, islice

# primeSumsOfTwoConsecutiveIntegers :: [Int]
def primeSumsOfTwoConsecutiveIntegers():
'''As a list comprehension.'''
return (
(n, (a, b))
for a in count(1)
if isPrime(n := a + (b := 1 + a))
)

# ------------------------- TEST -------------------------
# main :: IO ()
def main():
'''First 20 prime sums of two consecutive integers.'''
print(
'\n'.join([
f'{n} = {a} + {b}' for n, (a, b) in islice(
primeSumsOfTwoConsecutiveIntegers(),
20
)
])
)

# ----------------------- GENERIC ------------------------

# isPrime :: Int -> Bool
def isPrime(n):
'''True if n is prime.'''
if n in (2, 3):
return True
if 2 > n or 0 == n % 2:
return False
if 9 > n:
return True
if 0 == n % 3:
return False

def p(x):
return 0 == n % x or 0 == n % (2 + x)

return not any(map(p, range(5, 1 + int(n ** 0.5), 6)))

# MAIN ---
if __name__ == '__main__':
main()
```
Output:
```3 = 1 + 2
5 = 2 + 3
7 = 3 + 4
11 = 5 + 6
13 = 6 + 7
17 = 8 + 9
19 = 9 + 10
23 = 11 + 12
29 = 14 + 15
31 = 15 + 16
37 = 18 + 19
41 = 20 + 21
43 = 21 + 22
47 = 23 + 24
53 = 26 + 27
59 = 29 + 30
61 = 30 + 31
67 = 33 + 34
71 = 35 + 36
73 = 36 + 37```

## Quackery

Every odd number is the sum of two adjacent numbers, so this task is "the first twenty odd primes" or "the first twenty primes after 2".

`isprime` is defined at Primality by trial division#Quackery.

```  [ 1 [] rot
[ over size
over != while
dip
[ over isprime if
[ over join ]
dip [ 2 + ] ]
again ]
drop nip ]           is oddprimes ( n --> [ )

[ oddprimes
witheach
[ dup 2 /
dup echo
say " + "
1+ echo
say " = "
echo cr ] ]       is task       ( n -->   )

Output:
```1 + 2 = 3
2 + 3 = 5
3 + 4 = 7
5 + 6 = 11
6 + 7 = 13
8 + 9 = 17
9 + 10 = 19
11 + 12 = 23
14 + 15 = 29
15 + 16 = 31
18 + 19 = 37
20 + 21 = 41
21 + 22 = 43
23 + 24 = 47
26 + 27 = 53
29 + 30 = 59
30 + 31 = 61
33 + 34 = 67
35 + 36 = 71
36 + 37 = 73
```

## Raku

```my @n-n1-triangular = map { \$_, \$_ + 1, \$_ + (\$_ + 1) }, ^Inf;

my @wanted = @n-n1-triangular.grep: *.[2].is-prime;

printf "%2d + %2d = %2d\n", |.list for @wanted.head(20);
```
Output:
``` 1 +  2 =  3
2 +  3 =  5
3 +  4 =  7
5 +  6 = 11
6 +  7 = 13
8 +  9 = 17
9 + 10 = 19
11 + 12 = 23
14 + 15 = 29
15 + 16 = 31
18 + 19 = 37
20 + 21 = 41
21 + 22 = 43
23 + 24 = 47
26 + 27 = 53
29 + 30 = 59
30 + 31 = 61
33 + 34 = 67
35 + 36 = 71
36 + 37 = 73
```

## Ring

```load "stdlibcore.ring"
see "working..." + nl
n = 0
num = 0

while true
n++
sum = 2*n+1
if isprime(sum)
num++
if num < 21
? "" + n + " + " + (n+1) + " = " + sum
else
exit
ok
ok
end

see "done..." + nl```
Output:
```working...
1 + 2 = 3
2 + 3 = 5
3 + 4 = 7
5 + 6 = 11
6 + 7 = 13
8 + 9 = 17
9 + 10 = 19
11 + 12 = 23
14 + 15 = 29
15 + 16 = 31
18 + 19 = 37
20 + 21 = 41
21 + 22 = 43
23 + 24 = 47
26 + 27 = 53
29 + 30 = 59
30 + 31 = 61
33 + 34 = 67
35 + 36 = 71
36 + 37 = 73
done...
```

## Sidef

```var wanted = (1..Inf -> lazy.map  {|n| [n, n+1, n+(n+1)] }\
.grep { .tail.is_prime })

wanted.first(20).each_2d {|a,b,c|
printf("%2d + %2d = %2d\n", a,b,c)
}
```
Output:
``` 1 +  2 =  3
2 +  3 =  5
3 +  4 =  7
5 +  6 = 11
6 +  7 = 13
8 +  9 = 17
9 + 10 = 19
11 + 12 = 23
14 + 15 = 29
15 + 16 = 31
18 + 19 = 37
20 + 21 = 41
21 + 22 = 43
23 + 24 = 47
26 + 27 = 53
29 + 30 = 59
30 + 31 = 61
33 + 34 = 67
35 + 36 = 71
36 + 37 = 73
```

## Wren

Translation of: Go
Library: Wren-math
Library: Wren-fmt
```import "./math" for Int
import "./fmt" for Fmt

var limit = (1e7.log * 1e7 * 1.2).floor  // should be more than enough
var primes = Int.primeSieve(limit)
System.print("The first 20 pairs of natural numbers whose sum is prime are:")
for (i in 1..20) {
var p = primes[i]
var hp = (p/2).floor
Fmt.print("\$2d + \$2d = \$2d", hp, hp + 1, p)
}
System.print("\nThe 10 millionth such pair is:")
var p = primes[1e7]
var hp = (p/2).floor
Fmt.print("\$2d + \$2d = \$2d", hp, hp + 1, p)
```
Output:
```The first 20 pairs of natural numbers whose sum is prime are:
1 +  2 =  3
2 +  3 =  5
3 +  4 =  7
5 +  6 = 11
6 +  7 = 13
8 +  9 = 17
9 + 10 = 19
11 + 12 = 23
14 + 15 = 29
15 + 16 = 31
18 + 19 = 37
20 + 21 = 41
21 + 22 = 43
23 + 24 = 47
26 + 27 = 53
29 + 30 = 59
30 + 31 = 61
33 + 34 = 67
35 + 36 = 71
36 + 37 = 73

The 10 millionth such pair is:
89712345 + 89712346 = 179424691
```

## XPL0

Translation of: Ring
```include xpllib;
int N, Num, Sum;
[Text(0, "Working...^M^J");
N:= 0;
Num:= 0;
loop
[N:= N+1;
Sum:= 2*N + 1;
if IsPrime(Sum) then
[Num:= Num+1;
if Num < 21 then
[Text(0,"N = "); IntOut(0,N); Text(0,"  Sum = "); IntOut(0,Sum); CrLf(0)]
else
quit
]
];
Text(0, "Done...^M^J");
]```
Output:
```Working...
N = 1  Sum = 3
N = 2  Sum = 5
N = 3  Sum = 7
N = 5  Sum = 11
N = 6  Sum = 13
N = 8  Sum = 17
N = 9  Sum = 19
N = 11  Sum = 23
N = 14  Sum = 29
N = 15  Sum = 31
N = 18  Sum = 37
N = 20  Sum = 41
N = 21  Sum = 43
N = 23  Sum = 47
N = 26  Sum = 53
N = 29  Sum = 59
N = 30  Sum = 61
N = 33  Sum = 67
N = 35  Sum = 71
N = 36  Sum = 73
Done...
```