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.

Task

Show on this page the first 20 numbers and sum of two adjacent numbers which sum is prime.

Extra credit

Show the ten millionth such prime sum.

ALGOL 68

Library: ALGOL 68-primes

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 n WHILE p count < 20 DO
        INT n1 = n + 1;
        INT p  = n + n1;
        IF prime[ p ] THEN
            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

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

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

Python

#!/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

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...