Special neighbor primes

Special neighbor 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.

Let   (p1,  p2)   are neighbor primes.

Find and show here in base ten if   p1+ p2 -1   is prime,   where   p1,   p2  <  100.

11l

Translation of: Nim
```F is_prime(n)
I n == 2
R 1B
I n < 2 | n % 2 == 0
R 0B
L(i) (3 .. Int(sqrt(n))).step(2)
I n % i == 0
R 0B
R 1B

V primes = (0.<100).filter(n -> is_prime(n))

L(i) 0 .< primes.len - 1
V p1 = primes[i]
V p2 = primes[i + 1]
I is_prime(p1 + p2 - 1)
print((p1, p2))```
Output:
```(3, 5)
(5, 7)
(7, 11)
(11, 13)
(13, 17)
(19, 23)
(29, 31)
(31, 37)
(41, 43)
(43, 47)
(61, 67)
(67, 71)
(73, 79)
```

Action!

```INCLUDE "H6:SIEVE.ACT"

INT FUNC GetNextPrime(INT i BYTE ARRAY primes)
DO
i==+1
UNTIL primes(i)
OD
RETURN (i)

PROC Main()
DEFINE MAXPRIME="99"
DEFINE MAX="200"
BYTE ARRAY primes(MAX+1)
INT i,p

Put(125) PutE() ;clear the screen
Sieve(primes,MAX+1)
FOR i=2 TO MAXPRIME
DO
IF primes(i) THEN
p=GetNextPrime(i,primes)
IF p<=MAXPRIME AND primes(i+p-1)=1 THEN
PrintF("%I+%I-1=%I%E",i,p,i+p-1)
FI
FI
OD
RETURN```
Output:
```3+5-1=7
5+7-1=11
7+11-1=17
11+13-1=23
13+17-1=29
19+23-1=41
29+31-1=59
31+37-1=67
41+43-1=83
43+47-1=89
61+67-1=127
67+71-1=137
73+79-1=151
```

ALGOL 68

Very similar to The ALGOL 68 sample in the Neighbour primes task

```BEGIN  # find adjacent primes p1, p2 such that p1 + p2 - 1 is also prime #
INT max prime = 100;
[]BOOL prime    = PRIMESIEVE ( max prime * 2 );                      # sieve the primes to max prime * 2        #
[]INT low prime = EXTRACTPRIMESUPTO max prime FROMPRIMESIEVE prime;  # get a list of the primes up to max prime #
# find the adjacent primes p1, p2 such that p1 + p2 - 1 is prime #
FOR i TO UPB low prime - 1 DO
IF   INT p1 plus p2 minus 1 = ( low prime[ i ] + low prime[ i + 1 ] ) - 1;
prime[ p1 plus p2 minus 1 ]
THEN print( ( "(",         whole( low prime[ i     ], -3 )
, " +",        whole( low prime[ i + 1 ], -3 )
, " ) - 1 = ", whole( p1 plus p2 minus 1, -3 )
, newline
)
)
FI
OD
END```
Output:
```(  3 +  5 ) - 1 =   7
(  5 +  7 ) - 1 =  11
(  7 + 11 ) - 1 =  17
( 11 + 13 ) - 1 =  23
( 13 + 17 ) - 1 =  29
( 19 + 23 ) - 1 =  41
( 29 + 31 ) - 1 =  59
( 31 + 37 ) - 1 =  67
( 41 + 43 ) - 1 =  83
( 43 + 47 ) - 1 =  89
( 61 + 67 ) - 1 = 127
( 67 + 71 ) - 1 = 137
( 73 + 79 ) - 1 = 151
```

Arturo

```primesBelow100: select 1..100 => prime?

loop 1..dec size primesBelow100 'p [
p1: primesBelow100\[p-1]
p2: primesBelow100\[p]
if prime? dec p1 + p2 ->
print ["(" p1 "," p2 ")"]
]```
Output:
```( 3 , 5 )
( 5 , 7 )
( 7 , 11 )
( 11 , 13 )
( 13 , 17 )
( 19 , 23 )
( 29 , 31 )
( 31 , 37 )
( 41 , 43 )
( 43 , 47 )
( 61 , 67 )
( 67 , 71 )
( 73 , 79 )```

AWK

```# syntax: GAWK -f SPECIAL_NEIGHBOR_PRIMES.AWK
BEGIN {
start = 3
stop = 99
old_prime = 2
for (n=start; n<=stop; n++) {
if (is_prime(n) && is_prime(old_prime)) {
sum = old_prime + n - 1
if (is_prime(sum)) {
count++
printf("%d,%d -> %d\n",old_prime,n,sum)
}
old_prime = n
}
}
printf("Special neighbor primes %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:
```3,5 -> 7
5,7 -> 11
7,11 -> 17
11,13 -> 23
13,17 -> 29
19,23 -> 41
29,31 -> 59
31,37 -> 67
41,43 -> 83
43,47 -> 89
61,67 -> 127
67,71 -> 137
73,79 -> 151
Special neighbor primes 3-99: 13
```

BASIC

FreeBASIC

```#include"isprime.bas"

function nextprime( n as uinteger ) as uinteger
'finds the next prime after n
if n = 0 then return 2
if n < 3 then return n + 1
dim as integer q = n + 2
while not isprime(q)
q+=2
wend
return q
end function

dim as uinteger p1, p2

for p1 = 3 to 100 step 2
p2 = nextprime(p1)
if isprime(p1) andalso p2<100 andalso isprime( p1 + p2 - 1 ) then
print p1, p2, p1 + p2 - 1
end if
next p1
```
Output:
```
3             5             7
5             7             11
7             11            17
11            13            23
13            17            29
19            23            41
29            31            59
31            37            67
41            43            83
43            47            89
61            67            127
67            71            137
73            79            151

```

GW-BASIC

Works with: BASICA
```10 FOR P = 3 TO 99 STEP 2
20 GOSUB 130
30 IF Q = 0 THEN GOTO 110
40 GOSUB 220
50 IF P2>100 THEN END
60 T = P
70 P = P2 + T - 1
80 GOSUB 130
90 IF Q = 1 THEN PRINT USING "## + ## - 1 = ###";T;P2;P
100 P=T
110 NEXT P
120 END
130 REM tests if a number is prime
140 Q=0
150 IF P=3 THEN Q=1:RETURN
160 I=1
170 I=I+1
180 IF INT(P/I)*I = P THEN RETURN
190 IF I*I<=P THEN GOTO 170
200 Q = 1
210 RETURN
220 REM finds the next prime after P, result in P2
230 IF P = 0 THEN P2 = 2: RETURN
240 IF P<3 THEN P2 = P + 1: RETURN
250 T = P
260 P = P + 1
270 GOSUB 130
280 IF Q = 1 THEN P2 = P: P = T: RETURN
290 GOTO 260
```
Output:
``` 3 +  5 - 1 =   7
5 +  7 - 1 =  11
7 + 11 - 1 =  17
11 + 13 - 1 =  23
13 + 17 - 1 =  29
19 + 23 - 1 =  41
29 + 31 - 1 =  59
31 + 37 - 1 =  67
41 + 43 - 1 =  83
43 + 47 - 1 =  89
61 + 67 - 1 = 127
67 + 71 - 1 = 137
73 + 79 - 1 = 151
```

Tiny BASIC

```    REM B = SECOND OF THE NEIGBOURING PRIMES
REM C = P + B - 1
REM I = index variable
REM P = INPUT TO NEXTPRIME ROUTINE AND ISPRIME ROUTINE, also first of the two primes
REM T = Temporary variable, multiple uses
REM Z = OUTPUT OF ISPRIME, 1=prime, 0=not

LET P = 1
20 LET P = P + 2
IF P > 100 THEN END
GOSUB 100
IF Z = 0 THEN GOTO 20
GOSUB 120
IF B > 100 THEN END
LET T = P
LET P = P + B - 1
GOSUB 100
LET C = P
LET P = T
IF Z = 0 THEN GOTO 20
PRINT P," + ",B," - 1 = ", C
GOTO 20

100 REM PRIMALITY BY TRIAL DIVISION
LET Z = 1
LET I = 2
110 IF (P/I)*I = P THEN LET Z = 0
IF Z = 0 THEN RETURN
LET I = I + 1
IF I*I <= P THEN GOTO 110
RETURN

120 REM next prime after P
IF P < 2 THEN LET B = 2
IF P = 2 THEN LET B = 3
IF P < 3 THEN RETURN
LET T = P
130 LET P = P + 1
GOSUB 100
IF Z = 1 THEN GOTO 140
GOTO 130
140 LET B = P
LET P = T
RETURN```
Output:
```
3 + 5 - 1 = 7
5 + 7 - 1 = 11
7 + 11 - 1 = 17
11 + 13 - 1 = 23
13 + 17 - 1 = 29
19 + 23 - 1 = 41
29 + 31 - 1 = 59
31 + 37 - 1 = 67
41 + 43 - 1 = 83
43 + 47 - 1 = 89
61 + 67 - 1 = 127
67 + 71 - 1 = 137
73 + 79 - 1 = 151

```

C

```#include<stdio.h>
#include<stdlib.h>

int isprime( int p ) {
int i;
if(p==2) return 1;
if(!(p%2)) return 0;
for(i=3; i*i<=p; i+=2) {
if(!(p%i)) return 0;
}
return 1;
}

int nextprime( int p ) {
int i=0;
if(p==0) return 2;
if(p<3) return p+1;
while(!isprime(++i + p));
return i+p;
}

int main(void) {
int p1, p2;
for(p1=3;p1<=99;p1+=2) {
p2=nextprime(p1);
if(p2<100&&isprime(p1)&&isprime(p2+p1-1)) {
printf( "%d + %d - 1 = %d\n", p1, p2, p1+p2-1 );
}
}
return 0;
}
```
Output:
```3 + 5 - 1 = 7
5 + 7 - 1 = 11
7 + 11 - 1 = 17
11 + 13 - 1 = 23
13 + 17 - 1 = 29
19 + 23 - 1 = 41
29 + 31 - 1 = 59
31 + 37 - 1 = 67
41 + 43 - 1 = 83
43 + 47 - 1 = 89
61 + 67 - 1 = 127
67 + 71 - 1 = 137
73 + 79 - 1 = 151

```

F#

This task uses Extensible Prime Generator (F#)

```// Special neighbor primes. Nigel Galloway: August 6th., 2021
pCache|>Seq.pairwise|>Seq.takeWhile(snd>>(>)100)|>Seq.filter(fun(n,g)->isPrime(n+g-1))|>Seq.iter(printfn "%A")
```
Output:
```(3, 5)
(5, 7)
(7, 11)
(11, 13)
(13, 17)
(19, 23)
(29, 31)
(31, 37)
(41, 43)
(43, 47)
(61, 67)
(67, 71)
(73, 79)
```

Factor

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

lprimes dup cdr lzip [ sum 1 - prime? ] lfilter
[ second 100 < ] lwhile [ . ] leach
```
Output:
```{ 3 5 }
{ 5 7 }
{ 7 11 }
{ 11 13 }
{ 13 17 }
{ 19 23 }
{ 29 31 }
{ 31 37 }
{ 41 43 }
{ 43 47 }
{ 61 67 }
{ 67 71 }
{ 73 79 }
```

Fermat

```Func Nextprime(p) =
q:=1;
while not Isprime(p+q)=1 do
q:=q + 1;
od;
p+q.;

for p1 = 3 to 99 by 2 do
p2:=Nextprime(p1);
if p2<100 and Isprime(p1)=1 and Isprime(p1+p2-1) then
!!(p1,' +',p2,' - 1 =',p1+p2-1);
fi;
od;```
Output:
```
3 +  5 - 1 =  7
5 +  7 - 1 =  11
7 +  11 - 1 =  17
11 +  13 - 1 =  23
13 +  17 - 1 =  29
19 +  23 - 1 =  41
29 +  31 - 1 =  59
31 +  37 - 1 =  67
41 +  43 - 1 =  83
43 +  47 - 1 =  89
61 +  67 - 1 =  127
67 +  71 - 1 =  137
73 +  79 - 1 =  151

```

Go

Translation of: Wren
Library: Go-rcu
```package main

import (
"fmt"
"rcu"
)

const MAX = 1e7 - 1

var primes = rcu.Primes(MAX)

func specialNP(limit int, showAll bool) {
if showAll {
fmt.Println("Neighbor primes, p1 and p2, where p1 + p2 - 1 is prime:")
}
count := 0
for i := 1; i < len(primes); i++ {
p2 := primes[i]
if p2 >= limit {
break
}
p1 := primes[i-1]
p3 := p1 + p2 - 1
if rcu.IsPrime(p3) {
if showAll {
fmt.Printf("(%2d, %2d) => %3d\n", p1, p2, p3)
}
count++
}
}
ccount := rcu.Commatize(count)
climit := rcu.Commatize(limit)
fmt.Printf("\nFound %s special neighbor primes under %s.\n", ccount, climit)
}

func main() {
specialNP(100, true)
var pow = 1000
for i := 3; i < 8; i++ {
specialNP(pow, false)
pow *= 10
}
}
```
Output:
```Neighbor primes, p1 and p2, where p1 + p2 - 1 is prime:
( 3,  5) =>   7
( 5,  7) =>  11
( 7, 11) =>  17
(11, 13) =>  23
(13, 17) =>  29
(19, 23) =>  41
(29, 31) =>  59
(31, 37) =>  67
(41, 43) =>  83
(43, 47) =>  89
(61, 67) => 127
(67, 71) => 137
(73, 79) => 151

Found 13 special neighbor primes under 100.

Found 71 special neighbor primes under 1,000.

Found 367 special neighbor primes under 10,000.

Found 2,165 special neighbor primes under 100,000.

Found 14,526 special neighbor primes under 1,000,000.

Found 103,611 special neighbor primes under 10,000,000.
```

J

```   (,._1++/"1)p: 0 1+/~I.1 p: _1++/ p: _1 0 +/}.i. p:inv 100
3  5   7
5  7  11
7 11  17
11 13  23
13 17  29
19 23  41
29 31  59
31 37  67
41 43  83
43 47  89
61 67 127
67 71 137
73 79 151
```

jq

Works with: jq

Works with gojq, the Go implementation of jq

This entry uses `is_prime` as defined at Erdős-primes#jq.

```# Assumes . > 2
def next_prime:
first(range(.+2; infinite) | select(is_prime));

def specialNP(\$savePairs):
. as \$limit
| {p1: 2, p2: 3}
| until( .p2 >= \$limit;
if (.p1 + .p2 - 1 | is_prime)
then .pcount += 1
| if \$savePairs then .neighbors = .neighbors + [[.p1, .p2]] else . end
else .
end
| .p1 = .p2
| .p2 = (.p1|next_prime)
)
| if \$savePairs then {pcount, neighbors} else {pcount} end;

100|specialNP(true)```
Output:
```{"pcount":13,"neighbors":[[3,5],[5,7],[7,11],[11,13],[13,17],[19,23],[29,31],[31,37],[41,43],[43,47],[61,67],[67,71],[73,79]]}
```

Julia

```using Primes

function specialneighbors(N, savepairs=true)
neighbors, p1, pcount = Pair{Int}[], 2, 0
while (p2 = nextprime(p1 + 1)) < N
if isprime(p2 + p1 - 1)
savepairs && push!(neighbors, p1 => p2)
pcount += 1
end
p1 = p2
end
return neighbors, pcount
end

spn, n = specialneighbors(100)
println("\$n special neighbor prime pairs under 100:")
println("p1   p2   p1 + p2 - 1\n--------------------------")
for (p1, p2) in specialneighbors(100)[1]
end

print("\nCount of such prime pairs under 1,000,000,000: ",
specialneighbors(1_000_000_000, false)[2])
```
Output:
```13 special neighbor prime pairs under 100:
p1   p2   p1 + p2 - 1
--------------------------
3   5      7
5   7      11
7   11     17
11   13     23
13   17     29
19   23     41
29   31     59
31   37     67
41   43     83
43   47     89
61   67     127
67   71     137
73   79     151

Count of such prime pairs under 1,000,000,000: 6041231
```

Mathematica/Wolfram Language

```p = Prime@Range@PrimePi[100];
Select[Partition[p, 2, 1], Total/*(# - 1 &)/*PrimeQ]
```
Output:
`{{3, 5}, {5, 7}, {7, 11}, {11, 13}, {13, 17}, {19, 23}, {29, 31}, {31, 37}, {41, 43}, {43, 47}, {61, 67}, {67, 71}, {73, 79}}`

Nim

```import strutils, sugar

const Max = 100 - 1

func isPrime(n: Positive): bool =
if n == 1: return false
if n mod 2 == 0: return n == 2
for d in countup(3, n, 2):
if d * d > n: break
if n mod d == 0: return false
result = true

const Primes = collect(newSeq):
for n in 2..Max:
if n.isPrime: n

let list = collect(newSeq):
for i in 0..<Primes.high:
let p1 = Primes[i]
let p2 = Primes[i + 1]
if (p1 + p2 - 1).isPrime: (p1, p2)

echo "Found \$1 special neighbor primes less than \$2:".format(list.len, Max + 1)
echo list.join(", ")
```
Output:
```Found 13 special neighbor primes less than 100:
(3, 5), (5, 7), (7, 11), (11, 13), (13, 17), (19, 23), (29, 31), (31, 37), (41, 43), (43, 47), (61, 67), (67, 71), (73, 79)```

PARI/GP

`for(p1=1,100,p2=nextprime(p1+1); if(isprime(p1)&&p2<100&&isprime(p1+p2-1),print(p1," ",p2," ",p1+p2-1)))`

Perl

```#!/usr/bin/perl

use strict; # https://rosettacode.org/wiki/Special_neighbor_primes
use warnings;
use ntheory qw( primes is_prime );

my @primes = @{ primes(100) };
for ( 1 .. \$#primes )
{
is_prime( \$@ = \$primes[\$_-1] + \$primes[\$_] - 1 ) and
printf "%2d + %2d - 1 = %3d\n", \$primes[\$_-1], \$primes[\$_], \$@;
}
```
Output:
``` 3 +  5 - 1 =   7
5 +  7 - 1 =  11
7 + 11 - 1 =  17
11 + 13 - 1 =  23
13 + 17 - 1 =  29
19 + 23 - 1 =  41
29 + 31 - 1 =  59
31 + 37 - 1 =  67
41 + 43 - 1 =  83
43 + 47 - 1 =  89
61 + 67 - 1 = 127
67 + 71 - 1 = 137
73 + 79 - 1 = 151
```

Phix

```with javascript_semantics
function np(integer n) return is_prime(get_prime(n)+get_prime(n+1)-1) end function
function npt(integer p) return filter(tagset(length(get_primes_le(p))-1),np) end function
sequence s = npt(100)
printf(1,"Found %d special neighbour primes < 100:\n",length(s))
for i=1 to length(s) do
integer si = s[i],
pi = get_prime(si),
pj = get_prime(si+1)
printf(1," (%2d,%2d) => %d\n",{pi,pj,pi+pj-1})
end for
printf(1,"\n")
for i=2 to 7 do
integer p = power(10,i),
l = length(npt(p))
printf(1,"Found %,d special neighbour primes < %,d\n",{l,p})
end for
```
Output:
```Found 13 special neighbour primes < 100:
( 3, 5) => 7
( 5, 7) => 11
( 7,11) => 17
(11,13) => 23
(13,17) => 29
(19,23) => 41
(29,31) => 59
(31,37) => 67
(41,43) => 83
(43,47) => 89
(61,67) => 127
(67,71) => 137
(73,79) => 151

Found 13 special neighbour primes < 100
Found 71 special neighbour primes < 1,000
Found 367 special neighbour primes < 10,000
Found 2,165 special neighbour primes < 100,000
Found 14,526 special neighbour primes < 1,000,000
Found 103,611 special neighbour primes < 10,000,000
```

PL/0

PL/0 can only output a single integer per line, so to avoid confusing output, this sample just shows the first prime of each pair.
This is almost identical to the PL/0 sample in the Neighbour primes task

```var   n, p1, p2, prime;
procedure isnprime;
var p;
begin
prime := 1;
if n < 2 then prime := 0;
if n > 2 then begin
prime := 0;
if odd( n ) then prime := 1;
p := 3;
while p * p <= n * prime do begin
if n - ( ( n / p ) * p ) = 0 then prime := 0;
p := p + 2;
end
end
end;
begin
p1 := 3;
p2 := 5;
while p2 < 100 do begin
n := ( p1 + p2 ) - 1;
call isnprime;
if prime = 1 then ! p1;
n  := p2 + 2;
call isnprime;
while prime = 0 do begin
n := n + 2;
call isnprime;
end;
p1 := p2;
p2 := n;
end
end.
```
Output:
```          3
5
7
11
13
19
29
31
41
43
61
67
73
```

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

def nextPrime(n):
#finds the next prime after n
if n == 0:
return 2
if n < 3:
return n + 1
q = n + 2
while not isPrime(q):
q += 2
return q

if __name__ == "__main__":
for p1 in range(3,100,2):
p2 = nextPrime(p1)
if isPrime(p1) and p2 < 100 and isPrime(p1 + p2 - 1):
print(p1,'\t', p2,'\t', p1 + p2 - 1)
```
Output:
```3 	 5 	 7
5 	 7 	 11
7 	 11 	 17
11 	 13 	 23
13 	 17 	 29
19 	 23 	 41
29 	 31 	 59
31 	 37 	 67
41 	 43 	 83
43 	 47 	 89
61 	 67 	 127
67 	 71 	 137
73 	 79 	 151```

Raku

```# 20210809 Raku programming solution

for (grep {.is-prime}, 3..*).rotor(2 => -1) -> (\P1,\P2) {
last if P2 ≥ Ⅽ;
(\$_ = P1+P2-1).is-prime and printf "%2d, %2d => %3d\n", P1, P2, \$_
}
```
Output:
``` 3,  5 =>   7
5,  7 =>  11
7, 11 =>  17
11, 13 =>  23
13, 17 =>  29
19, 23 =>  41
29, 31 =>  59
31, 37 =>  67
41, 43 =>  83
43, 47 =>  89
61, 67 => 127
67, 71 => 137
73, 79 => 151```

REXX

A little extra code was added to present the results in a grid-like format.

```/*REXX pgm finds special neighbor primes:  P1, P2, P1+P2-1  are prime, and P1 and P2<100*/
parse arg hi cols .                              /*obtain optional argument from the CL.*/
if   hi=='' |   hi==","  then   hi=  100         /*Not specified?  Then use the default.*/
if cols=='' | cols==","  then cols=    5         /* "      "         "   "   "     "    */
call genP hi                                     /*build semaphore array for low primes.*/
do p=1  while @.p<hi
end  /*p*/;           lim= p-1;   q= p+1    /*set LIM to prime for P; calc. 2nd HI.*/
#m= # - 1
call genP @.# + @.#m  -  1                       /*build semaphore array for high primes*/
w= 20                                            /*width of a number in any column.     */
title= ' special neighbor primes:  P1, P2, P1+P2-1  are primes,  and P1 and P2 < ' ,
commas(hi)
if cols>0 then say ' index │'center(title,   1 + cols*(w+1)     )
if cols>0 then say '───────┼'center(""   ,   1 + cols*(w+1), '─')
found= 0;                  idx= 1                /*initialize # neighbor primes & index.*/
\$=                                               /*a list of  neighbor  primes (so far).*/
do j=1  to  lim;      jp= j+1;   q= @.jp    /*look for neighbor primes within range*/
y= @.j + q  -  1;     if \!.y  then iterate /*is X also a prime?  No, then skip it.*/
found= found + 1                            /*bump the number of  neighbor primes. */
if cols==0            then iterate          /*Build the list  (to be shown later)? */
\$= \$  right( @.j','q"──►"y, w)              /*add neighbor prime ──► the  \$  list. */
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   /*j*/

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: !.= 0;            parse arg limit          /*placeholders for primes (semaphores).*/
@.1=2;  @.2=3;  @.3=5;  @.4=7;  @.5=11     /*define some low primes.              */
!.2=1;  !.3=1;  !.5=1;  !.7=1;  !.11=1     /*   "     "   "    "     flags.       */
#=5;     sq.#= @.# **2   /*number of primes so far;     prime². */
/* [↓]  generate more  primes  ≤  high.*/
do j=@.#+2  by 2  to limit               /*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# */
end          /*j*/;               return
```
output   when using the default inputs:
``` index │               special neighbor primes:  P1, P2, P1+P2-1  are primes,  and P1 and P2 <  100
───────┼──────────────────────────────────────────────────────────────────────────────────────────────────────────
1   │              3,5──►7             5,7──►11            7,11──►17           11,13──►23           13,17──►29
6   │           19,23──►41           29,31──►59           31,37──►67           41,43──►83           43,47──►89
11   │          61,67──►127          67,71──►137          73,79──►151
───────┴──────────────────────────────────────────────────────────────────────────────────────────────────────────

Found  13  special neighbor primes:  P1, P2, P1+P2-1  are primes,  and P1 and P2 <  100
```

Ring

```load "stdlib.ring"

see "working..." + nl
see "Special neighbor primes are:" + nl
row = 0
oldPrime = 2

for n = 3 to 100
if isprime(n) and isprime(oldPrime)
sum = oldPrime + n - 1
if isprime(sum)
row++
see "" + oldPrime + "," + n + " => " + sum + nl
ok
oldPrime = n
ok
next

see "Found " + row + " special neighbor primes"
see "done..." + nl```
Output:
```working...
Special neighbor primes are:
3,5 => 7
5,7 => 11
7,11 => 17
11,13 => 23
13,17 => 29
19,23 => 41
29,31 => 59
31,37 => 67
41,43 => 83
43,47 => 89
61,67 => 127
67,71 => 137
73,79 => 151
Found 13 special neighbor primes
done...
```

Sidef

```func special_neighbor_primes(upto) {
var list = []
upto.primes.each_cons(2, {|p1,p2|
var n = (p1 + p2 - 1)
if (n.is_prime) {
list << [p1, p2, n]
}
})
return list
}

with (100) {|n|
var list = special_neighbor_primes(n)
say "Found #{list.len} special neighbour primes < n:"
list.each_2d {|p1,p2,q|
printf(" (%2s, %2s) => %s\n", p1, p2, q)
}
}

say ''

for n in (1..7) {
var list = special_neighbor_primes(10**n)
say "Found #{list.len} special neighbour primes < 10^#{n}"
}
```
Output:
```Found 13 special neighbour primes < n:
( 3,  5) => 7
( 5,  7) => 11
( 7, 11) => 17
(11, 13) => 23
(13, 17) => 29
(19, 23) => 41
(29, 31) => 59
(31, 37) => 67
(41, 43) => 83
(43, 47) => 89
(61, 67) => 127
(67, 71) => 137
(73, 79) => 151

Found 2 special neighbour primes < 10^1
Found 13 special neighbour primes < 10^2
Found 71 special neighbour primes < 10^3
Found 367 special neighbour primes < 10^4
Found 2165 special neighbour primes < 10^5
Found 14526 special neighbour primes < 10^6
Found 103611 special neighbour primes < 10^7
```

Wren

Library: Wren-math
Library: Wren-fmt

I assume that 'neighbor' primes means pairs of successive primes.

Anticipating a likely stretch goal.

```import "/math" for Int
import "/fmt" for Fmt

var max = 1e7 - 1
var primes = Int.primeSieve(max)

var specialNP = Fn.new { |limit, showAll|
if (showAll) System.print("Neighbor primes, p1 and p2, where p1 + p2 - 1 is prime:")
var count = 0
var p3
for (i in 1...primes.where { |p| p < limit }.count) {
var p2 = primes[i]
var p1 = primes[i-1]
if (Int.isPrime(p3 = p1 + p2 - 1)) {
if (showAll) Fmt.print("(\$2d, \$2d) => \$3d", p1, p2, p3)
count = count + 1
}
}
Fmt.print("\nFound \$,d special neighbor primes under \$,d.", count, limit)
}

specialNP.call(100, true)
for (i in 3..7) {
specialNP.call(10.pow(i), false)
}
```
Output:
```Neighbor primes, p1 and p2, where p1 + p2 - 1 is prime:
( 3,  5) =>   7
( 5,  7) =>  11
( 7, 11) =>  17
(11, 13) =>  23
(13, 17) =>  29
(19, 23) =>  41
(29, 31) =>  59
(31, 37) =>  67
(41, 43) =>  83
(43, 47) =>  89
(61, 67) => 127
(67, 71) => 137
(73, 79) => 151

Found 13 special neighbor primes under 100.

Found 71 special neighbor primes under 1,000.

Found 367 special neighbor primes under 10,000.

Found 2,165 special neighbor primes under 100,000.

Found 14,526 special neighbor primes under 1,000,000.

Found 103,611 special neighbor primes under 10,000,000.
```

XPL0

```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;
];

int P, P1, P2;
[P:= 2;
loop    [P1:= P;
repeat  P:= P+1;
if P >= 100 then quit;
until   IsPrime(P);
P2:= P;
if IsPrime(P1+P2-1) then
[IntOut(0, P1);  ChOut(0, ^ );
IntOut(0, P2);  ChOut(0, ^ );
IntOut(0, P1+P2-1);  CrLf(0);
];
];
]```
Output:
```3 5 7
5 7 11
7 11 17
11 13 23
13 17 29
19 23 41
29 31 59
31 37 67
41 43 83
43 47 89
61 67 127
67 71 137
73 79 151
```