Neighbour primes

Neighbour 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

Find and show primes p such that p*q+2 is prime, where q is next prime after p and p < 500

FreeBASIC

<lang freebasic>#include "isprime.bas"

dim as uinteger q

print "p q pq+2" print "--------------------------------" for p as uinteger = 2 to 499

    if not isprime(p) then continue for
    q = p + 1
    while not isprime(q)
        q+=1
    wend
    if not isprime( 2 + p*q ) then continue for
    print p,q,2+p*q

next p</lang>

Output:

p             q             pq+2
--------------------------------
3             5             17
5             7             37
7             11            79
13            17            223
19            23            439
67            71            4759
149           151           22501
179           181           32401
229           233           53359
239           241           57601
241           251           60493
269           271           72901
277           281           77839
307           311           95479
313           317           99223
397           401           159199
401           409           164011
419           421           176401
439           443           194479
487           491           239119

Julia

<lang julia>using Primes

isneiprime(known) = isprime(known) && isprime(known * nextprime(known + 1) + 2) println(filter(isneiprime, primes(500)))

</lang>

Output:

[3, 5, 7, 13, 19, 67, 149, 179, 229, 239, 241, 269, 277, 307, 313, 397, 401, 419, 439, 487]

Raku

<lang perl6>my @primes = grep &is-prime, ^Inf; my $last_p = @primes.first: :k, * >= 500; my $last_q = $last_p + 1;

my @cousins = @primes.head( $last_q )

                    .rotor( 2 => -1 )
                    .map(-> (\p, \q) { p, q, p*q+2 } )
                    .grep( *.[2].is-prime );

say .fmt('%6d') for @cousins;</lang>

Output:

     3      5     17
     5      7     37
     7     11     79
    13     17    223
    19     23    439
    67     71   4759
   149    151  22501
   179    181  32401
   229    233  53359
   239    241  57601
   241    251  60493
   269    271  72901
   277    281  77839
   307    311  95479
   313    317  99223
   397    401 159199
   401    409 164011
   419    421 176401
   439    443 194479
   487    491 239119

Ring

<lang ring> load "stdlib.ring" see "working..." + nl see "Neighbour primes are:" + nl see "p q p*q+2" + nl

row = 0 num = 0 pr = 0 limit = 100 Primes = []

while true

   pr = pr + 1
   if isprime(pr)
      add(Primes,pr)
      num = num + 1
      if num = limit 
         exit
      ok
   ok

end

for n = 1 to limit-1

   prim = Primes[n]*Primes[n+1]+2
   if isprime(prim)
      row = row + 1
      see "" + Primes[n] + " " + Primes[n+1] + " " + prim + nl
   ok

next

see "Found " + row + " neighbour primes" + nl see "done..." + nl </lang>

Output:

working...
Neighbour primes are:
p q p*q+2
3 5 17
5 7 37
7 11 79
13 17 223
19 23 439
67 71 4759
149 151 22501
179 181 32401
229 233 53359
239 241 57601
241 251 60493
269 271 72901
277 281 77839
307 311 95479
313 317 99223
397 401 159199
401 409 164011
419 421 176401
439 443 194479
487 491 239119
Found 20 neighbour primes
done...