Neighbour primes
- Task
Find and show primes p such that p*q+2 is prime, where q is next prime after p and p < 500
ALGOL W
<lang algolw>begin % find some primes where ( p*q ) + 2 is also a prime ( where p and q are adjacent primes ) %
% sets p( 1 :: n ) to a sieve of primes up to n %
procedure sieve ( logical array p( * ) ; integer value n ) ;
begin
p( 1 ) := false; p( 2 ) := true;
for i := 3 step 2 until n do p( i ) := true;
for i := 4 step 2 until n do p( i ) := false;
for i := 3 step 2 until truncate( sqrt( n ) ) do begin
integer ii; ii := i + i;
if p( i ) then for np := i * i step ii until n do p( np ) := false
end for_i ;
end sieve ;
integer MAX_NUMBER, MAX_PRIME;
MAX_NUMBER := 500;
MAX_PRIME := MAX_NUMBER * MAX_NUMBER;
begin
logical array prime( 1 :: MAX_PRIME );
integer pCount, thisPrime, nextPrime;
% sieve the primes to MAX_PRIME %
sieve( prime, MAX_PRIME );
% find the neighbour primes %
pCount := 0;
thisPrime := 2; % 2 is the lowest prime %
while thisPrime > 0 do begin
% find the next prime after this one %
nextPrime := thisPrime + 1;
while nextPrime <= MAX_NUMBER and not prime( nextPrime ) do nextPrime := nextPrime + 1;
if nextPrime > MAX_NUMBER then thisPrime := 0
else begin
if prime( ( thisPrime * nextPrime ) + 2 ) then begin
% have another neighbour prime %
writeon( i_w := 1, s_w := 0, " ", thisPrime );
pCount := pCount + 1
end if_prime__thisPrime_x_nextPrime_plus_2 ;
thisPrime := nextPrime
end if_nextPrime_gt_MAX_NUMBER__
end while_thisPrime_gt_0 ;
write( i_w := 1, s_w := 0, "Found ", pCount, " neighbour primes up to 500" )
end
end.</lang>
- Output:
3 5 7 13 19 67 149 179 229 239 241 269 277 307 313 397 401 419 439 487 Found 20 neighbour primes up to 500
AppleScript
<lang applescript>on isPrime(n)
if (n < 6) then return ((n > 1) and (n is not 4))
if ((n mod 2 = 0) or (n mod 3 = 0) or (n mod 5 = 0)) then return false
repeat with i from 7 to (n ^ 0.5) div 1 by 30
if (n mod i = 0) or (n mod (i + 4) = 0) or (n mod (i + 6) = 0) or (n mod (i + 10) = 0) or ¬
(n mod (i + 12) = 0) or (n mod (i + 16) = 0) or (n mod (i + 22) = 0) or (n mod (i + 24) = 0) then ¬
return false
end repeat
return true
end isPrime
on neighbourPrimes(max)
set output to {}
repeat with p from 3 to max by 2
if (isPrime(p)) then
set q to p + 2
repeat until (isPrime(q))
set q to q + 2
end repeat
if (isPrime(p * q + 2)) then set end of output to p
end if
end repeat
return output
end neighbourPrimes
neighbourPrimes(499)</lang>
- Output:
<lang applescript>{3, 5, 7, 13, 19, 67, 149, 179, 229, 239, 241, 269, 277, 307, 313, 397, 401, 419, 439, 487}</lang>
Arturo
<lang rebol>primesUpTo500: select 1..500 => prime?
print [pad "p" 5 pad "q" 4 pad "p*q+2" 7] print "--------------------" i: 0 while [i < dec size primesUpTo500][
p: primesUpTo500\[i]
q: primesUpTo500\[i+1]
if prime? 2 + p * q [
prints pad to :string p 5
prints pad to :string q 5
print pad to :string 2 + p * q 8
]
i: i + 1
]</lang>
- Output:
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
AWK
<lang AWK>
- syntax: GAWK -f NEIGHBOUR_PRIMES.AWK
BEGIN {
print(" p q p*q+2")
print("---- ---- ------")
start = 1
stop = 499
for (p=start; p<=stop; p++) {
if (!is_prime(p)) { continue }
q = p + 1
while (!is_prime(q)) {
q++
}
if (!is_prime(p*q+2)) { continue }
printf("%4d %4d %6d\n",p,q,p*q+2)
count++
}
printf("Neighbour 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)
} </lang>
- Output:
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 Neighbour primes 1-499: 20
C#
How about some other offsets besides + 2 ?
<lang fsharp>using System; using System.Collections.Generic;
using System.Linq; using static System.Console; using System.Collections;
class Program {
static void Main(string[] args) {
WriteLine ("Multiply two consecutive prime numbers, add an even number," +
" see if the result is a prime number (up to a limit).");
int c, lim = 500; var pr = PG.Primes(lim * lim).ToList();
pr = pr.TakeWhile(x => x < lim).ToList();
var Lst = new[]{ Tuple.Create(2, 2), Tuple.Create(-20, 20) };
foreach (var pair in Lst) {
bool sho = pair.Item1 == pair.Item2;
for (int ofs = pair.Item1; ofs <= pair.Item2; ofs += ofs == -2 ? 4 : 2) {
c = 0; string s = ofs.ToString("+0;-#").Insert(1, " ");
for (int i = 0, j = 1, k; j < pr.Count; i = j++)
if (PG.isPr(k = pr[i] * pr[j] + ofs))
if (sho) WriteLine (" {0,3} * {1,3} {2} = {3,-6}",
pr[i], pr[j], s, k, c++);
else c++;
WriteLine("{0,2} found under {1} for \" {2} \"", c, lim, s);
} WriteLine (); } } }
class PG { static bool[] flags; public static bool isPr(int x) {
if (x < 2) return false; return !flags[x]; }
public static IEnumerable<int> Primes(int lim) {
flags = new bool[lim + 1]; int j = 3;
for (int d = 8, sq = 9; sq <= lim; j += 2, sq += d += 8)
if (!flags[j]) { yield return j;
for (int k = sq, i=j<<1; k<=lim; k += i) flags[k] = true; }
for (; j <= lim; j += 2) if (!flags[j]) yield return j; } }</lang>
- Output:
Multiply two consecutive prime numbers, add an even number, see if the result is a prime number (up to a limit).
3 * 5 + 2 = 17
5 * 7 + 2 = 37
7 * 11 + 2 = 79
13 * 17 + 2 = 223
19 * 23 + 2 = 439
67 * 71 + 2 = 4759
149 * 151 + 2 = 22501
179 * 181 + 2 = 32401
229 * 233 + 2 = 53359
239 * 241 + 2 = 57601
241 * 251 + 2 = 60493
269 * 271 + 2 = 72901
277 * 281 + 2 = 77839
307 * 311 + 2 = 95479
313 * 317 + 2 = 99223
397 * 401 + 2 = 159199
401 * 409 + 2 = 164011
419 * 421 + 2 = 176401
439 * 443 + 2 = 194479
487 * 491 + 2 = 239119
20 found under 500 for " + 2 "
5 found under 500 for " - 20 "
26 found under 500 for " - 18 "
22 found under 500 for " - 16 "
10 found under 500 for " - 14 "
22 found under 500 for " - 12 "
21 found under 500 for " - 10 "
13 found under 500 for " - 8 "
32 found under 500 for " - 6 "
20 found under 500 for " - 4 "
5 found under 500 for " - 2 "
20 found under 500 for " + 2 "
9 found under 500 for " + 4 "
36 found under 500 for " + 6 "
18 found under 500 for " + 8 "
11 found under 500 for " + 10 "
27 found under 500 for " + 12 "
20 found under 500 for " + 14 "
8 found under 500 for " + 16 "
17 found under 500 for " + 18 "
25 found under 500 for " + 20 "
F#
This task uses Extensible Prime Generator (F#) <lang fsharp> // Nigel Galloway. April 13th., 2021 primes32()|>Seq.pairwise|>Seq.takeWhile(fun(n,_)->n<500)|>Seq.filter(fun(n,g)->isPrime(n*g+2))|>Seq.iter(fun(n,g)->printfn "%d*%d=%d" n g (n*g+2)) </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 Real: 00:00:00.029
Factor
<lang factor>USING: formatting io kernel math math.primes ;
"p q p*q+2" print 2 3 [ over 500 < ] [
2dup * 2 + dup prime? [ 3dup "%-4d %-4d %-6d\n" printf ] when drop nip dup next-prime
] while 2drop</lang>
- Output:
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
Fermat
<lang fermat>for i = 1 to 95 do if Isprime(2+Prime(i)*Prime(i+1)) then !!Prime(i) fi od</lang>
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
Fōrmulæ
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In this page you can see the program(s) related to this task and their results.
Go
<lang go>package main
import (
"fmt" "rcu"
)
func main() {
primes := rcu.Primes(504)
var nprimes []int
fmt.Println("Neighbour primes < 500:")
for i := 0; i < len(primes)-1; i++ {
p := primes[i]*primes[i+1] + 2
if rcu.IsPrime(p) {
nprimes = append(nprimes, primes[i])
}
}
rcu.PrintTable(nprimes, 10, 3, false)
fmt.Println("\nFound", len(nprimes), "such primes.")
}</lang>
- Output:
Neighbour primes < 500: 3 5 7 13 19 67 149 179 229 239 241 269 277 307 313 397 401 419 439 487 Found 20 such primes.
jq
Works with gojq, the Go implementation of jq
This entry uses `is_prime` as defined at Erdős-primes#jq. <lang jq>def next_prime:
if . == 2 then 3 else first(range(.+2; infinite; 2) | select(is_prime)) end;
- (not actually used)
def is_neighbour_prime:
is_prime and ((. * next_prime) + 2 | is_prime);
- The task, implemented using only `next_prime` for efficiency
{p: 2} | while (.p < 500;
(.p|next_prime) as $np
| .emit = false
| if (.p * $np) + 2 | is_prime
then .emit = .p
else .
end
| .p = $np )
| select(.emit).emit
</lang>
- Output:
3 5 7 13 19 67 149 179 229 239 241 269 277 307 313 397 401 419 439 487
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]
Ksh
<lang ksh>
- !/bin/ksh
- Find and show primes p such that p*q+2 is prime, where q is next prime after p and p<500
- # Variables:
integer MAX_PRIME=500
typeset -a parr
- # Functions:
- # Function _isprime(n) return 1 for prime, 0 for not prime
function _isprime { typeset _n ; integer _n=$1 typeset _i ; integer _i
(( _n < 2 )) && return 0 for (( _i=2 ; _i*_i<=_n ; _i++ )); do (( ! ( _n % _i ) )) && return 0 done return 1 }
- # Function _neighbourprime(n) return p*q+2 if prime; 0 if not
function _neighbourprime { typeset _indx ; integer _indx=$1 typeset _arr ; nameref _arr="$2" typeset _neighbor
(( _neighbor = _arr[_indx] * _arr[_indx+1] + 2 )) _isprime ${_neighbor} (( $? )) && echo ${_neighbor} && return echo 0 }
######
- main #
######
for ((i=2; i<MAX_PRIME; i++)); do _isprime ${i} ; (( $? )) && parr+=( ${i} ) done
printf "%3s %3s %6s\n" p q p*q+2 printf "%3s %3s %6s\n" --- --- ----- for ((i=0; i<$((${#parr[*]}-1)); i++)); do np=$(_neighbourprime ${i} parr) (( np > 0 )) && printf "%3d %3d %6d\n" ${parr[i]} ${parr[i+1]} ${np}
done</lang>
- Output:
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
Mathematica /Wolfram Language
<lang Mathematica>p = Prime@Range@PrimePi[499]; Select[p, PrimeQ[# NextPrime[#] + 2] &]</lang>
- Output:
{3, 5, 7, 13, 19, 67, 149, 179, 229, 239, 241, 269, 277, 307, 313, 397, 401, 419, 439, 487}
Nim
<lang Nim>import strformat, sugar
const
Max1 = 499 # Maximum for first prime. Max2 = 251_000 # Maximum for sieve (in fact 250_999 = 499 * 503 + 2).
- Sieve of Erathosthenes: false (default) is composite.
var composite: array[3..Max2, bool] # Ignore 2 as 2 * 3 + 8 is not prime. var n = 3 while true:
let n2 = n * n
if n2 > Max2: break
if not composite[n]:
for k in countup(n2, Max2, 2 * n):
composite[k] = true
inc n, 2
template isPrime(n: int): bool = not composite[n]
let primes = collect(newSeq):
for n in countup(3, Max2, 2):
if n.isPrime: n
var p = primes[0] var i = 0 while p <= Max1:
inc i
let q = primes[i]
if (p * q + 2).isPrime:
echo &"{p:3} {q:3} {p*q+2:6}"
p = q</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
PARI/GP
Cheats a little in the sense that it requires knowing the 95th prime is 499 beforehand. <lang parigp>for(i=1, 95, if(isprime(2+prime(i)*prime(i+1)),print(prime(i))))</lang>
Perl
<lang perl>use strict; use warnings; use ntheory <next_prime is_prime>;
my $p = 2; do {
my $q = next_prime($p); printf "%3d%5d%8d\n", $p, $q, $p*$q+2 if is_prime $p*$q+2; $p = $q;
} until $p >= 500;</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
Phix
function np(integer p) return is_prime(get_prime(p)*get_prime(p+1)+2) end function constant N = length(get_primes_le(500)) sequence res = apply(apply(filter(tagset(N),np),get_prime),sprint) printf(1,"Found %d such primes: %s\n",{length(res),join(shorten(res,"",5),", ")})
- Output:
Found 20 such primes: 3, 5, 7, 13, 19, ..., 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
REXX
Neighbor primes can also be spelled neighbour primes. <lang rexx>/*REXX program finds neighbor primes: P, Q, P*Q+2 are primes, and P < some specified N.*/ parse arg hi cols . /*obtain optional argument from the CL.*/ if hi== | hi=="," then hi= 500 /*Not specified? Then use the default.*/ if cols== | cols=="," then cols= 10 /* " " " " " " */ call genP hi+50 /*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.*/
call genP @.p * @.q + 2 /*build semaphore array for high primes*/ w= 10 /*width of a number in any column. */
@neig= ' neighbor primes: p, q, p*q+2 are primes, and p < ' commas(hi)
if cols>0 then say ' index │'center(@neig, 1 + cols*(w+1) ) if cols>0 then say '───────┼'center("" , 1 + cols*(w+1), '─') Nprimes= 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*/
x= @.j * q + 2; if \!.x then iterate /*is X also a prime? No, then skip it.*/
Nprimes= Nprimes + 1 /*bump the number of neighbor primes. */
if cols==0 then iterate /*Build the list (to be shown later)? */
$= $ right( commas(@.j), w) /*add neighbor prime ──► the $ list. */
if Nprimes//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(Nprimes) @neig 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; s.#= @.# **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 s.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; s.#= j*j; !.j= 1 /*bump # of Ps; assign next P; P²; P# */
end /*j*/; return</lang>
- output when using the default inputs:
index │ neighbor primes: p, q, p*q+2 are primes, and p < 500 ───────┼─────────────────────────────────────────────────────────────────────────────────────────────────────────────── 1 │ 3 5 7 13 19 67 149 179 229 239 11 │ 241 269 277 307 313 397 401 419 439 487 ───────┴─────────────────────────────────────────────────────────────────────────────────────────────────────────────── Found 20 neighbor primes: p, q, p*q+2 are primes, and p < 500
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...
Sidef
<lang ruby>500.primes.grep {|p| p * p.next_prime + 2 -> is_prime }.say</lang>
- Output:
[3, 5, 7, 13, 19, 67, 149, 179, 229, 239, 241, 269, 277, 307, 313, 397, 401, 419, 439, 487]
Wren
<lang ecmascript>import "/math" for Int import "/seq" for Lst import "/fmt" for Fmt
var primes = Int.primeSieve(504) var nprimes = [] System.print("Neighbour primes < 500:") for (i in 0...primes.count-1) {
var p = primes[i] * primes[i+1] + 2 if (Int.isPrime(p)) nprimes.add(primes[i])
} for (chunk in Lst.chunks(nprimes, 10)) Fmt.print("$3d", chunk) System.print("\nFound %(nprimes.count) such primes.")</lang>
- Output:
Neighbour primes < 500: 3 5 7 13 19 67 149 179 229 239 241 269 277 307 313 397 401 419 439 487 Found 20 such primes.
XPL0
<lang 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 Count, P, Q; [Count:= 0; P:= 2; Q:= 3; repeat if IsPrime(Q) then
[if IsPrime(P*Q+2) then
[IntOut(0, P);
ChOut(0, ^ );
Count:= Count+1;
];
P:= Q;
];
Q:= Q+2;
until P >= 500; CrLf(0); IntOut(0, Count); Text(0, " neighbour primes found below 500. "); ]</lang>
- Output:
3 5 7 13 19 67 149 179 229 239 241 269 277 307 313 397 401 419 439 487 20 neighbour primes found below 500.