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# Find prime numbers of the form n*n*n+2

Find prime numbers of the form n*n*n+2 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.
Find prime numbers of the form   n3+2,   where 0 < n < 200

procedure Find_Primes is

type Number is new Long_Integer range 0 .. Long_Integer'Last;
package Number_Io is new Ada.Text_Io.Integer_Io (Number);

function Is_Prime (A : Number) return Boolean is
D : Number;
begin
if A < 2 then return False; end if;
if A in 2 .. 3 then return True; end if;
if A mod 2 = 0 then return False; end if;
if A mod 3 = 0 then return False; end if;
D := 5;
while D * D < A loop
if A mod D = 0 then
return False;
end if;
D := D + 2;
if A mod D = 0 then
return False;
end if;
D := D + 4;
end loop;
return True;
end Is_Prime;

P : Number;
begin
for N in Number range 1 .. 199 loop
P := N**3 + 2;
if Is_Prime (P) then
Number_Io.Put (N, Width => 3); Ada.Text_Io.Put (" ");
Number_Io.Put (P, Width => 7);
end if;
end loop;
end Find_Primes;
Output:
N   N**3+2
------------
1        3
3       29
5      127
29    24391
45    91127
63   250049
65   274627
69   328511
71   357913
83   571789
105  1157627
113  1442899
123  1860869
129  2146691
143  2924209
153  3581579
171  5000213
173  5177719
189  6751271

## ALGOL 68

BEGIN # Find n such that n^3 + 2 is a prime for n < 200                      #
FOR n TO 199 DO
INT candidate = ( n * n * n ) + 2;
# there will only be 199 candidates, so a primality check by trial #
# division should be OK #
BOOL is prime := TRUE;
FOR f FROM 2 TO ENTIER sqrt( candidate )
WHILE is prime := candidate MOD f /= 0
DO SKIP OD;
IF is prime THEN
# n^3 + 2 is prime #
print( ( whole( n, -4 ), ": ", whole( candidate, -8 ), newline ) )
FI
OD
END
Output:
1:        3
3:       29
5:      127
29:    24391
45:    91127
63:   250049
65:   274627
69:   328511
71:   357913
83:   571789
105:  1157627
113:  1442899
123:  1860869
129:  2146691
143:  2924209
153:  3581579
171:  5000213
173:  5177719
189:  6751271

## ALGOL W

begin % Find n such that n^3 + 2 is a prime for n < 200                      %
for n := 1 until 199 do begin
integer candidate;
logical isPrime;
candidate := ( n * n * n ) + 2;
% there will only be 199 candidates, so a primality check by trial  %
% division should be OK  %
isPrime := true;
for f := 2 until entier( sqrt( candidate ) ) do begin
isPrime := candidate rem f not = 0;
if not isPrime then goto endPrimalityCheck
end for_f ;
endPrimalityCheck:
if isPrime then begin
% n^3 + 2 is prime  %
write( i_w := 4, s_w := 0, n, ": ", i_w := 8, candidate )
end if_isPrime
end for_n
end.
Output:
1:        3
3:       29
5:      127
29:    24391
45:    91127
63:   250049
65:   274627
69:   328511
71:   357913
83:   571789
105:  1157627
113:  1442899
123:  1860869
129:  2146691
143:  2924209
153:  3581579
171:  5000213
173:  5177719
189:  6751271

## AWK

# syntax: GAWK -f FIND_PRIME_NUMBERS_OF_THE_FORM_NNN2.AWK
BEGIN {
start = 1
stop = 200
for (n=start; n<=stop; n++) {
p = n*n*n + 2
if (is_prime(p)) {
printf("%3d %'10d\n",n,p)
count++
}
}
printf("Prime numbers %d-%d of the form n*n*n+2: %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:
1          3
3         29
5        127
29     24,391
45     91,127
63    250,049
65    274,627
69    328,511
71    357,913
83    571,789
105  1,157,627
113  1,442,899
123  1,860,869
129  2,146,691
143  2,924,209
153  3,581,579
171  5,000,213
173  5,177,719
189  6,751,271
Prime numbers 1-200 of the form n*n*n+2: 19

## C

Translation of: Wren
#include <stdio.h>
#include <stdbool.h>
#include <locale.h>

bool isPrime(int n) {
int d;
if (n < 2) return false;
if (!(n%2)) return n == 2;
if (!(n%3)) return n == 3;
d = 5;
while (d*d <= n) {
if (!(n%d)) return false;
d += 2;
if (!(n%d)) return false;
d += 4;
}
return true;
}

int main() {
int n, p;
const int limit = 200;
setlocale(LC_ALL, "");
for (n = 1; n < limit; ++n) {
p = n*n*n + 2;
if (isPrime(p)) {
printf("n = %3d => n³ + 2 = %'9d\n", n, p);
}
}
return 0;
}
Output:
n =   1 => n³ + 2 =         3
n =   3 => n³ + 2 =        29
n =   5 => n³ + 2 =       127
n =  29 => n³ + 2 =    24,391
n =  45 => n³ + 2 =    91,127
n =  63 => n³ + 2 =   250,049
n =  65 => n³ + 2 =   274,627
n =  69 => n³ + 2 =   328,511
n =  71 => n³ + 2 =   357,913
n =  83 => n³ + 2 =   571,789
n = 105 => n³ + 2 = 1,157,627
n = 113 => n³ + 2 = 1,442,899
n = 123 => n³ + 2 = 1,860,869
n = 129 => n³ + 2 = 2,146,691
n = 143 => n³ + 2 = 2,924,209
n = 153 => n³ + 2 = 3,581,579
n = 171 => n³ + 2 = 5,000,213
n = 173 => n³ + 2 = 5,177,719
n = 189 => n³ + 2 = 6,751,271

## C++

#include <iomanip>
#include <iostream>

bool is_prime(unsigned int n) {
if (n < 2)
return false;
if (n % 2 == 0)
return n == 2;
if (n % 3 == 0)
return n == 3;
for (unsigned int p = 5; p * p <= n; p += 4) {
if (n % p == 0)
return false;
p += 2;
if (n % p == 0)
return false;
}
return true;
}

int main() {
for (unsigned int n = 1; n < 200; n += 2) {
auto p = n * n * n + 2;
if (is_prime(p))
std::cout << std::setw(3) << n << std::setw(9) << p << '\n';
}
}
Output:
1        3
3       29
5      127
29    24391
45    91127
63   250049
65   274627
69   328511
71   357913
83   571789
105  1157627
113  1442899
123  1860869
129  2146691
143  2924209
153  3581579
171  5000213
173  5177719
189  6751271

## CLU

is_prime = proc (n: int) returns (bool)
if n<2 then return(false) end
if n//2=0 then return(n=2) end
if n//3=0 then return(n=3) end

d: int := 5
while d*d <= n do
if n//d=0 then return(false) end
d := d+2
if n//d=0 then return(false) end
d := d+4
end
return(true)
end is_prime

n3plus2_primes = iter (max: int) yields (int,int)
for n: int in int\$from_to(1, max) do
p: int := n**3 + 2
if is_prime(p) then yield(n,p) end
end
end n3plus2_primes

start_up = proc ()
po: stream := stream\$primary_output()
for n, p: int in n3plus2_primes(200) do
stream\$puts(po, "n = ")
stream\$putright(po, int\$unparse(n), 3)
stream\$puts(po, " => n^3 + 2 = ")
stream\$putright(po, int\$unparse(p), 7)
stream\$putl(po, "")
end
end start_up
Output:
n =   1 => n^3 + 2 =       3
n =   3 => n^3 + 2 =      29
n =   5 => n^3 + 2 =     127
n =  29 => n^3 + 2 =   24391
n =  45 => n^3 + 2 =   91127
n =  63 => n^3 + 2 =  250049
n =  65 => n^3 + 2 =  274627
n =  69 => n^3 + 2 =  328511
n =  71 => n^3 + 2 =  357913
n =  83 => n^3 + 2 =  571789
n = 105 => n^3 + 2 = 1157627
n = 113 => n^3 + 2 = 1442899
n = 123 => n^3 + 2 = 1860869
n = 129 => n^3 + 2 = 2146691
n = 143 => n^3 + 2 = 2924209
n = 153 => n^3 + 2 = 3581579
n = 171 => n^3 + 2 = 5000213
n = 173 => n^3 + 2 = 5177719
n = 189 => n^3 + 2 = 6751271

## COBOL

IDENTIFICATION DIVISION.
PROGRAM-ID. N3-PLUS-2-PRIMES.

DATA DIVISION.
WORKING-STORAGE SECTION.
01 VARIABLES.
03 N PIC 9(3).
03 N3PLUS2 PIC 9(7).
03 DIVISOR PIC 9(4).
03 DIV-SQ PIC 9(8).
03 DIV-CHECK PIC 9(4)V9(4).
03 FILLER REDEFINES DIV-CHECK.
05 FILLER PIC 9(4).
05 FILLER PIC 9(4).
88 DIVISIBLE VALUE ZERO.
03 FILLER REDEFINES N3PLUS2.
05 FILLER PIC 9(6).
05 FILLER PIC 9.
88 EVEN VALUE 0, 2, 4, 6, 8.
03 PRIME-FLAG PIC X.
88 PRIME VALUE '*'.

01 FORMAT.
03 FILLER PIC X(4) VALUE "N = ".
03 N-OUT PIC ZZ9.
03 FILLER PIC X(17) VALUE " => N ** 3 + 2 = ".
03 N3PLUS2-OUT PIC Z(6)9.

PROCEDURE DIVISION.
BEGIN.
PERFORM TRY-N VARYING N FROM 1 BY 1
UNTIL N IS GREATER THAN 200.
STOP RUN.

TRY-N.
COMPUTE N3PLUS2 = N ** 3 + 2.
PERFORM CHECK-PRIME.
IF PRIME,
MOVE N TO N-OUT,
MOVE N3PLUS2 TO N3PLUS2-OUT,
DISPLAY FORMAT.

CHECK-PRIME SECTION.
BEGIN.
MOVE SPACE TO PRIME-FLAG.
IF N3PLUS2 IS LESS THAN 5, GO TO TRIVIAL.
IF EVEN, GO TO CHECK-PRIME-DONE.
DIVIDE N3PLUS2 BY 3 GIVING DIV-CHECK.
IF DIVISIBLE, GO TO CHECK-PRIME-DONE.
MOVE ZERO TO DIV-SQ.
MOVE 5 TO DIVISOR.
MOVE '*' TO PRIME-FLAG.
PERFORM CHECK-DIVISOR
UNTIL NOT PRIME OR DIV-SQ IS GREATER THAN N3PLUS2.
GO TO CHECK-PRIME-DONE.

CHECK-DIVISOR.
MULTIPLY DIVISOR BY DIVISOR GIVING DIV-SQ.
DIVIDE N3PLUS2 BY DIVISOR GIVING DIV-CHECK.
IF DIVISIBLE, MOVE SPACE TO PRIME-FLAG.
DIVIDE N3PLUS2 BY DIVISOR GIVING DIV-CHECK.
IF DIVISIBLE, MOVE SPACE TO PRIME-FLAG.

TRIVIAL.
IF N3PLUS2 IS EQUAL TO 2 OR EQUAL TO 3,
MOVE '*' TO PRIME-FLAG.

CHECK-PRIME-DONE.
EXIT.
Output:
N =   1 => N ** 3 + 2 =       3
N =   3 => N ** 3 + 2 =      29
N =   5 => N ** 3 + 2 =     127
N =  29 => N ** 3 + 2 =   24391
N =  45 => N ** 3 + 2 =   91127
N =  63 => N ** 3 + 2 =  250049
N =  65 => N ** 3 + 2 =  274627
N =  69 => N ** 3 + 2 =  328511
N =  71 => N ** 3 + 2 =  357913
N =  83 => N ** 3 + 2 =  571789
N = 105 => N ** 3 + 2 = 1157627
N = 113 => N ** 3 + 2 = 1442899
N = 123 => N ** 3 + 2 = 1860869
N = 129 => N ** 3 + 2 = 2146691
N = 143 => N ** 3 + 2 = 2924209
N = 153 => N ** 3 + 2 = 3581579
N = 171 => N ** 3 + 2 = 5000213
N = 173 => N ** 3 + 2 = 5177719
N = 189 => N ** 3 + 2 = 6751271

## Cowgol

include "cowgol.coh";

sub is_prime(n: uint32): (p: uint8) is
p := 0;
if n<=4 then
if n==2 or n==3 then
p := 1;
end if;
return;
end if;
if n&1 == 0 or n%3 == 0 then
return;
end if;

var d: uint32 := 5;
while d*d <= n loop
if n%d==0 then return; end if;
d := d+2;
if n%d==0 then return; end if;
d := d+4;
end loop;
p := 1;
end sub;

var n: uint32 := 1;
while n < 200 loop
var p: uint32 := n*n*n + 2;
if is_prime(p) != 0 then
print("n = ");
print_i32(n);
print("\t=> n^3 + 2 = ");
print_i32(p);
print_nl();
end if;
n := n+1;
end loop;
Output:
n =   1 => n^3 + 2 =       3
n =   3 => n^3 + 2 =      29
n =   5 => n^3 + 2 =     127
n =  29 => n^3 + 2 =   24391
n =  45 => n^3 + 2 =   91127
n =  63 => n^3 + 2 =  250049
n =  65 => n^3 + 2 =  274627
n =  69 => n^3 + 2 =  328511
n =  71 => n^3 + 2 =  357913
n =  83 => n^3 + 2 =  571789
n = 105 => n^3 + 2 = 1157627
n = 113 => n^3 + 2 = 1442899
n = 123 => n^3 + 2 = 1860869
n = 129 => n^3 + 2 = 2146691
n = 143 => n^3 + 2 = 2924209
n = 153 => n^3 + 2 = 3581579
n = 171 => n^3 + 2 = 5000213
n = 173 => n^3 + 2 = 5177719
n = 189 => n^3 + 2 = 6751271

## Delphi

Library: PrimTrial

program Find_prime_numbers_of_the_form_n_n_n_plus_2;

{\$APPTYPE CONSOLE}

uses
System.SysUtils,
PrimTrial;

function Commatize(n: NativeInt): string;
var
fmt: TFormatSettings;
begin
fmt := TFormatSettings.Create('en-Us');
Result := double(n).ToString(ffNumber, 64, 0, fmt);
end;

const
limit = 200;

begin
for var n := 1 to limit - 1 do
begin
var p := n * n * n + 2;
if isPrime(p) then
writeln('n = ', n: 3, ' => n^3 + 2 = ', Commatize(p): 9);
end;
end.
Output:
n =   1 => n^3 + 2 =         3
n =   3 => n^3 + 2 =        29
n =   5 => n^3 + 2 =       127
n =  29 => n^3 + 2 =    24,391
n =  45 => n^3 + 2 =    91,127
n =  63 => n^3 + 2 =   250,049
n =  65 => n^3 + 2 =   274,627
n =  69 => n^3 + 2 =   328,511
n =  71 => n^3 + 2 =   357,913
n =  83 => n^3 + 2 =   571,789
n = 105 => n^3 + 2 = 1,157,627
n = 113 => n^3 + 2 = 1,442,899
n = 123 => n^3 + 2 = 1,860,869
n = 129 => n^3 + 2 = 2,146,691
n = 143 => n^3 + 2 = 2,924,209
n = 153 => n^3 + 2 = 3,581,579
n = 171 => n^3 + 2 = 5,000,213
n = 173 => n^3 + 2 = 5,177,719
n = 189 => n^3 + 2 = 6,751,271

## F#

This task uses Extensible Prime Generator (F#).

[1..2..200]|>Seq.filter(fun n->isPrime(2+pown n 3))|>Seq.iter(fun n->printfn "n=%3d -> %d" n (2+pown n 3))

Output:
n=  1 -> 3
n=  3 -> 29
n=  5 -> 127
n= 29 -> 24391
n= 45 -> 91127
n= 63 -> 250049
n= 65 -> 274627
n= 69 -> 328511
n= 71 -> 357913
n= 83 -> 571789
n=105 -> 1157627
n=113 -> 1442899
n=123 -> 1860869
n=129 -> 2146691
n=143 -> 2924209
n=153 -> 3581579
n=171 -> 5000213
n=173 -> 5177719
n=189 -> 6751271

## Factor

Using the parity optimization from the Wren entry:

Works with: Factor version 0.99 2021-02-05
USING: formatting kernel math math.functions math.primes
math.ranges sequences tools.memory.private ;

1 199 2 <range> [
dup 3 ^ 2 + dup prime?
[ commas "n = %3d => n³ + 2 = %9s\n" printf ] [ 2drop ] if
] each

Or, using local variables:

Translation of: Wren
Works with: Factor version 0.99 2021-02-05
USING: formatting kernel math math.primes math.ranges sequences
tools.memory.private ;

[let
199 :> limit
1 limit 2 <range> [| n |
n n n * * 2 + :> p
p prime?
[ n p commas "n = %3d => n³ + 2 = %9s\n" printf ] when
] each
]
Output:
n =   1 => n³ + 2 =         3
n =   3 => n³ + 2 =        29
n =   5 => n³ + 2 =       127
n =  29 => n³ + 2 =    24,391
n =  45 => n³ + 2 =    91,127
n =  63 => n³ + 2 =   250,049
n =  65 => n³ + 2 =   274,627
n =  69 => n³ + 2 =   328,511
n =  71 => n³ + 2 =   357,913
n =  83 => n³ + 2 =   571,789
n = 105 => n³ + 2 = 1,157,627
n = 113 => n³ + 2 = 1,442,899
n = 123 => n³ + 2 = 1,860,869
n = 129 => n³ + 2 = 2,146,691
n = 143 => n³ + 2 = 2,924,209
n = 153 => n³ + 2 = 3,581,579
n = 171 => n³ + 2 = 5,000,213
n = 173 => n³ + 2 = 5,177,719
n = 189 => n³ + 2 = 6,751,271

## Fermat

for n=1,199 do if Isprime(n^3+2)=1 then !!(n,n^3+2) fi od
Output:
1 3
3 29
5 127
29 24391
45 91127
63 250049
65 274627
69 328511
71 357913
83 571789
105 1157627
113 1442899
123 1860869
129 2146691
143 2924209
153 3581579
171 5000213
173 5177719
189 6751271

## Forth

Works with: Gforth
: prime? ( n -- flag )
dup 2 < if drop false exit then
dup 2 mod 0= if 2 = exit then
dup 3 mod 0= if 3 = exit then
5
begin
2dup dup * >=
while
2dup mod 0= if 2drop false exit then
2 +
2dup mod 0= if 2drop false exit then
4 +
repeat
2drop true ;

: main
200 1 do
i i i * * 2 + dup prime? if
i 3 .r 9 .r cr
else
drop
then
2 +loop ;

main
bye
Output:
1        3
3       29
5      127
29    24391
45    91127
63   250049
65   274627
69   328511
71   357913
83   571789
105  1157627
113  1442899
123  1860869
129  2146691
143  2924209
153  3581579
171  5000213
173  5177719
189  6751271

## FreeBASIC

Use the code from Primality by trial division#FreeBASIC as an include.

#include"isprime.bas"

for n as uinteger = 1 to 200
if isprime(n^3+2) then
print n, n^3+2
end if
next n
Output:
1              3
3              29
5              127
29             24391
45             91127
63             250049
65             274627
69             328511
71             357913
83             571789
105            1157627
113            1442899
123            1860869
129            2146691
143            2924209
153            3581579
171            5000213
173            5177719
189            6751271

## Frink

for n = 1 to 199
if isPrime[n^3 + 2]
println["\$n\t" + n^3+2]
Output:
1	3
3	29
5	127
29	24391
45	91127
63	250049
65	274627
69	328511
71	357913
83	571789
105	1157627
113	1442899
123	1860869
129	2146691
143	2924209
153	3581579
171	5000213
173	5177719
189	6751271

## Fōrmulæ

Fōrmulæ programs are not textual, visualization/edition of programs is done showing/manipulating structures but not text. Moreover, there can be multiple visual representations of the same program. Even though it is possible to have textual representation —i.e. XML, JSON— they are intended for storage and transfer purposes more than visualization and edition.

Programs in Fōrmulæ are created/edited online in its website, However they run on execution servers. By default remote servers are used, but they are limited in memory and processing power, since they are intended for demonstration and casual use. A local server can be downloaded and installed, it has no limitations (it runs in your own computer). Because of that, example programs can be fully visualized and edited, but some of them will not run if they require a moderate or heavy computation/memory resources, and no local server is being used.

## Go

package main

import "fmt"

func isPrime(n int) bool {
switch {
case n < 2:
return false
case n%2 == 0:
return n == 2
case n%3 == 0:
return n == 3
default:
d := 5
for d*d <= n {
if n%d == 0 {
return false
}
d += 2
if n%d == 0 {
return false
}
d += 4
}
return true
}
}

func commatize(n int) string {
s := fmt.Sprintf("%d", n)
if n < 0 {
s = s[1:]
}
le := len(s)
for i := le - 3; i >= 1; i -= 3 {
s = s[0:i] + "," + s[i:]
}
if n >= 0 {
return s
}
return "-" + s
}

func main() {
const limit = 200
for n := 1; n < limit; n++ {
p := n*n*n + 2
if isPrime(p) {
fmt.Printf("n = %3d => n³ + 2 = %9s\n", n, commatize(p))
}
}
}
Output:
n =   1 => n³ + 2 =         3
n =   3 => n³ + 2 =        29
n =   5 => n³ + 2 =       127
n =  29 => n³ + 2 =    24,391
n =  45 => n³ + 2 =    91,127
n =  63 => n³ + 2 =   250,049
n =  65 => n³ + 2 =   274,627
n =  69 => n³ + 2 =   328,511
n =  71 => n³ + 2 =   357,913
n =  83 => n³ + 2 =   571,789
n = 105 => n³ + 2 = 1,157,627
n = 113 => n³ + 2 = 1,442,899
n = 123 => n³ + 2 = 1,860,869
n = 129 => n³ + 2 = 2,146,691
n = 143 => n³ + 2 = 2,924,209
n = 153 => n³ + 2 = 3,581,579
n = 171 => n³ + 2 = 5,000,213
n = 173 => n³ + 2 = 5,177,719
n = 189 => n³ + 2 = 6,751,271

## J

([,.2+]^3:)@([#~1:p:2+]^3:) }.i.200x
Output:
1       3
3      29
5     127
29   24391
45   91127
63  250049
65  274627
69  328511
71  357913
83  571789
105 1157627
113 1442899
123 1860869
129 2146691
143 2924209
153 3581579
171 5000213
173 5177719
189 6751271

## jq

Works with: jq

Works with gojq, the Go implementation of jq

Using a definition of `is_prime` such as can be found at Safe_primes_and_unsafe_primes:

range(1;200) | pow(.; 3) + 2 | select(is_prime)

Output:
3
29
127
24391
91127
250049
274627
328511
357913
571789
1157627
1442899
1860869
2146691
2924209
3581579
5000213
5177719
6751271

## Julia

# Formatting output as in Go example.
using Primes, Formatting

isncubedplus2prime(x) = begin fx = x * x * x + 2; (isprime(fx), fx) end

tostring(x, fx) = "n = " * lpad(x, 3) * " => n³ + 2 =" * lpad(format(fx, commas=true), 10)

function filterprintresults(x_to_bool_and_fx, start, stop, stringify=(x, fx)->"\$x \$fx", doprint=true)
ncount = 0
println("Filtering \$x_to_bool_and_fx for integers between \$start and \$stop:\n")
for n in start+1:stop-1
isone, result = x_to_bool_and_fx(n)
if isone
doprint && println(stringify(n, result))
ncount += 1
end
end
println("\nThe total found was \$ncount.")
end

filterprintresults(isncubedplus2prime, 0, 200, tostring)

Output:
n =   1 => n³ + 2 =         3
n =   3 => n³ + 2 =        29
n =   5 => n³ + 2 =       127
n =  29 => n³ + 2 =    24,391
n =  45 => n³ + 2 =    91,127
n =  63 => n³ + 2 =   250,049
n =  65 => n³ + 2 =   274,627
n =  69 => n³ + 2 =   328,511
n =  71 => n³ + 2 =   357,913
n =  83 => n³ + 2 =   571,789
n = 105 => n³ + 2 = 1,157,627
n = 113 => n³ + 2 = 1,442,899
n = 123 => n³ + 2 = 1,860,869
n = 129 => n³ + 2 = 2,146,691
n = 143 => n³ + 2 = 2,924,209
n = 153 => n³ + 2 = 3,581,579
n = 171 => n³ + 2 = 5,000,213
n = 173 => n³ + 2 = 5,177,719
n = 189 => n³ + 2 = 6,751,271

The total found was 19.

### One-liner version

using Primes; println(filter(isprime, map(x -> x^3 + 2, 1:199)))
Output:

[3, 29, 127, 24391, 91127, 250049, 274627, 328511, 357913, 571789, 1157627, 1442899, 1860869, 2146691, 2924209, 3581579, 5000213, 5177719, 6751271]

NORMAL MODE IS INTEGER

INTERNAL FUNCTION(P)
ENTRY TO PRIME.
WHENEVER P.L.2, FUNCTION RETURN 0B
WHENEVER P.E.P/2*2, FUNCTION RETURN P.E.2
WHENEVER P.E.P/3*3, FUNCTION RETURN P.E.3
D = 5
CHKDIV WHENEVER D*D.LE.P
WHENEVER P.E.P/D*D, FUNCTION RETURN 0B
D = D+2
WHENEVER P.E.P/D*D, FUNCTION RETURN 0B
D = D+4
TRANSFER TO CHKDIV
END OF CONDITIONAL
FUNCTION RETURN 1B
END OF FUNCTION

VECTOR VALUES FMT = \$4HN = ,I3,S4,12HN*N*N + 2 = ,I7*\$

THROUGH LOOP, FOR N=1, 1, N.GE.200
M = N*N*N + 2
WHENEVER PRIME.(M)
PRINT FORMAT FMT,N,M
END OF CONDITIONAL
LOOP CONTINUE
END OF PROGRAM
Output:
N =   1    N*N*N + 2 =       3
N =   3    N*N*N + 2 =      29
N =   5    N*N*N + 2 =     127
N =  29    N*N*N + 2 =   24391
N =  45    N*N*N + 2 =   91127
N =  63    N*N*N + 2 =  250049
N =  65    N*N*N + 2 =  274627
N =  69    N*N*N + 2 =  328511
N =  71    N*N*N + 2 =  357913
N =  83    N*N*N + 2 =  571789
N = 105    N*N*N + 2 = 1157627
N = 113    N*N*N + 2 = 1442899
N = 123    N*N*N + 2 = 1860869
N = 129    N*N*N + 2 = 2146691
N = 143    N*N*N + 2 = 2924209
N = 153    N*N*N + 2 = 3581579
N = 171    N*N*N + 2 = 5000213
N = 173    N*N*N + 2 = 5177719
N = 189    N*N*N + 2 = 6751271

## Mathematica/Wolfram Language

Select[Range[199]^3 + 2, PrimeQ]
Output:
{3, 29, 127, 24391, 91127, 250049, 274627, 328511, 357913, 571789, 1157627, 1442899, 1860869, 2146691, 2924209, 3581579, 5000213, 5177719, 6751271}

## Nim

import strutils

func isPrime(n: Positive): bool =
if n < 2: return false
if n mod 2 == 0: return n == 2
if n mod 3 == 0: return n == 3
var d = 5
while d * d <= n:
if n mod d == 0: return false
inc d, 2
if n mod d == 0: return false
inc d, 4
result = true

for n in 1..<200:
let p = n * n * n + 2
if p.isPrime:
echo (\$n).align(3), " → ", p
Output:
1 → 3
3 → 29
5 → 127
29 → 24391
45 → 91127
63 → 250049
65 → 274627
69 → 328511
71 → 357913
83 → 571789
105 → 1157627
113 → 1442899
123 → 1860869
129 → 2146691
143 → 2924209
153 → 3581579
171 → 5000213
173 → 5177719
189 → 6751271

## PARI/GP

for(N=1,200,if(isprime(N^3+2),print(N," ",N^3+2)))
Output:
1 3
3 29
5 127
29 24391
45 91127
63 250049
65 274627
69 328511
71 357913
83 571789
105 1157627
113 1442899
123 1860869
129 2146691
143 2924209
153 3581579
171 5000213
173 5177719
189 6751271

## Perl

use strict;
use warnings;
use feature 'say';

say join ' ', grep { is_prime \$_ } map { \$_**3 + 2 } grep { 0 != \$_%2 } 1..199;

# generalize a bit, how many primes over a range of exponents and offsets?
use Math::AnyNum ':all'; # in order to handle large values
say ' ' . sprintf '%4d'x11 , 1..10;
for my \$e (1..10) {
printf '%2d ', \$e;
for my \$o (1..10) {
printf '%4d', scalar grep { is_prime \$_ } map { \$_**\$e + \$o } 1..199;
}
print "\n";
}
Output:
3 29 127 24391 91127 250049 274627 328511 357913 571789 1157627 1442899 1860869 2146691 2924209 3581579 5000213 5177719 6751271

1   2   3   4   5   6   7   8   9  10
1   46  45  44  44  43  43  42  42  42  42
2   34  17  29  34  12  19  49  19  24  32
3    1  19  26  25  23  17  18   0  28  20
4   30  13  20   1   7  12  28   7   6  11
5    1  12  14  14  11   7  15  17  12   3
6    1   5  19  11   3   2  24   0   7  11
7    1  10   8   8   7   9   7   6   9   8
8    7   7   7   1   2   5   9   5   1   8
9    1   5   7   7   5   5   6   0   9   6
10    1   3   3   9   3   1   5   3   2   1

## Phix

function pn3p2(integer n)
integer n3p2 = power(n,3)+2
return iff(is_prime(n3p2)?{n,n3p2}:0)
end function
sequence res = filter(apply(tagset(199,1,2),pn3p2),"!=",0)
printf(1,"Found %d primes of the form n^3+2:\n",length(res))
papply(true,printf,{1,{"n = %3d => n^3+2 = %,9d\n"},res})
Output:
Found 19 primes of the form n^3+2:
n =   1 => n^3+2 =         3
n =   3 => n^3+2 =        29
n =   5 => n^3+2 =       127
n =  29 => n^3+2 =    24,391
n =  45 => n^3+2 =    91,127
n =  63 => n^3+2 =   250,049
n =  65 => n^3+2 =   274,627
n =  69 => n^3+2 =   328,511
n =  71 => n^3+2 =   357,913
n =  83 => n^3+2 =   571,789
n = 105 => n^3+2 = 1,157,627
n = 113 => n^3+2 = 1,442,899
n = 123 => n^3+2 = 1,860,869
n = 129 => n^3+2 = 2,146,691
n = 143 => n^3+2 = 2,924,209
n = 153 => n^3+2 = 3,581,579
n = 171 => n^3+2 = 5,000,213
n = 173 => n^3+2 = 5,177,719
n = 189 => n^3+2 = 6,751,271

## Plain English

To run:
Start up.
Put 1 into a counter.
Loop.
Put the counter into a number.
Raise the number to 3.
If the number is prime, write the counter then " " then the number on the console.
If the counter is greater than 200, break.
Repeat.
Write "Done." on the console.
Wait for the escape key.
Shut down.
Output:
1 3
3 29
5 127
29 24391
45 91127
63 250049
65 274627
69 328511
71 357913
83 571789
105 1157627
113 1442899
123 1860869
129 2146691
143 2924209
153 3581579
171 5000213
173 5177719
189 6751271
Done.

## 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__':
for n in range(1, 200):
if isPrime(n**3+2):
print(f'{n}\t{n**3+2}');
Output:
1	3
3	29
5	127
29	24391
45	91127
63	250049
65	274627
69	328511
71	357913
83	571789
105	1157627
113	1442899
123	1860869
129	2146691
143	2924209
153	3581579
171	5000213
173	5177719
189	6751271

## Raku

# 20210315 Raku programming solution

say ((1199)»³ »+»2).grep: *.is-prime
Output:
(3 29 127 24391 91127 250049 274627 328511 357913 571789 1157627 1442899 1860869 2146691 2924209 3581579 5000213 5177719 6751271)

## REXX

Since REXX doesn't have a   isPrime   function,   this REXX program generates a number of primes such that some
numbers can be tested for primality directly,   other numbers have to be tested by trial division for primality.

A suitable number was calculated to generate a number of primes such that about half of the computing time used to
test the numbers for primality could be directly determined their primality,   the other half of the computing time used
would use trial division.

Since the task's requirements are pretty straight-forward and easy,   a little extra code was added for presentation
(title and title separator line,   the count of primes found,   and commatization of the numbers).

/*REXX program  finds and displays      n       primes of the form:     n**3  +  2.     */
parse arg LO HI hp . /*obtain optional argument from the CL.*/
if LO=='' | LO=="," then LO= 0 /*Not specified? Then use the default.*/
if HI=='' | HI=="," then HI= 200 /* " " " " " " */
if hp=='' | hp=="," then hp= 19 /* " " " " " " */
h= max(iSqrt(HI**3), hp**3) /*a high prime to generate primes to. */
w= length( commas(HI**3) ) + 3
call genP /*build array of semaphores for primes.*/
say right('n', 20) ' (n**3 + 2)' /*display a title for the output list. */
say left('', 20 + w + 20, '─') /*display a sep for the output list. */
finds= 0 /*# of triplet strange primes (so far).*/

do j=LO+1 by 2 to HI-1 /*look for primes of form of: n**3 + 2 */
x= j**3 + 2
if x<[email protected].# then if \!.x then iterate /*Not a semaphore prime? Then skip it.*/
else nop /*the NOP matches up with the "THEN".*/
else do k=2 while @.k**2<=x /*perform a primality test by division.*/
if x//@.k==0 then iterate j
end /*k*/
finds= finds + 1 /*bump # primes found of form: n**3+2 */
say right(commas(j), 20) right( commas(x), w)
end /*j*/

say left('', 20 + w + 20, '─'); say /*display a sep for the output list. */
say 'Found ' commas(finds) ' primes in the form of: n**3 + 2'
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 ?
/*──────────────────────────────────────────────────────────────────────────────────────*/
iSqrt: procedure; parse arg x; r=0; q=1; do while q<=x; q=q*4; end
do while q>1; q=q%4; _=x-r-q; r=r%2; if _>=0 then do;x=_;r=r+q; end; end
return r
/*──────────────────────────────────────────────────────────────────────────────────────*/
genP: !.= 0; pad= left('', 9) /*placeholders for primes; width of #'s*/
@.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 ≤ h. */
do [email protected].#+2 by 2 to h /*find odd primes from here on. */
parse var j '' -1 _; if _==5 then iterate /*J divisible by 5? (right dig)*/
if j// 3==0 then iterate /*" " " 3? */
if j// 7==0 then iterate /*" " " 7? */
/* [↑] the above five lines saves time*/
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
output   when using the default inputs:
n   (n**3 + 2)
────────────────────────────────────────────────────
1            3
3           29
5          127
29       24,391
45       91,127
63      250,049
65      274,627
69      328,511
71      357,913
83      571,789
105    1,157,627
113    1,442,899
123    1,860,869
129    2,146,691
143    2,924,209
153    3,581,579
171    5,000,213
173    5,177,719
189    6,751,271
────────────────────────────────────────────────────

Found  19  primes in the form of:  n**3 + 2

## Ring

see "working..." + nl

for n = 1 to 200 step 2
pr = pow(n,3)+2
if isprime(pr)
see "n = " + n + " => n³+2 = " + pr + nl
ok
next

see "done..." + nl

Output:
working...
n = 1 => n³+2 = 3
n = 3 => n³+2 = 29
n = 5 => n³+2 = 127
n = 29 => n³+2 = 24391
n = 45 => n³+2 = 91127
n = 63 => n³+2 = 250049
n = 65 => n³+2 = 274627
n = 69 => n³+2 = 328511
n = 71 => n³+2 = 357913
n = 83 => n³+2 = 571789
n = 105 => n³+2 = 1157627
n = 113 => n³+2 = 1442899
n = 123 => n³+2 = 1860869
n = 129 => n³+2 = 2146691
n = 143 => n³+2 = 2924209
n = 153 => n³+2 = 3581579
n = 171 => n³+2 = 5000213
n = 173 => n³+2 = 5177719
n = 189 => n³+2 = 6751271
done...

## Rust

// 202100327 Rust programming solution

use primes::is_prime;

fn main() {

let mut count = 0;
let begin = 0;
let end = 200;

println!("Find prime numbers of the form");
println!(" n => n³ + 2 ");

for n in begin+1..end-1 {
let m = n*n*n+2;
if is_prime(m) {
println!("{:4} => {}", n, m);
count += 1;
}
}

println!("Found {} such prime numbers where {} < n < {}.", count,begin,end);
}
Output:
Find prime numbers of the form
n => n³ + 2
1 => 3
3 => 29
5 => 127
29 => 24391
45 => 91127
63 => 250049
65 => 274627
69 => 328511
71 => 357913
83 => 571789
105 => 1157627
113 => 1442899
123 => 1860869
129 => 2146691
143 => 2924209
153 => 3581579
171 => 5000213
173 => 5177719
189 => 6751271
Found 19 such prime numbers where 0 < n < 200.

## Seed7

Credit for isPrime function: [1]

\$ include "seed7_05.s7i";

const func boolean: isPrime (in integer: number) is func
result
var boolean: prime is FALSE;
local
var integer: upTo is 0;
var integer: testNum is 3;
begin
if number = 2 then
prime := TRUE;
elsif odd(number) and number > 2 then
upTo := sqrt(number);
while number rem testNum <> 0 and testNum <= upTo do
testNum +:= 2;
end while;
prime := testNum > upTo;
end if;
end func;

const proc: main is func
local
const integer: limit is 199;
var integer: n is 1;
var integer: p is 0;
begin
writeln(" n n**3+2");
writeln("------------");
for n range 1 to limit step 2 do
p := n ** 3 + 2;
if isPrime(p) then
end if;
end for;
end func;
Output:
n   n**3+2
------------
1        3
3       29
5      127
29    24391
45    91127
63   250049
65   274627
69   328511
71   357913
83   571789
105  1157627
113  1442899
123  1860869
129  2146691
143  2924209
153  3581579
171  5000213
173  5177719
189  6751271

## Sidef

1..^200 -> map { _**3 + 2 }.grep {.is_prime}.say
Output:
[3, 29, 127, 24391, 91127, 250049, 274627, 328511, 357913, 571789, 1157627, 1442899, 1860869, 2146691, 2924209, 3581579, 5000213, 5177719, 6751271]

## Swift

import Foundation

func isPrime(_ n: Int) -> Bool {
if n < 2 {
return false
}
if n % 2 == 0 {
return n == 2
}
if n % 3 == 0 {
return n == 3
}
var p = 5
while p * p <= n {
if n % p == 0 {
return false
}
p += 2
if n % p == 0 {
return false
}
p += 4
}
return true
}

for n in 1...200 {
let p = n * n * n + 2
if isPrime(p) {
print(String(format: "%3d%9d", n, p))
}
}
Output:
1        3
3       29
5      127
29    24391
45    91127
63   250049
65   274627
69   328511
71   357913
83   571789
105  1157627
113  1442899
123  1860869
129  2146691
143  2924209
153  3581579
171  5000213
173  5177719
189  6751271

## Wren

Library: Wren-math
Library: Wren-trait
Library: Wren-fmt

If n is even then n³ + 2 is also even, so we only need to examine odd values of n here.

import "/math" for Int
import "/trait" for Stepped
import "/fmt" for Fmt

var limit = 200
for (n in Stepped.new(1...limit, 2)) {
var p = n*n*n + 2
if (Int.isPrime(p)) Fmt.print("n = \$3d => n³ + 2 = \$,9d", n, p)
}
Output:
n =   1 => n³ + 2 =         3
n =   3 => n³ + 2 =        29
n =   5 => n³ + 2 =       127
n =  29 => n³ + 2 =    24,391
n =  45 => n³ + 2 =    91,127
n =  63 => n³ + 2 =   250,049
n =  65 => n³ + 2 =   274,627
n =  69 => n³ + 2 =   328,511
n =  71 => n³ + 2 =   357,913
n =  83 => n³ + 2 =   571,789
n = 105 => n³ + 2 = 1,157,627
n = 113 => n³ + 2 = 1,442,899
n = 123 => n³ + 2 = 1,860,869
n = 129 => n³ + 2 = 2,146,691
n = 143 => n³ + 2 = 2,924,209
n = 153 => n³ + 2 = 3,581,579
n = 171 => n³ + 2 = 5,000,213
n = 173 => n³ + 2 = 5,177,719
n = 189 => n³ + 2 = 6,751,271

## 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 N;
[for N:= 1 to 199 do
[if IsPrime(N*N*N+2) then
[IntOut(0, N);
ChOut(0, 9\tab\);
IntOut(0, N*N*N+2);
CrLf(0);
]
];
]
Output:
1       3
3       29
5       127
29      24391
45      91127
63      250049
65      274627
69      328511
71      357913
83      571789
105     1157627
113     1442899
123     1860869
129     2146691
143     2924209
153     3581579
171     5000213
173     5177719
189     6751271