Primality by trial division

Write a boolean function that tells whether a given integer is prime. Remember that 1 and all non-positive numbers are not prime.

Use trial division. Even numbers over two may be eliminated right away. A loop from 3 to √n will suffice, but other loops are allowed.

Related task: Sieve of Eratosthenes, Prime decomposition.

ACL2

<lang Lisp>(defun is-prime-r (x i)

  (declare (xargs :measure (nfix (- x i))))
  (if (zp (- (- x i) 1))
      t
      (and (/= (mod x i) 0)
           (is-prime-r x (1+ i)))))

(defun is-prime (x)

  (or (= x 2)
      (is-prime-r x 2)))</lang>

<lang ActionScript>function isPrime(n:int):Boolean { if(n < 2) return false; if(n == 2) return true; if((n & 1) == 0) return false; for(var i:int = 3; i <= Math.sqrt(n); i+= 2) if(n % i == 0) return false; return true; }</lang>

Ada

<lang ada>function Is_Prime(Item : Positive) return Boolean is

  Result : Boolean := True;
  Test : Natural;

begin

  if Item /= 2 and Item mod 2 = 0 then
     Result := False;
  else
     Test := 3;
     while Test < Integer(Sqrt(Float(Item))) loop
        if Item mod Test = 0 then
           Result := False;
           exit;
        end if;
        Test := Test + 2;
     end loop;
 end if;
 return Result;

end Is_Prime;</lang>

ALGOL 68

Works with: ALGOL 68 version Revision 1 - no extensions to language used

Works with: ALGOL 68G version Any - tested with release 1.18.0-9h.tiny

COMMENT
  This routine is used in more than one place, and is essentially a
  template that can by used for many different types, eg INT, LONG INT...
USAGE
  MODE ISPRIMEINT = INT, LONG INT, etc
  PR READ "prelude/is_prime.a68" PR
END COMMENT

PROC is prime = ( ISPRIMEINT p )BOOL:
  IF p <= 1 OR ( NOT ODD p AND p/= 2) THEN
    FALSE
  ELSE
    BOOL prime := TRUE;
    FOR i FROM 3 BY 2 TO ENTIER sqrt(p)
      WHILE prime := p MOD i /= 0 DO SKIP OD;
    prime
  FI

<lang algol68>main:(

 INT upb=100;
 printf(($" The primes up to "g(-3)" are:"l$,upb));
 FOR i TO upb DO 
   IF is prime(i) THEN
     printf(($g(-4)$,i))
   FI
 OD;
 printf($l$)

)</lang>

Output:

The primes up to 100 are:
   2   3   5   7  11  13  17  19  23  29  31  37  41  43  47  53  59  61  67  71  73  79  83  89  97

AutoHotkey

Discussion <lang autohotkey>MsgBox % IsPrime(1995937) Loop

  MsgBox % A_Index-3 . " is " . (IsPrime(A_Index-3) ? "" : "not ") . "prime."

IsPrime(n,k=2) { ; testing primality with trial divisors not multiple of 2,3,5, up to sqrt(n)

  d := k+(k<7 ? 1+(k>2) : SubStr("6-----4---2-4---2-4---6-----2",Mod(k,30),1)) 
  Return n < 3 ? n>1 : Mod(n,k) ? (d*d <= n ? IsPrime(n,d) : 1) : 0

}</lang>

AutoIT

cs ----------------------------------------------------------------------------

AutoIt Version: 3.3.8.1
Author:         Alexander Alvonellos

Script Function:

Perform primality test on a given integer $number. RETURNS: TRUE/FALSE

ce ----------------------------------------------------------------------------

Func main() ConsoleWrite("The primes up to 100 are: " & @LF) For $i = 1 To 100 Step 1 If(isPrime($i)) Then If($i <> 97) Then ConsoleWrite($i & ", ") Else ConsoleWrite($i) EndIf EndIf Next EndFunc

Func isPrime($n) If($n < 2) Then Return False If($n = 2) Then Return True If(BitAnd($n, 1) = 0) Then Return False For $i = 3 To Sqrt($n) Step 2 If(Mod($n, $i) = 0) Then Return False Next Return True EndFunc main() </lang>

AWK

$ awk 'func prime(n){for(d=2;d<=sqrt(n);d++)if(!(n%d)){return 0};return 1}{print prime($1)}'

Or more legibly, and with n <= 1 handling

<lang AWK>function prime(n) {

   if (n <= 1) return 0
   for (d = 2; d <= sqrt(n); d++)
       if (!(n % d)) return 0
   return 1

}

{print prime($1)}</lang>

BASIC

Works with: QuickBasic version 4.5

Returns 1 for prime, 0 for non-prime <lang QBasic>FUNCTION prime% (n!)

 STATIC i AS INTEGER
 IF n = 2 THEN
   prime = 1
 ELSEIF n <= 1 OR n MOD 2 = 0 THEN
   prime = 0
 ELSE
   prime = 1
   FOR i = 3 TO INT(SQR(n)) STEP 2
     IF n MOD i = 0 THEN
       prime = 0
       EXIT FUNCTION
     END IF
   NEXT i
 END IF

END FUNCTION

' Test and display primes 1 .. 50 DECLARE FUNCTION prime% (n!) FOR n = 1 TO 50

 IF prime(n) = 1 THEN PRINT n;

NEXT n</lang>

Output:

 2  3  5  7  11  13  17  19  23  29  31  37  41  43  47

ZX Spectrum Basic

<lang ZXBasic>10 LET n=0: LET p=0 20 INPUT "Enter number: ";n 30 GO SUB 1000 40 IF p=0 THEN PRINT n;" is not prime!" 50 IF p<>0 THEN PRINT n;" is prime!" 60 GO TO 10 1000 REM *************** 1001 REM * PRIME CHECK * 1002 REM *************** 1010 LET p=0 1020 IF n/2=INT (n/2) THEN RETURN 1030 LET p=1 1040 FOR i=3 TO SQR (n) STEP 2 1050 IF n/i=INT (n/i) THEN LET p=0: LET i= SQR (n) 1060 NEXT i 1070 RETURN </lang>

Output:

15 is not prime!
17 is prime!
119 is not prime!
137 is prime!

Bracmat

<lang bracmat> ( prime

 =   incs n I inc
   .   4 2 4 2 4 6 2 6:?incs
     & 2:?n
     & 1 2 2 !incs:?I
     &   whl
       ' ( !n*!n:~>!arg
         & div$(!arg.!n)*!n:~!arg
         & (!I:%?inc ?I|!incs:%?inc ?I)
         & !n+!inc:?n
         )
     & !n*!n:>!arg
 )

& 100000000000:?p & whl

 ' ( !p+1:<100000000100:?p
   & (   prime$!p
       & out$!p
     | 
     )
   )

& ;</lang>

Output:

100000000003
100000000019
100000000057
100000000063
100000000069
100000000073
100000000091

C

<lang c>int is_prime(unsigned int n) { unsigned int p; if (!(n & 1) || n < 2 ) return n == 2;

/* comparing p*p <= n can overflow */ for (p = 3; p <= n/p; p += 2) if (!(n % p)) return 0; return 1; }</lang>

C++

<lang cpp>#include <cmath>

bool is_prime(unsigned int n) {

   if (n <= 1)
       return false;
   if (n == 2)
       return true;
   for (unsigned int i = 2; i <= sqrt(n); ++i)
       if (n % i == 0)
           return false;
   return true;

}</lang>

C#

<lang csharp>static bool isPrime(int n)

       {
           if (n <= 1) return false;
           for (int i = 2; i * i <= n; i++)            
               if (n % i == 0) return false;            
           return true;
       }</lang>

Clojure

The symbol # is a shortcut for creating lambda functions; the arguments in such a function are %1, %2, %3... (or simply % if there is only one argument). Thus, #(< (* % %) n) is equivalent to (fn [x] (< (* x x) n)) or more mathematically f(x) = x * x < n. <lang clojure>(defn divides? [k n] (= (rem n k) 0))

(defn prime? [n]

 (if (< n 2)
   false
   (empty? (filter #(divides? % n) (take-while #(<= (* % %) n) (range 2 n))))))</lang>

CMake

<lang cmake># Prime predicate: does n be a prime number? Sets var to true or false. function(primep var n)

 if(n GREATER 2)
   math(EXPR odd "${n} % 2")
   if(odd)
     # n > 2 and n is odd.
     set(factor 3)
     # Loop for odd factors from 3, while factor <= n / factor.
     math(EXPR quot "${n} / ${factor}")
     while(NOT factor GREATER quot)
       math(EXPR rp "${n} % ${factor}")        # Trial division
       if(NOT rp)
         # factor divides n, so n is not prime.
         set(${var} false PARENT_SCOPE)
         return()
       endif()
       math(EXPR factor "${factor} + 2")       # Next odd factor
       math(EXPR quot "${n} / ${factor}")
     endwhile(NOT factor GREATER quot)
     # Loop found no factor, so n is prime.
     set(${var} true PARENT_SCOPE)
   else()
     # n > 2 and n is even, so n is not prime.
     set(${var} false PARENT_SCOPE)
   endif(odd)
 elseif(n EQUAL 2)
   set(${var} true PARENT_SCOPE)       # 2 is prime.
 else()
   set(${var} false PARENT_SCOPE)      # n < 2 is not prime.
 endif()

endfunction(primep)</lang>

<lang cmake># Quick example. foreach(i -5 1 2 3 37 39)

 primep(b ${i})
 if(b)
   message(STATUS "${i} is prime.")
 else()
   message(STATUS "${i} is _not_ prime.")
 endif(b)

endforeach(i)</lang>

CoffeeScript

<lang coffeescript>is_prime = (n) ->

 # simple prime detection using trial division, works
 # for all integers
 return false if n <= 1 # by definition
 p = 2
 while p * p <= n
   return false if n % p == 0
   p += 1
 true

for i in [-1..100]

 console.log i if is_prime i</lang>

Common Lisp

<lang Lisp> (defun primep (a)

 (cond ((= a 2) T)
       ((or (<= a 1) (= (mod a 2) 0)) nil)
       ((loop for i from 3 to (sqrt a) by 2 do
               (if (= (mod a i) 0)
                   (return nil))) nil)
       (T T)))</lang>

<lang Lisp>(defun primep (n)

 "Is N prime?"
 (and (> n 1) 
      (or (= n 2) (oddp n))
      (loop for i from 3 to (isqrt n) by 2

never (zerop (rem n i)))))</lang>

D

Simple Version

<lang d>import std.stdio, std.algorithm, std.range, std.math;

bool isPrime(in int n) pure nothrow {

   if (n == 2)
       return true;
   if (n <= 1 || (n & 1) == 0)
       return false;

   for(int i = 3; i <= sqrt(cast(real)n); i += 2)
       if (n % i == 0)
           return false;
   return true;

}

void main() { // demo code

   iota(2, 40).filter!isPrime().writeln();

}</lang>

Output:

[2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37]

Version with excluded multiplies of 2 and 3

Same output. <lang d>import std.stdio, std.algorithm, std.range;

bool isPrime2(Integer)(in Integer number) pure nothrow {

 // Adapted from: http://www.devx.com/vb2themax/Tip/19051
 // manually test 1, 2, 3 and multiples of 2 and 3
 if (number == 2 || number == 3)
   return true;
 else if (number < 2 || number % 2 == 0 || number % 3 == 0)
   return false;

 /* we can now avoid to consider multiples
  * of 2 and 3. This can be done really simply
  * by starting at 5 and incrementing by 2 and 4
  * alternatively, that is:
  *    5, 7, 11, 13, 17, 19, 23, 25, 29, 31, 35, 37, ...
  * we don't need to go higher than the square root of the number */
 for (Integer divisor = 5, increment = 2; divisor*divisor <= number;
      divisor += increment, increment = 6 - increment)
   if (number % divisor == 0)
     return false;

 return true; // if we get here, the number is prime

}

void main() { // demo code

   iota(2, 40).filter!isPrime2().writeln();

}</lang>

Two Way Test

Odd divisors is generated both from increasing and decreasing sequence, may improve performance for numbers that have large minimum factor. Same output. <lang d>import std.stdio, std.algorithm, std.range, std.math;

bool isPrime3(T)(in T n) pure nothrow {

   if (n % 2 == 0 || n <= 1)
       return n == 2;
   T head = 3, tail = (cast(T)sqrt(cast(real)n) / 2) * 2 + 1;
   for ( ; head <= tail ; head +=2, tail -= 2)
       if ((n % head) == 0 || (n % tail) == 0)
           return false;
   return true;

}

void main() { // demo code

   iota(2, 40).filter!isPrime3().writeln();

}</lang>

Delphi

<lang Delphi>function IsPrime(aNumber: Integer): Boolean; var

 I: Integer;

begin

 Result:= True;
 if(aNumber = 2) then Exit;

 Result:= not ((aNumber mod 2 = 0)  or
               (aNumber <= 1));
 if not Result then Exit;

 for I:=3 to Trunc(Sqrt(aNumber)) do
 if(aNumber mod I = 0) then
 begin
   Result:= False;
   Break;
 end;

end;</lang>

E

Translation of: D

<lang e>def isPrime(n :int) {

   if (n == 2) {
       return true
   } else if (n <= 1 || n %% 2 == 0) {
       return false
   } else {
       def limit := (n :float64).sqrt().ceil()
       var divisor := 1
       while ((divisor += 2) <= limit) {
           if (n %% divisor == 0) {
               return false
           }
       }
       return true
   }

}</lang>

Erlang

<lang erlang>is_prime(N) when N == 2 -> true; is_prime(N) when N < 2 orelse N rem 2 == 0 -> false; is_prime(N) -> is_prime(N,3).

is_prime(N,K) when K*K > N -> true; is_prime(N,K) when N rem K == 0 -> false; is_prime(N,K) -> is_prime(N,K+2).</lang>

Euphoria

<lang euphoria>function is_prime(integer n)

   if n<=2 or remainder(n,2)=0 then
       return 0
   else
       for i=3 to sqrt(n) by 2 do
           if remainder(n,i)=0 then
               return 0
           end if
       end for
       return 1
   end if

end function</lang>

Factor

<lang factor>USING: combinators kernel math math.functions math.ranges sequences ;

prime? ( n -- ? )

   {
       { [ dup 2 < ] [ drop f ] }
       { [ dup even? ] [ 2 = ] }
       [ 3 over sqrt 2 <range> [ mod 0 > ] with all? ]
   } cond ;</lang>

FALSE

<lang false>[0\$2=$[@~@@]?~[$$2>\1&&[\~\

  3[\$@$@1+\$*>][\$@$@$@$@\/*=[%\~\$]?2+]#%

]?]?%]p:</lang>

Forth

<lang forth>: prime? ( n -- ? )

       dup 2 < if      drop false
   else dup 2 = if      drop true
   else dup 1 and 0= if drop false
   else 3
       begin 2dup dup * >=
       while 2dup mod 0=
             if       2drop false exit
             then 2 +
       repeat         2drop true
   then then then ;</lang>

Fortran

Works with: Fortran version 90 and later

<lang fortran> FUNCTION isPrime(number)

  LOGICAL :: isPrime
  INTEGER, INTENT(IN) :: number
  INTEGER :: i

  IF(number==2) THEN
     isPrime = .TRUE.
  ELSE IF(number < 2 .OR. MOD(number,2) == 0) THEN
     isPRIME = .FALSE.
  ELSE
     isPrime = .TRUE.
     DO i = 3, INT(SQRT(REAL(number))), 2
        IF(MOD(number,i) == 0) THEN
           isPrime = .FALSE.
           EXIT
        END IF
     END DO
  END IF
END FUNCTION</lang>

GAP

<lang gap>IsPrimeTrial := function(n)

  local k, m;
  if n < 5 then
     return (n = 2) or (n = 3);
  fi;
  if RemInt(n, 2) = 0 then
     return false;
  fi;
  m := RootInt(n);
  k := 3;
  while k <= m do
     if RemInt(n, k) = 0 then
        return false;
     fi;
     k := k + 2;
  od;
  return true;

end;

Filtered([1 .. 100], IsPrimeTrial);

[ 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97 ]</lang>

Go

<lang go>func isPrime(p int) bool {

   if p < 2 {return false}
   if p % 2 == 0 { return false }

   for i := 3; i*i < p; i += 2 {
       if p % i == 0 { return false }
   }
   return true

}</lang>

Groovy

<lang groovy>def isPrime = {

   it == 2 || 
   it > 1 &&
   (2..Math.max(2, (int) Math.sqrt(it))).every{ k -> it % k != 0 }

}

(0..20).grep(isPrime)</lang>

Output:

[2, 3, 5, 7, 11, 13, 17, 19]

Haskell

The basic divisibility test by odd numbers: <lang haskell>isPrime n = n==2 || n>2 && all ((> 0).rem n) (2:[3,5..floor.sqrt(fromIntegral n+1)])</lang>

Testing by prime numbers only is faster. Primes list is saved for reuse. Precalculation of primes pays off if testing more than just a few numbers: <lang haskell>noDivsBy factors n = foldr (\f r-> f*f>n || (rem n f /= 0 && r)) True factors

-- primeNums = filter (noDivsBy [2..]) [2..] primeNums = 2 : 3 : filter (noDivsBy $ tail primeNums) [5,7..]

isPrime n = n > 1 && noDivsBy primeNums n</lang> Any increasing unbounded primes source can be used with the testing function noDivsBy to define isPrime function, say one from Sieve of Eratosthenes, or noDivsBy itself can be used to define primeNums as shown above, because it stops when the square root is reached.

Trial division can be used to produce short ranges of primes: <lang haskell>primesFromTo n m = filter isPrime [n..m]</lang> This code, when inlined, rearranged and optimized, leads to segmented "generate-and-test" sieve by trial division.

Sieve by trial division

Filtering of candidate numbers by prime at a time is a kind of sieving. One often sees a "naive" version presented as an unbounded sieve of Eratosthenes, similar to David Turner's 1975 SASL code, <lang haskell>primes = sieve [2..] where

  sieve (p:xs) = p : sieve [x | x <- xs, rem x p /= 0]</lang>

As is shown in Melissa O'Neill's "The Genuine Sieve of Eratosthenes", this is rather a sub-optimal trial division algorithm. Its complexity is quadratic in number of primes produced whereas that of optimal trial division is $O(n^{1.5}/(\log n)^{0.5})$ , and of true SoE it is $O(n\log n\log \log n)$ , in n primes produced.

Bounded sieve by trial division

Bounded formulation has normal trial division complexity, because it can stop early through an explicit guard: <lang haskell>primesTo m = 2 : sieve [3,5..m] where

  sieve (p:xs) | p*p > m = p : xs
               | True    = p : sieve [x | x <- xs, rem x p /= 0]</lang>

Postponed sieve by trial division

To make it unbounded, the guard cannot be simply discarded, the firing up of a filter by a prime should be postponed until its square is seen amongst the candidates: <lang haskell>primesT = 2 : 3 : sieve [5,7..] 9 (tail primesT) where

  sieve (x:xs) q ps@(p:t) | x < q = x : sieve xs q ps
                          | True  = sieve [x | x <- xs, rem x p /= 0] (head t^2) t</lang>

Segmented Generate and Test

Explicating the list of filters as a list of factors to test by on each segment between the consecutive squares of primes (so that no testing is done prematurely), and rearranging to avoid recalculations, leads to this code: <lang haskell>primesST = 2 : 3 : sieve 5 9 (drop 2 primesST) 0 where

  sieve x q ps k = let fs = take k (tail primesST) in
     filter (\x-> all ((/=0).rem x) fs) [x,x+2..q-2]
     ++ sieve (q+2) (head ps^2) (tail ps) (k+1)</lang>

Runs at empirical $O(n^{1.4...})$ time complexity, in n primes produced. Can be used as a framework for unbounded segmented sieves, replacing divisibility testing with proper sieve of Eratosthenes, etc.

HicEst

<lang HicEst> DO n = 1, 1E6

    Euler = n^2 + n + 41
    IF( Prime(Euler) == 0 ) WRITE(Messagebox) Euler, ' is NOT prime for n =', n
  ENDDO                            ! e.g.    1681 = 40^2 + 40 + 41 is NOT prime

END

FUNCTION Prime(number)

  Prime = number == 2
  IF( (number > 2) * MOD(number,2) ) THEN
    DO i = 3, number^0.5, 2
      IF(MOD(number,i) == 0) THEN
        Prime = 0
        RETURN
      ENDIF
    ENDDO
    Prime = 1
  ENDIF

END</lang>

Icon and Unicon

Procedure shown without a main program. <lang Icon>procedure isprime(n) #: return n if prime (using trial division) or fail if not n = integer(n) | n < 2 then fail # ensure n is an integer greater than 1 every if 0 = (n % (2 to sqrt(n))) then fail return n end</lang>

J

Generally, this task should be accomplished in J using 1&p:. Here we take an approach that's more comparable with the other examples on this page.

<lang j>isprime=: 3 : 'if. 3>:y do. 1<y else. 0 *./@:< y|~2+i.<.%:y end.'</lang>

Java

<lang java>public static boolean prime(long a){

  if(a == 2){
     return true;
  }else if(a <= 1 || a % 2 == 0){
     return false;
  }
  long max = (long)Math.sqrt(a);
  for(long n= 3; n <= max; n+= 2){
     if(a % n == 0){ return false; }
  }
  return true;

}</lang>

By Regular Expression

<lang java>public static boolean prime(int n) {

   return !new String(new char[n]).matches(".?|(..+?)\\1+");

}</lang>

JavaScript

<lang javascript>function isPrime(n) {

 if (n == 2) {
   return true;
 } else if ((n < 2) || (n % 2 == 0)) {
   return false;
 } else {
   for (var i = 3; i < Math.ceil(Math.sqrt(n)); i += 2) {
     if (n % i == 0)
       return false;
   }
   return true;
 }

}</lang>

Joy

From here <lang joy>DEFINE prime ==

       2
       [ [dup * >] nullary  [rem 0 >] dip  and ]
       [ succ ]
       while
       dup * < .</lang>

K

<lang K> isprime:{(x>1)&&/x!'2_!1+_sqrt x}

  &isprime'!100

2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 73 79 83 89 97</lang>

Liberty BASIC

<lang lb>for n =1 to 50

 if prime( n) = 1 then print n; " is prime."

next n

wait

function prime( n)

 if n =2 then
   prime =1
 else
   if ( n <=1) or ( n mod 2 =0) then
       prime =0
   else
       prime =1
       for i = 3 to int( sqr( n)) step 2
           if ( n MOD i) =0 then prime = 0: exit function
       next i
   end if
 end if

end function

end</lang>

Logo

<lang logo>to prime? :n

  if :n < 2 [output "false]
  if :n = 2 [output "true]
  if equal? 0 modulo :n 2 [output "false]
  for [i 3 [sqrt :n] 2] [if equal? 0 modulo :n :i [output "false]]
  output "true

end</lang>

LSE64

<lang LSE64>over : 2 pick

2dup : over over
even? : 1 & 0 =

# trial n d yields "n d 0/1 false" or "n d+2 true"
trial : 2 +                 true
trial : 2dup % 0 =   then 0 false
trial : 2dup dup * < then 1 false
trial-loop : trial &repeat

# prime? n yields flag
prime? : 3 trial-loop >flag drop drop
prime? : dup even? then drop false
prime? : dup 2 =   then drop true
prime? : dup 2 <   then drop false</lang>

Lua

<lang Lua>function IsPrime( n )

   if n <= 1 or ( n ~= 2 and n % 2 == 0 ) then
       return false
   end

   for i = 3, math.sqrt(n), 2 do

if n % i == 0 then

 	    return false

end

end

   return true

end</lang>

M4

<lang M4>define(`testnext',

  `ifelse(eval($2*$2>$1),1,
     1,
     `ifelse(eval($1%$2==0),1,
        0,
        `testnext($1,eval($2+2))')')')

define(`isprime',

  `ifelse($1,2,
     1,
     `ifelse(eval($1<=1 || $1%2==0),1,
        0,
        `testnext($1,3)')')')

isprime(9) isprime(11)</lang>

Output:

0
1

Mathematica

<lang mathematica>IsPrime[n_Integer] :=

Module[{k = 2},
 If[n <= 1, Return False];
 If[n == 2, Return True];
 While[k <= Sqrt[n],
  If[Mod[n, k] == 0, Return[False], k++]
  ];
 Return[True]
]</lang>

MATLAB

<lang MATLAB>function isPrime = primalityByTrialDivision(n)

   if n == 2
       isPrime = true;
       return
   elseif (mod(n,2) == 0) || (n <= 1)
       isPrime = false;
       return
   end
   
   %First n mod (3 to sqrt(n)) is taken. This will be a vector where the
   %first element is equal to n mod 3 and the last element is equal to n
   %mod sqrt(n). Then the all function is applied to that vector. If all
   %of the elements of this vector are non-zero (meaning n is prime) then
   %all() returns true. Otherwise, n is composite, so it returns false.
   %This return value is then assigned to the variable isPrime.
   isPrime = all(mod(n, (3:round(sqrt(n))) ));

end</lang>

Sample output:

<lang MATLAB>>> arrayfun(@primalityByTrialDivision,(1:14))

ans =

    0     1     1     0     1     0     1     0     0     0     1     0     1     0</lang>

MAXScript

<lang MAXScript> fn isPrime n =

(
    if n == 2 then
    (
        return true
    )
    else if (n <= 1) OR (mod n 2 == 0) then
    (
        return false
    )

    for i in 3 to (sqrt n) by 2 do
    (
        if mod n i == 0 then return false
    )
    true
)</lang>

MUMPS

<lang MUMPS>ISPRIME(N)

QUIT:(N=2) 1
NEW I,TP
SET TP=+'$PIECE((N/2),".",2)
IF 'TP FOR I=3:2:(N**.5) SET TP=+'$PIECE((N/I),".",2) Q:TP
KILL I
QUIT 'TP</lang>

Usage (0 is false, nonzero is true):

USER>W $$ISPRIME^ROSETTA(2)
1
USER>W $$ISPRIME^ROSETTA(4)
0
USER>W $$ISPRIME^ROSETTA(7)
1
USER>W $$ISPRIME^ROSETTA(97)
1
USER>W $$ISPRIME^ROSETTA(99)
0

NetRexx

<lang NetRexx>/* NetRexx */

options replace format comments java crossref savelog symbols nobinary

parse arg nbr rangeBegin rangeEnd .

if nbr = | nbr = '.' then do

 if rangeBegin =  | rangeBegin = '.' then rangeBegin = 1
 if rangeEnd   =  | rangeEnd   = '.' then rangeEnd   = 100
 if rangeEnd > rangeBegin then direction = 1
                          else direction = -1

 say 'List of prime numbers from' rangeBegin 'to' rangeEnd':'
 primes = 
 loop nn = rangeBegin to rangeEnd by direction
   if isPrime(nn) then primes = primes nn
   end nn
   primes = primes.strip
   say '  'primes.changestr(' ', ',')
   say '  Total number of primes:' primes.words
 end

else do

 if isPrime(nbr) then say nbr.right(20) 'is prime'
                 else say nbr.right(20) 'is not prime'
 end

return

method isPrime(nbr = long) public static binary returns boolean

 ip = boolean

 select
   when nbr < 2 then do
     ip = isFalse
     end
   when '2 3 5 7'.wordpos(Rexx(nbr)) \= 0 then do
     -- crude shortcut ripped from the Rexx example
     ip = isTrue
     end
   when  nbr // 2 == 0 | nbr // 3 == 0 then do
     -- another shortcut permitted by the one above
     ip = isFalse
     end
   otherwise do
     nn = long
     nnStartTerm = long 3 -- a reasonable start term - nn <= 2 is never prime
     nnEndTerm = long Math.ceil(Math.sqrt(nbr)) -- a reasonable end term
     ip = isTrue -- prime the pump (pun intended)
     loop nn = nnStartTerm to nnEndTerm by 2
        -- Note: in Rexx and NetRexx "//" is the 'remainder of division operator' (which is not the same as modulo)
       if nbr // nn = 0 then do
         ip = isFalse
         leave nn
         end
       end nn
     end
   end

 return ip

method isTrue public static returns boolean

 return 1 == 1

method isFalse public static returns boolean

 return \isTrue</lang>

Output:

$ java -cp . RCPrimality 
List of prime numbers from 1 to 100:
  2,3,5,7,11,13,17,19,23,29,31,37,41,43,47,53,59,61,67,71,73,79,83,89,97
  Total number of primes: 25

$ java -cp . RCPrimality 91
                  91 is not prime

$ java -cp . RCPrimality 101
                 101 is prime

$ java -cp . RCPrimality . . 25
List of prime numbers from 1 to 25:
  2,3,5,7,11,13,17,19,23
  Total number of primes: 9

$ java -cp . RCPrimality . 9900 10010
List of prime numbers from 9900 to 10010:
  9901,9907,9923,9929,9931,9941,9949,9967,9973,10007,10009
  Total number of primes: 11

$ java -cp . RCPrimality . -57 1
List of prime numbers from -57 to 1:
  
  Total number of primes: 0

$ java -cp . RCPrimality . 100 -57
List of prime numbers from 100 to -57:
  97,89,83,79,73,71,67,61,59,53,47,43,41,37,31,29,23,19,17,13,11,7,5,3,2
  Total number of primes: 25

Rexx version reimplemented in NetRexx

Translation of: REXX

<lang NetRexx>/* NetRexx */

options replace format comments java crossref savelog symbols nobinary

/*REXX program tests for primality using (kinda smartish) trial division*/

parse arg n . /*let user choose how many, maybe*/ if n== then n=10000 /*if not, then assume the default*/ p=0 /*a count of primes (so far). */

                                      /*I like Heinz's 57 varieties... */
 loop j=-57 to n                      /*start in the cellar and work up*/
 if \isprime(j) then iterate          /*if not prime, keep looking.    */
 p=p+1                                /*bump the jelly bean counter.   */
 if j.length>2 then iterate           /*only show two-digit primes.    */
 say j.right(20) 'is prime.'          /*Just blab about the wee primes.*/
 end

say say "there're" p "primes up to" n '(inclusive).' exit

/*-------------------------------------ISPRIME subroutine---------------*/ method isprime(x) public static returns boolean --isprime: procedure; arg x /*get the number in question*/ if '2 3 5 7'.wordpos(x)\==0 then return 1 /*is it a teacher's pet? */ if x<2 | x//2==0 | x//3==0 then return 0 /*weed out the riff-raff. */

                                           /*Note:   //  is modulus.   */
  loop j=5 by 6 until j*j>x                /*skips multiples of three. */
  if x//j==0 | x//(j+2)==0 then return 0   /*do a pair of divides (mod)*/
  end

return 1 /*I'm exhausted, it's prime!*/</lang>

Objeck

<lang objeck>function : IsPrime(n : Int) ~ Bool {

 if(n <= 1) {
   return false;
 };
 
 for(i := 2; i * i <= n; i += 1;) {           
   if(n % i = 0) {
     return false;
   };
 };
 
 return true;

}</lang>

OCaml

<lang ocaml>let is_prime n =

 if n = 2 then true
 else if n < 2 || n mod 2 = 0 then false
 else
   let rec loop k =
     if k * k > n then true
     else if n mod k = 0 then false
     else loop (k+2)
   in loop 3</lang>

Octave

This function works on vectors and matrix. <lang octave>function b = isthisprime(n)

 for r = 1:rows(n)
   for c = 1:columns(n)
     b(r,c) = false;
     if ( n(r,c) == 2 )

b(r,c) = true;

     elseif ( (n(r,c) < 2) || (mod(n(r,c),2) == 0) )

b(r,c) = false;

     else

b(r,c) = true; for i = 3:2:sqrt(n(r,c)) if ( mod(n(r,c), i) == 0 ) b(r,c) = false; break; endif endfor

     endif
   endfor
 endfor

endfunction

% as test, print prime numbers from 1 to 100 p = [1:100]; pv = isthisprime(p); disp(p( pv ));</lang>

PARI/GP

<lang parigp>trial(n)={

 if(n < 4, return(n > 1)); /* Handle negatives */
 forprime(p=2,sqrt(n),
   if(n%p == 0, return(0))
 );
 1

};</lang>

Pascal

Translation of: BASIC

<lang Pascal>program primes;

function prime(n: integer): boolean; var

 i: integer; max: real;

begin

 if n = 2 then
   prime := true
 else if (n <= 1) or (n mod 2 = 0) then
   prime := false
 else begin
   prime := true; i := 3; max := sqrt(n);
   while i <= max do begin
     if n mod i = 0 then begin
       prime := false; exit
     end;
     i := i + 2
   end
 end

end;

{ Test and display primes 0 .. 50 } var

 n: integer;

begin

 for n := 0 to 50 do
   if (prime(n)) then
     write(n, ' ');

end.</lang>

Output:

2 3 5 7 11 13 17 19 23 29 31 37 41 43 47

Perl

A more idiomatic solution than the RE-based version below: <lang perl>sub prime { my $n = shift || $_;

   $n % $_ or return for 2 .. sqrt $n;
   $n > 1

}

print join(', ' => grep prime, 1..100), "\n";</lang>

By Regular Expression

JAPH by Abigail 1999 [1] in conference slides 2000 [2] <lang perl>sub isprime {

 ('1' x shift) !~ /^1?$|^(11+?)\1+$/

}

A quick test

print join(', ', grep(isprime($_), 0..39)), "\n";</lang>

Perl 6

Works with: Rakudo Star version 2010.09

Here we use a "none" junction which will autothread through the %% "is divisible by" operator to assert that $n is not divisible by 2 or any of the odd numbers up to its square root. Read it just as you would English: "Integer $n is prime if it is greater than one and is divisible by none of 2, 3, whatever + 2, up to but not including whatever is greater than the square root of $n." <lang perl6>sub prime (Int $n --> Bool) {

   $n > 1 and $n %% none 2, 3, *+2 ...^ * > sqrt $n;

}</lang> (No pun indented.)

Testing: <lang perl6>say "$_ is{ "n't" x !prime($_) } prime." for 1 .. 100;</lang>

PHP

<lang php><?php function prime($a) {

 if (($a % 2 == 0 && $a != 2) || $a < 2)
   return false;
 $limit = sqrt($a);
 for ($i = 2; $i <= $limit; $i++)
   if ($a % $i == 0)
     return false;
 return true;

}

foreach (range(1, 100) as $x)

 if (prime($x)) echo "$x\n";

?></lang>

By Regular Expression

<lang php><?php function prime($a) {

 return !preg_match('/^1?$|^(11+?)\1+$/', str_repeat('1', $a));

} ?></lang>

PicoLisp

<lang PicoLisp>(de prime? (N)

  (or
     (= N 2)
     (and
        (> N 1)
        (bit? 1 N)
        (for (D 3  T  (+ D 2))
           (T (> D (sqrt N)) T)
           (T (=0 (% N D)) NIL) ) ) ) )</lang>

PL/I

<lang PL/I>is_prime: procedure (n) returns (bit(1));

  declare n fixed (15);
  declare i fixed (10);

  if n < 2 then return ('0'b);
  if mod(n, 2) = 0 then return ('0'b);

  do i = 3 to sqrt(n) by 2;
     if mod(n, i) = 0 then return ('0'b);
  end;
  return ('1'b);

end is_prime;</lang>

PowerShell

<lang powershell>function isPrime ($n) {

   if ($n -eq 1) {
       return $false
   } else {
       return (@(2..[Math]::Sqrt($n) | Where-Object { $n % $_ -eq 0 }).Length -eq 0)
   }

}</lang>

Prolog

<lang Prolog>prime(N) :- integer(N), N > 1,

   M is floor(sqrt(N+1)),    % round-off paranoia 
   N mod 2 > 0,              % is odd
   check(N, M, 3).

check(N, M, F) :- F>M. check(N, M, F) :- F=<M,

   N mod F > 0,
   F1 is F + 2,              % test by odds only
   check(N, M, F1).</lang>

PureBasic

<lang PureBasic>Procedure.i IsPrime(n)

  Protected k

  If n = 2
     ProcedureReturn #True
  EndIf

  If n <= 1 Or n % 2 = 0
     ProcedureReturn #False
  EndIf
  
  For k = 3 To Int(Sqr(n)) Step 2
     If n % k = 0
        ProcedureReturn #False
     EndIf
  Next

  ProcedureReturn #True

EndProcedure</lang>

Python

The simplest primality test, using trial division:

Works with: Python version 2.5

<lang python>def prime(a):

   return not (a < 2 or any(a % x == 0 for x in xrange(2, int(a**0.5) + 1)))</lang>

Another test. Exclude even numbers first: <lang python>def prime2(a):

   if a == 2: return True
   if a < 2 or a % 2 == 0: return False
   return not any(a % x == 0 for x in xrange(3, int(a**0.5) + 1, 2))</lang>

Yet another test. Exclude multiples of 2 and 3, see http://www.devx.com/vb2themax/Tip/19051:

Works with: Python version 2.4

<lang python>def prime3(a):

   if a < 2: return False
   if a == 2 or a == 3: return True # manually test 2 and 3   
   if a % 2 == 0 or a % 3 == 0: return False # exclude multiples of 2 and 3

   maxDivisor = a**0.5
   d, i = 5, 2
   while d <= maxDivisor:
       if a % d == 0: return False
       d += i 
       i = 6 - i # this modifies 2 into 4 and viceversa

   return True</lang>

By Regular Expression

Regular expression by "Abigail".
(An explanation is given in "The Story of the Regexp and the Primes"). <lang python>>>> import re >>> def isprime(n):

   return not re.match(r'^1?$|^(11+?)\1+$', '1' * n)

>>> # A quick test >>> [i for i in range(40) if isprime(i)] [2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37]</lang>

Qi

<lang Qi>(define prime?-0

 K N -> true where (> (* K K) N)
 K N -> false where (= 0 (MOD N K))
 K N -> (prime?-0 (+ K 2) N))

(define prime?

 1 -> false
 2 -> true
 N -> false where (= 0 (MOD N 2))
 N -> (prime?-0 3 N))</lang>

R

<lang R>isPrime <- function(n) {

 if (n == 2) return(TRUE)
 if ( (n <= 1) || ( n %% 2 == 0 ) ) return(FALSE)
 for( i in 3:sqrt(n) ) {
   if ( n %% i == 0 ) return(FALSE)
 }
 TRUE

}</lang>

<lang R>print(lapply(1:100, isPrime))</lang>

REBOL

<lang REBOL>prime?: func [n] [

   case [
       n = 2 [ true  ]
       n <= 1 or (n // 2 = 0) [ false ]
       true [
           for i 3 round square-root n 2 [
               if n // i = 0 [ return false ]
           ]
           true
       ]
   ]

]</lang>

<lang REBOL>repeat i 100 [ print [i prime? i]]</lang>

REXX

<lang rexx>/*REXX program tests for primality using (kinda smartish) trial division*/

parse arg n . /*let user choose how many, maybe*/ if n== then n=10000 /*if not, then assume the default*/ p=0 /*a count of primes (so far). */

                                      /*I like Heinz's 57 varieties... */
 do j=-57 to n                        /*start in the cellar and work up*/
 if \isprime(j) then iterate          /*if not prime, keep looking.    */
 p=p+1                                /*bump the jelly bean counter.   */
 if length(j)>2 then iterate          /*only show two-digit primes.    */
 say right(j,20) 'is prime.'          /*Just blab about the wee primes.*/
 end

say say "there're" p "primes up to" n '(inclusive).' exit /*─────────────────────────────────────ISPRIME subroutine───────────────*/ isprime: procedure; arg x /*get the number in question*/ if wordpos(x,'2 3 5 7')\==0 then return 1 /*is it a teacher's pet? */ if x<2 | x//2==0 | x//3==0 then return 0 /*weed out the riff-raff. */

                                           /*Note:   //  is modulus.   */
  do j=5 by 6 until j*j>x                  /*skips multiples of three. */
  if x//j==0 | x//(j+2)==0 then return 0   /*do a pair of divides (mod)*/
  end

return 1 /*I'm exhausted, it's prime!*/</lang>

Output:

                   2 is prime.
                   3 is prime.
                   5 is prime.
                   7 is prime.
                  11 is prime.
                  13 is prime.
                  17 is prime.
                  19 is prime.
                  23 is prime.
                  29 is prime.
                  31 is prime.
                  37 is prime.
                  41 is prime.
                  43 is prime.
                  47 is prime.
                  53 is prime.
                  59 is prime.
                  61 is prime.
                  67 is prime.
                  71 is prime.
                  73 is prime.
                  79 is prime.
                  83 is prime.
                  89 is prime.
                  97 is prime.

there're 1229 primes up to 10000 (inclusive).

Ruby

<lang ruby>def prime(a)

 if a == 2
   true
 elsif a <= 1 || a % 2 == 0
   false
 else
   divisors = Enumerable::Enumerator.new(3..Math.sqrt(a), :step, 2)
   # this also creates an enumerable object:  divisors = (3..Math.sqrt(a)).step(2)
   !divisors.any? { |d| a % d == 0 }
 end

end</lang> The mathn package in the stdlib for Ruby 1.9.2 contains this compact Prime#prime? method: <lang ruby> def prime?(value, generator = Prime::Generator23.new)

   return false if value < 2
   for num in generator
     q,r = value.divmod num
     return true if q < num
     return false if r == 0
   end
 end</lang>

Without any fancy stuff: <lang ruby>def primes(limit)

 (enclose = lambda { |primes| 
   primes.last.succ.upto(limit) do |trial_pri|
     if primes.find { |pri| (trial_pri % pri).zero? }.nil?
       return enclose.call(primes << trial_pri)
     end
   end
   primes
 }).call([2])

end</lang>

By Regular Expression

<lang ruby>def isprime(n)

 '1'*n !~ /^1?$|^(11+?)\1+$/

end</lang>

Run BASIC

<lang runbasic>' Test and display primes 1 .. 50 for i = 1 TO 50

 if prime(i) <> 0 then print i;" ";

next i

FUNCTION prime(n) if n < 2 then prime = 0 : goto [exit] if n = 2 then prime = 1 : goto [exit] if n mod 2 = 0 then prime = 0 : goto [exit] prime = 1 for i = 3 to int(n^.5) step 2

if n mod i = 0 then  prime = 0 : goto [exit]

next i [exit]

END FUNCTION</lang>

2 3 5 7 11 13 17 19 23 25 29 31 37 41 43 47 49

SAS

<lang sas>data primes; do n=1 to 1000;

 link primep;
 if primep then output;

end; stop;

primep: if n < 4 then do;

 primep=n=2 or n=3;
 return;

end; primep=0; if mod(n,2)=0 then return; do k=3 to sqrt(n) by 2;

 if mod(n,k)=0 then return;

end; primep=1; return; keep n; run;</lang>

Scala

<lang scala>def isPrime(n: Int) = n > 1 && (Iterator.from(2) takeWhile (d => d * d <= n) forall (n % _ != 0))</lang>

Scheme

Works with: Scheme version R

^{5}

RS

<lang scheme>(define (prime? number)

 (define (*prime? divisor)
   (or (> (* divisor divisor) number)
       (and (> (modulo number divisor) 0)
            (*prime? (+ divisor 1)))))
 (and (> number 1)
      (*prime? 2)))</lang>

<lang scheme>; twice faster, testing only odd divisors (define (prime? n)

 (if (< n 4) (> n 1)
     (and (odd? n)

(let loop ((k 3)) (or (> (* k k) n) (and (positive? (remainder n k)) (loop (+ k 2))))))))</lang>

Seed7

<lang seed7>const func boolean: is_prime (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 number rem 2 = 0 or number <= 1 then
     prime := FALSE;
   else
     upTo := sqrt(number);
     while number rem testNum <> 0 and testNum <= upTo do
       testNum +:= 2;
     end while;
     prime := testNum > upTo;
   end if;
 end func;</lang>

Original source: [3]

SNOBOL4

<lang SNOBOL4>define('isprime(n)i,max') :(isprime_end) isprime isprime = n

       le(n,1) :s(freturn)
       eq(n,2) :s(return)
       eq(remdr(n,2),0) :s(freturn)
       max = sqrt(n); i = 1

isp1 i = le(i + 2,max) i + 2 :f(return)

       eq(remdr(n,i),0) :s(freturn)f(isp1)

isprime_end</lang>

By Patterns

Using the Abigail regex transated to Snobol patterns. <lang SNOBOL4> define('rprime(n)str,pat1,pat2,m1') :(end_rprime) rprime str = dupl('1',n); rprime = n

       pat1 = ('1' | )
       pat2 = ('11' arbno('1')) $ m1 (*m1 arbno(*m1))
       str pos(0) (pat1 | pat2) rpos(0) :s(freturn)f(return)

end_rprime

# Test and display primes 0 .. 50

loop rprimes = rprimes rprime(n) ' '

       n = lt(n,50) n + 1 :s(loop)
       output = rprimes

end</lang>

Output:

2 3 5 7 11 13 17 19 23 29 31 37 41 43 47

Standard ML

<lang sml>fun is_prime n =

 if n = 2 then true
 else if n < 2 orelse n mod 2 = 0 then false
 else let
   fun loop k =
     if k * k > n then true
     else if n mod k = 0 then false
     else loop (k+2)
   in loop 3
 end</lang>

Tcl

<lang tcl>proc is_prime n {

   if {$n <= 1} {return false}
   if {$n == 2} {return true}
   if {$n % 2 == 0} {return false}
   for {set i 3} {$i <= sqrt($n)} {incr i 2} {
       if {$n % $i == 0} {return false}
   }
   return true

}</lang>

TI-83 BASIC

Prompt A
If A=2:Then
Disp "PRIME"
Stop
End

If (fPart(A/2)=0 and A>0) or A<2:Then
Disp "NOT PRIME"
Stop
End

1→P
For(B,3,int(√(A)))
If FPart(A/B)=0:Then
0→P
√(A)→B
End
B+1→B
End

If P=1:Then
Disp "PRIME"
Else
Disp "NOT PRIME"
End

UNIX Shell

Translation of: C

Works with: bash

Works with: ksh93

Works with: pdksh

Works with: zsh

<lang bash>function primep { typeset n=$1 p=3 (( n == 2 )) && return 0 # 2 is prime. (( n & 1 )) || return 1 # Other evens are not prime. (( n >= 3 )) || return 1

# Loop for odd p from 3 to sqrt(n). # Comparing p * p <= n can overflow. while (( p <= n / p )); do # If p divides n, then n is not prime. (( n % p )) || return 1 (( p += 2 )) done return 0 # Yes, n is prime. }</lang>

Works with: Bourne Shell

<lang bash>primep() { set -- "$1" 3 test "$1" -eq 2 && return 0 # 2 is prime. expr "$1" \% 2 >/dev/null || return 1 # Other evens are not prime. test "$1" -ge 3 || return 1

# Loop for odd p from 3 to sqrt(n). # Comparing p * p <= n can overflow. We use p <= n / p. while expr $2 \<= "$1" / $2 >/dev/null; do # If p divides n, then n is not prime. expr "$1" % $2 >/dev/null || return 1 set -- "$1" `expr $2 + 2` done return 0 # Yes, n is prime. }</lang>

Ursala

Excludes even numbers, and loops only up to the approximate square root or until a factor is found. <lang Ursala>#import std

import nat

prime = ~<{0,1}&& -={2,3}!| ~&h&& (all remainder)^Dtt/~& iota@K31</lang> Test program to try it on a few numbers: <lang Ursala>#cast %bL

test = prime* <5,6,7></lang>

Output:

<true,false,true>

V

Translation of: Joy

<lang v>[prime?

    2
    [[dup * >] [true] [false] ifte [% 0 >] dip and]
      [succ]
    while
    dup * <].</lang>

Using it:

<lang v>|11 prime? =true |4 prime? =false</lang>

XPL0

<lang XPL0>include c:\cxpl\codes; \intrinsic 'code' declarations

func Prime(N); \Return 'true' if N is a prime number int N; int I; [if N <= 1 then return false; for I:= 3 to sqrt(N) do

       if rem(N/I) = 0 then return false;

return true; ]; \Prime

int Num; repeat Num:= IntIn(0);

       Text(0, if Prime(Num) then "is " else "not ");
       Text(0, "prime^M^J");

until Num = 0</lang>

Example output:

777777777
not prime
777777773
is prime
0
not prime