Primality by trial division
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You are encouraged to solve this task according to the task description, using any language you may know.
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 tasks: Sieve of Eratosthenes, Prime decomposition, AKS test for primes.
ABAP
<lang ABAP>class ZMLA_ROSETTA definition
public create public .
public section.
types: enumber TYPE N LENGTH 60 . types: listof_enumber TYPE TABLE OF enumber .
class-methods IS_PRIME importing value(N) type ENUMBER returning value(OFLAG) type ABAP_BOOL . class-methods IS_PRIME_I importing value(N) type I returning value(OFLAG) type ABAP_BOOL . protected section. private section.
ENDCLASS.
CLASS ZMLA_ROSETTA IMPLEMENTATION.
- <SIGNATURE>---------------------------------------------------------------------------------------+
- | Static Public Method ZMLA_ROSETTA=>IS_PRIME
- +-------------------------------------------------------------------------------------------------+
- | [--->] N TYPE ENUMBER
- | [<-()] OFLAG TYPE ABAP_BOOL
- +--------------------------------------------------------------------------------------</SIGNATURE>
method IS_PRIME. IF n < 2. oflag = abap_false. RETURN. ENDIF. IF n = 2 or n = 3. oflag = abap_true. RETURN. ENDIF. IF n mod 2 = 0 or n mod 3 = 0. oflag = abap_false. RETURN. ENDIF. DATA: lim type enumber, d type enumber, i TYPE i VALUE 2. lim = sqrt( n ). d = 5. WHILE d <= lim. IF n mod d = 0. oflag = abap_false. RETURN. ENDIF. d = d + i. i = 6 - i. " this modifies 2 into 4 and viceversa ENDWHILE. oflag = abap_true. RETURN. endmethod.
- <SIGNATURE>---------------------------------------------------------------------------------------+
- | Static Public Method ZMLA_ROSETTA=>IS_PRIME_I
- +-------------------------------------------------------------------------------------------------+
- | [--->] N TYPE I
- | [<-()] OFLAG TYPE ABAP_BOOL
- +--------------------------------------------------------------------------------------</SIGNATURE>
method IS_PRIME_I. IF n < 2. oflag = abap_false. RETURN. ENDIF. IF n = 2 or n = 3. oflag = abap_true. RETURN. ENDIF. IF n mod 2 = 0 or n mod 3 = 0. oflag = abap_false. RETURN. ENDIF. DATA: lim type i, d type i, i TYPE i VALUE 2. lim = sqrt( n ). d = 5. WHILE d <= lim. IF n mod d = 0. oflag = abap_false. RETURN. ENDIF. d = d + i. i = 6 - i. " this modifies 2 into 4 and viceversa ENDWHILE. oflag = abap_true. RETURN. endmethod.
ENDCLASS.</lang>
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>
ActionScript
<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
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
<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($1)}</lang>
BASIC
Returns 1 for prime, 0 for non-prime <lang QBasic>FUNCTION prime% (n!)
STATIC i AS INTEGER IF n = 2 THEN prime = 1 ELSEIF n <= 1 OR n MOD 2 = 0 THEN prime = 0 ELSE prime = 1 FOR i = 3 TO INT(SQR(n)) STEP 2 IF n MOD i = 0 THEN prime = 0 EXIT FUNCTION END IF NEXT i END IF
END FUNCTION
' Test and display primes 1 .. 50 DECLARE FUNCTION prime% (n!) FOR n = 1 TO 50
IF prime(n) = 1 THEN PRINT n;
NEXT n</lang>
- Output:
2 3 5 7 11 13 17 19 23 29 31 37 41 43 47
ZX Spectrum Basic
<lang ZXBasic>10 LET n=0: LET p=0 20 INPUT "Enter number: ";n 30 GO SUB 1000 40 IF p=0 THEN PRINT n;" is not prime!" 50 IF p<>0 THEN PRINT n;" is prime!" 60 GO TO 10 1000 REM *************** 1001 REM * PRIME CHECK * 1002 REM *************** 1010 LET p=0 1020 IF n/2=INT (n/2) THEN RETURN 1030 LET p=1 1040 FOR i=3 TO SQR (n) STEP 2 1050 IF n/i=INT (n/i) THEN LET p=0: LET i= SQR (n) 1060 NEXT i 1070 RETURN </lang>
- Output:
15 is not prime! 17 is prime! 119 is not prime! 137 is prime!
BBC BASIC
<lang bbcbasic> FOR i% = -1 TO 100
IF FNisprime(i%) PRINT ; i% " is prime" NEXT END DEF FNisprime(n%) IF n% <= 1 THEN = FALSE IF n% <= 3 THEN = TRUE IF (n% AND 1) = 0 THEN = FALSE LOCAL t% FOR t% = 3 TO SQR(n%) STEP 2 IF n% MOD t% = 0 THEN = FALSE NEXT = TRUE</lang>
Output:
2 is prime 3 is prime 5 is prime 7 is prime 11 is prime 13 is prime 17 is prime 19 is prime 23 is prime 29 is prime 31 is prime 37 is prime 41 is prime 43 is prime 47 is prime 53 is prime 59 is prime 61 is prime 67 is prime 71 is prime 73 is prime 79 is prime 83 is prime 89 is prime 97 is prime
bc
<lang bc>/* Return 1 if n is prime, 0 otherwise */ define p(n) {
auto i
if (n < 2) return(0) if (n == 2) return(1) if (n % 2 == 0) return(0) for (i = 3; i * i <= n; i += 2) { if (n % i == 0) return(0) } return(1)
}</lang>
Bracmat
<lang bracmat> ( prime
= incs n I inc . 4 2 4 2 4 6 2 6:?incs & 2:?n & 1 2 2 !incs:?I & whl ' ( !n*!n:~>!arg & div$(!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
Burlesque
<lang burlesque> fcL[2== </lang>
Implicit trial division is done by the fc function. It checks if the number has exactly two divisors.
Version not using the fc function:
<lang burlesque> blsq ) 11^^2\/?dr@.%{0==}ayn! 1 blsq ) 12^^2\/?dr@.%{0==}ayn! 0 blsq ) 13^^2\/?dr@.%{0==}ayn! 1 </lang>
Explanation. Given n generates a block containing 2..n-1. Calculates a block of modolus and check if it contains 0. If it contains 0 it is not a prime.
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>
Chapel
<lang chapel>proc is_prime(n) {
if n == 2 then return true; if n <= 1 || n % 2 == 0 then return false; for i in 3..floor(sqrt(n)):int by 2 do if n % i == 0 then 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>
COBOL
<lang cobol> Identification Division.
Program-Id. Primality-By-Subdiv.
Data Division. Working-Storage Section. 78 True-Val Value 0. 78 False-Val Value 1.
Local-Storage Section. 01 lim Pic 9(10). 01 i Pic 9(10).
Linkage Section. 01 num Pic 9(10). 01 result Pic 9.
Procedure Division Using num result. Main. If num <= 1 Move False-Val To result Goback Else If num = 2 Move True-Val To result Goback End-If
Compute lim = Function Sqrt(num) + 1 Perform Varying i From 3 By 1 Until lim < i If Function Mod(num, i) = 0 Move False-Val To result Goback End-If End-Perform
Move True-Val To Result
Goback .</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 isPrime1(in int n) pure nothrow {
if (n == 2) return true; if (n <= 1 || (n & 1) == 0) return false;
for(int i = 3; i <= real(n).sqrt; i += 2) if (n % i == 0) return false; return true;
}
void main() {
iota(2, 40).filter!isPrime1.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>bool isPrime2(It)(in It n) pure nothrow {
// Adapted from: http://www.devx.com/vb2themax/Tip/19051 // Test 1, 2, 3 and multiples of 2 and 3: if (n == 2 || n == 3) return true; else if (n < 2 || n % 2 == 0 || n % 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 n. for (It div = 5, inc = 2; div ^^ 2 <= n; div += inc, inc = 6 - inc) if (n % div == 0) return false;
return true;
}
void main() {
import std.stdio, std.algorithm, std.range;
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)real(n).sqrt / 2) * 2 + 1; for ( ; head <= tail ; head +=2, tail -= 2) if ((n % head) == 0 || (n % tail) == 0) return false; return true;
}
void main() {
iota(2, 40).filter!isPrime3.writeln;
}</lang>
Delphi
First
<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>
Second
<lang Delphi>function IsPrime(const x: integer): Boolean; var
i: integer;
begin
i := 2; repeat if X mod i = 0 then begin Result := False; Exit; end; Inc(i); until i > Sqrt(x); Result := True;
end;</lang>
E
<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>
FBSL
The second function (included by not used) I would have thought would be faster because it lacks the SQR() function. As it happens, the first is over twice as fast. <lang qbasic>#APPTYPE CONSOLE
FUNCTION ISPRIME(n AS INTEGER) AS INTEGER
IF n = 2 THEN RETURN TRUE ELSEIF n <= 1 ORELSE n MOD 2 = 0 THEN RETURN FALSE ELSE FOR DIM i = 3 TO SQR(n) STEP 2 IF n MOD i = 0 THEN RETURN FALSE END IF NEXT RETURN TRUE END IF
END FUNCTION
FUNCTION ISPRIME2(N AS INTEGER) AS INTEGER
IF N <= 1 THEN RETURN FALSE DIM I AS INTEGER = 2 WHILE I * I <= N IF N MOD I = 0 THEN RETURN FALSE END IF I = I + 1 WEND RETURN TRUE
END FUNCTION
' Test and display primes 1 .. 50 DIM n AS INTEGER
FOR n = 1 TO 50
IF ISPRIME(n) THEN PRINT n, " "; END IF
NEXT
PAUSE </lang> Output
2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 Press any key to continue...
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
<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>
F#
<lang fsharp>open NUnit.Framework open FsUnit
let isPrime x =
match x with | 2 | 3 -> true | x when x % 2 = 0 -> false | _ -> let rec aux i = match i with | i when x % i = 0 -> false | i when x < i*i -> true | _ -> aux (i+2) aux 3
[<Test>] let ``Validate that 2 is prime`` () =
isPrime 2 |> should equal true
[<Test>] let ``Validate that 4 is not prime`` () =
isPrime 4 |> should equal false
[<Test>] let ``Validate that 3 is prime`` () =
isPrime 3 |> should equal true
[<Test>] let ``Validate that 9 is not prime`` () =
isPrime 9 |> should equal false
[<Test>] let ``Validate that 5 is prime`` () =
isPrime 5 |> should equal true
[<Test>] let ``Validate that 277 is prime`` () =
isPrime 277 |> should equal true</lang>
Shorter version using sequence and library function:
<lang fsharp> let isPrime x =
seq { 2 .. x/2 } |> Seq.exists (fun i -> x % i = 0) |> not
</lang>
and if you want to test really big numbers, use System.Numerics.BigInteger:
<lang fsharp> let isPrime x =
seq { 2I .. x/2I } |> Seq.exists (fun i -> x % i = 0I) |> not
</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 { return true } 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).mod 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 || (n `mod` 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, x `mod` 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 , and of true SoE it is , in n primes produced.
The above can be written on one line as <lang haskell>import Data.List (nubBy)
primes = nubBy (\p x -> x `mod` p == 0) [2..]</lang>
(The above works with the reference implementation of nubBy
in Haskell 98, but does not work with current versions of GHC due to a bug.)
A similar version, using gcd instead of modulus, is: <lang haskell>import Data.List (nubBy)
primes = nubBy (((>1).).gcd) [2..]</lang>
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 | otherwise = p : sieve [x | x <- xs, x `mod` 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 | otherwise = sieve [x | x <- xs, x `mod` 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).mod x) fs) [x,x+2..q-2] ++ sieve (q+2) (head ps^2) (tail ps) (k+1)</lang>
Runs at empirical 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
<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.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
Maple
This could be coded in myriad ways; here is one. <lang Maple> TrialDivision := proc( n :: integer )
if n <= 1 then false elif n = 2 then true elif type( n, 'even' ) then false else for local i from 3 by 2 while i * i <= n do if irem( n, i ) = 0 then return false end if end do; true end if
end proc: </lang> Using this to pick off the primes up to 30, we get: <lang Maple> > select( TrialDivision, [seq]( 1 .. 30 ) );
[2, 3, 5, 7, 11, 13, 17, 19, 23, 29]
</lang> Here is a way to check that TrialDivision above agrees with Maple's built-in primality test (isprime). <lang Maple> > N := 10000: evalb( select( TrialDivision, [seq]( 1 .. N ) ) = select( isprime, [seq]( 1 .. N ) ) );
true
</lang>
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>
Maxima
<lang Maxima>isprme(n):= catch(
for k: 2 thru sqrt(n) do if mod(n, k)=0 then throw(false), true);
map(isprme, [2, 3, 4, 65, 100, 181, 901]); /* [true, true, false, false, false, true, false] */</lang>
МК-61/52
<lang>П0 1 - x#0 34 2 - /-/ x<0 32 ИП0 2 / {x} x#0 34 3 П4 ИП0 ИП4 / {x} x#0 34 КИП4 КИП4 ИП0 КвКор ИП4 - x<0 16 1 С/П 0 С/П</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
<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
<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
Here we use a "none" junction which will autothread through the %% "is divisible by" operator to assert that $i is not divisible by 2 or any of the odd numbers up to its square root. Read it just as you would English: "Integer $i 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 $i." <lang perl6>sub prime (Int $i --> Bool) {
$i > 1 and $i %% none 2, 3, *+2 ...^ * >= sqrt $i;
}</lang> (No pun indented.)
This can easily be improved in two ways. First, we generate the primes so we only divide by those, instead of all odd numbers. Second, we memoize the result using the //= idiom of Perl, which does the right-hand calculation and assigns it only if the left side is undefined. We also try to avoid recalculating the square root each time. <lang perl6>my @primes := 2, 3, 5, -> $p { ($p+2, $p+4 ... &prime)[*-1] } ... *; my @isprime = False,False; # 0 and 1 are not prime by definition sub prime($i) { @isprime[$i] //= ($i %% none @primes ...^ * > $_ given $i.sqrt.floor) }</lang> Note the mutual dependency between the prime generator and the prime tester.
Testing: <lang perl6>say "$_ 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', $a));
} ?></lang>
PicoLisp
<lang PicoLisp>(de prime? (N)
(or (= N 2) (and (> N 1) (bit? 1 N) (for (D 3 T (+ D 2)) (T (> D (sqrt N)) T) (T (=0 (% N D)) NIL) ) ) ) )</lang>
PL/I
<lang PL/I>is_prime: procedure (n) returns (bit(1));
declare n fixed (15); declare i fixed (10);
if n < 2 then return ('0'b); if n = 2 then return ('1'b); if mod(n, 2) = 0 then return ('0'b);
do i = 3 to sqrt(n) by 2; if mod(n, i) = 0 then return ('0'b); end; return ('1'b);
end is_prime;</lang>
PowerShell
<lang powershell>function isPrime ($n) {
if ($n -eq 1) { return $false } else { return (@(2..[Math]::Sqrt($n) | Where-Object { $n % $_ -eq 0 }).Length -eq 0) }
}</lang>
Prolog
The following predicate showcases Prolog's support for writing predicates suitable for both testing and generating. In this case, assuming the Prolog implemenation supports indefinitely large integers, prime(N) can be used to generate primes until memory is exhausted. <lang Prolog>prime(2). prime(N) :-
between(3, inf, N), 1 is N mod 2, % odd M is floor(sqrt(N+1)), % round-off paranoia Max is (M-1) // 2, % integer division forall( between(1, Max, I), N mod (2*I+1) > 0 ).
</lang> Example using SWI-Prolog:
?- time( (bagof( P, (prime(P), ((P > 100000, !, fail); true)), Bag), length(Bag,N), last(Bag,Last), writeln( (N,Last) ) )). % 1,724,404 inferences, 1.072 CPU in 1.151 seconds (93% CPU, 1607873 Lips) Bag = [2, 3, 5, 7, 11, 13, 17, 19, 23|...], N = 9592, Last = 99991. ?- time( prime(99991) ). % 165 inferences, 0.000 CPU in 0.000 seconds (92% CPU, 1213235 Lips) true. ?-
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:
<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:
<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>
Racket
<lang Racket>#lang racket
(define (prime? number)
(cond ((not (positive? number)) #f) ((= 1 number) #f) ((even? number) (= 2 number)) (else (for/and ((i (in-range 3 (ceiling (sqrt number))))) (not (zero? (remainder number i)))))))</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
compact version
This version uses a technique which increments by six for testing primality (up to the √n). <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). */
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 j<99 then say right(j,20) 'is prime.' /*Just show wee primes.*/ end /*j*/
say say "There are" p "primes up to" n '(inclusive).' exit /*stick a fork in it, we're done.*/
/*──────────────────────────────────ISPRIME subroutine──────────────────*/ isprime: procedure; parse arg x /*get the number in question*/ if \datatype(x,'W') then return 0 /*X not an integer? ¬prime.*/ if wordpos(x,'2 3 5 7')\==0 then return 1 /*is number a teacher's pet?*/ if x<2 | x//2==0 | x//3==0 then return 0 /*weed out the riff-raff. */
do k=5 by 6 until k*k > x /*skips odd multiples of 3. */ if x//k==0 | x//(k+2)==0 then return 0 /*do a pair of divides (mod)*/ end /*k*/ /*Note: // is modulus. */
return 1 /*done dividing, it's prime.*/</lang> output when using the default input of 10000
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).
optimized version
This version separates multiple-clause IF statements, and also test for low primes,
making it about 8% faster.
<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). */
do j=-57 to n /*start in the cellar and work up*/ if \isprime(j) then iterate /*if not prime, then keep looking*/ p=p+1 /*bump the jelly bean counter. */ if j<99 then say right(j,20) 'is prime.' /*Just show wee primes.*/ end /*j*/
say; say "There are" p "primes up to" n '(inclusive).' exit /*stick a fork in it, we're done.*/
/*──────────────────────────────────ISPRIME subroutine──────────────────*/ isprime: procedure; parse arg x /*get integer to be investigated.*/ if \datatype(x,'W') then return 0 /*X isn't an integer? Not prime.*/ if x<11 then return wordpos(x,'2 3 5 7')\==0 /*is it a wee prime? */ if x//2==0 then return 0 /*eliminate the evens. */ if x//3==0 then return 0 /* ··· and eliminate the triples.*/
/*right dig test: faster than //.*/ do k=5 by 6 until k*k > x /*this skips odd multiples of 3. */ if x // k == 0 then return 0 /*perform a divide (modulus), */ if x // (k+2) == 0 then return 0 /* ··· and the next umpty one. */ end /*k*/ /*Note: // is modulus. */
return 1 /*did all divisions, it's prime. *//</lang> output is identical to the first version.
unrolled version
This version uses an unrolled version (of the 2nd version) of some multiple-clause IF statements, and
also an optimized version of the testing of low primes, making it about 22% faster.
Note that the DO ... UNTIL ... was changed to DO ... WHILE ....
<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). */
do j=-57 to n /*start in the cellar and work up*/ if \isprime(j) then iterate /*if not prime, then keep looking*/ p=p+1 /*bump the jelly bean counter. */ if j<99 then say right(j,20) 'is prime.' /*Just show wee primes.*/ end /*j*/
say; say "There are" p "primes up to" n '(inclusive).' exit /*stick a fork in it, we're done.*/
/*──────────────────────────────────ISPRIME subroutine──────────────────*/ isprime: procedure; parse arg x /*get integer to be investigated.*/ if \datatype(x,'W') then return 0 /*X isn't an integer? Not prime.*/ if x<107 then do /*test for (low) special cases. */
_='2 3 5 7 11 13 17 19 23 29 31 37 41 43 47', /*list of ··*/ '53 59 61 67 71 73 79 83 89 97 101 103' /*wee primes*/ return wordpos(x,_)\==0 /*is it a wee prime? ··· or not?*/ end
if x// 2 ==0 then return 0 /*eliminate the evens. */ if x// 3 ==0 then return 0 /* ··· and eliminate the triples.*/ if right(x,1) == 5 then return 0 /* ··· and eliminate the nickels.*/ if x// 7 ==0 then return 0 /* ··· and eliminate the luckies.*/ if x//11 ==0 then return 0 if x//13 ==0 then return 0 if x//17 ==0 then return 0 if x//19 ==0 then return 0 if x//23 ==0 then return 0 if x//29 ==0 then return 0 if x//31 ==0 then return 0 if x//37 ==0 then return 0 if x//41 ==0 then return 0 if x//43 ==0 then return 0 if x//47 ==0 then return 0 if x//53 ==0 then return 0 if x//59 ==0 then return 0 if x//61 ==0 then return 0 if x//67 ==0 then return 0 if x//71 ==0 then return 0 if x//73 ==0 then return 0 if x//79 ==0 then return 0 if x//83 ==0 then return 0 if x//89 ==0 then return 0 if x//97 ==0 then return 0 if x//101==0 then return 0 if x//103==0 then return 0
/*Note: // is modulus. */ do k=107 by 6 while k*k<=x /*this skips odd multiples of 3. */ if x // k == 0 then return 0 /*perform a divide (modulus), */ if x // (k+2) == 0 then return 0 /* ··· and the next umpty one. */ end /*k*/
return 1 /*after all that, ··· it's prime.*/</lang>
output is identical to the first version.
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
Simple version, robustified
<lang Scala> def isPrime(n: Int) = {
assume(n <= Int.MaxValue - 1) n > 1 && (Iterator.from(2) takeWhile (d => d * d <= n) forall (n % _ != 0)) }</lang>
Optimized version FP and parallel runabled
<lang Scala> def isPrime(n: Long) = {
n > 1 && ((n % 2 != 0) || (n == 2)) && ((3 until Math.sqrt(n).toInt by 2).par forall (n % _ != 0)) }
assert(isPrime(9223372036854775783L)) // Biggest 63-bit prime assert(!isPrime(Long.MaxValue))</lang>
Optimized version FP, robustified and tail recursion
<lang Scala> def isPrime(n: Int) = {
@tailrec def inner(n: Int, k: Int): Boolean = { if (k * k > n) true else if (n % k != 0) inner(n, k + 2) else false }
assume(n <= Int.MaxValue - 1) n > 1 && ((n % 2 != 0) || (n == 2)) && inner(n, 3)
}</lang>
- Output:
The testcases for all the Scala versions above
assert(!isPrime(9)) assert(isPrime(214748357)) assert(!isPrime(Int.MaxValue - 1))</lang>
Scheme
<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
SQL
<lang tsql> declare @number int set @number = 514229 -- number to check
- with cte(number) as
(
select 2 union all select number+1 from cte where number+1 < @number
) select
cast(@number as varchar(100)) + case when exists
( select * from ( select number, @number % number modNumber from cte ) tmp where tmp.modNumber = 0 ) then ' is composite' else ' is prime' end primalityTest option (maxrecursion 0) </lang>
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
<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>
<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
<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
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