Primality by trial division: Difference between revisions
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119 is not prime! |
119 is not prime! |
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137 is prime!</pre> |
137 is prime!</pre> |
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=={{header|BASIC256}}== |
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{{trans|FreeBASIC}} |
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<lang freebasic>for i = 1 to 99 |
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if isPrime(i) then print string(i); " "; |
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next i |
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end |
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function isPrime(v) |
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if v < 2 then return False |
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if v mod 2 = 0 then return v = 2 |
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if v mod 3 = 0 then return v = 3 |
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d = 5 |
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while d * d <= v |
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if v mod d = 0 then return False else d += 2 |
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end while |
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return True |
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end function</lang> |
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=={{header|BBC BASIC}}== |
=={{header|BBC BASIC}}== |
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Revision as of 17:44, 9 June 2022
You are encouraged to solve this task according to the task description, using any language you may know.
- Task
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 greater than 2 may be eliminated right away.
A loop from 3 to √ n will suffice, but other loops are allowed.
- Related tasks
- count in factors
- prime decomposition
- AKS test for primes
- factors of an integer
- Sieve of Eratosthenes
- factors of a Mersenne number
- trial factoring of a Mersenne number
- partition an integer X into N primes
- sequence of primes by Trial Division
11l
<lang 11l>F is_prime(n)
I n < 2
R 0B
L(i) 2..Int(sqrt(n))
I n % i == 0
R 0B
R 1B</lang>
360 Assembly
<lang 360asm>* Primality by trial division 26/03/2017 PRIMEDIV CSECT
USING PRIMEDIV,R13 base register
B 72(R15) skip savearea
DC 17F'0' savearea
STM R14,R12,12(R13) save previous context
ST R13,4(R15) link backward
ST R15,8(R13) link forward
LR R13,R15 set addressability
LA R10,PG pgi=0
LA R6,1 i=1
DO WHILE=(C,R6,LE,=F'50') do i=1 to 50
LR R1,R6 i
BAL R14,ISPRIME call isprime(i)
IF C,R0,EQ,=F'1' THEN if isprime(i) then
XDECO R6,XDEC edit i
MVC 0(4,R10),XDEC+8 output i
LA R10,4(R10) pgi+=4
ENDIF , endif
LA R6,1(R6) i++
ENDDO , enddo i
XPRNT PG,L'PG print buffer
L R13,4(0,R13) restore previous savearea pointer
LM R14,R12,12(R13) restore previous context
XR R15,R15 rc=0
BR R14 exit
- ------- ---- ----------------------------------------
ISPRIME EQU * function isprime(n)
IF C,R1,LE,=F'1' THEN if n<=1 then
LA R0,0 return(0)
BR R14 return
ENDIF , endif
IF C,R1,EQ,=F'2' THEN if n=2 then
LA R0,1 return(1)
BR R14 return
ENDIF , endif
LR R4,R1 n
N R4,=X'00000001' n and 1
IF LTR,R4,Z,R4 THEN if mod(n,2)=0 then
LA R0,0 return(0)
BR R14 return
ENDIF , endif
LA R7,3 j=3
LA R5,9 r5=j*j
DO WHILE=(CR,R5,LE,R1) do j=3 by 2 while j*j<=n
LR R4,R1 n
SRDA R4,32 ~
DR R4,R7 /j
IF LTR,R4,Z,R4 THEN if mod(n,j)=0 then
LA R0,0 return(0)
BR R14 return
ENDIF , endif
LA R7,2(R7) j+=2
LR R5,R7 j
MR R4,R7 r5=j*j
ENDDO , enddo j
LA R0,1 return(1)
BR R14 return
- ------- ---- ----------------------------------------
PG DC CL80' ' buffer XDEC DS CL12 temp for xdeco
YREGS
END PRIMEDIV</lang>
- Output:
2 3 5 7 11 13 17 19 23 29 31 37 41 43 47
68000 Assembly
<lang 68000devpac>isPrime:
- REG USAGE
- D0 = input (unsigned 32-bit integer)
- D1 = temp storage for D0
- D2 = candidates for possible factors
- D3 = temp storage for quotient/remainder
- D4 = total count of proper divisors.
MOVEM.L D1-D4,-(SP) ;push data regs except D0 MOVE.L #0,D1 MOVEM.L D1,D2-D4 ;clear regs D1 thru D4
CMP.L #0,D0 BEQ notPrime CMP.L #1,D0 BEQ notPrime CMP.L #2,D0 BEQ wasPrime
MOVE.L D0,D1 ;D1 will be used for temp storage. AND.L #1,D1 ;is D1 even? BEQ notPrime ;if so, it's not prime!
MOVE.L D0,D1 ;restore D1
MOVE.L #3,D2 ;start with 3
computeFactors: DIVU D2,D1 ;remainder is in top 2 bytes, quotient in bottom 2. MOVE.L D1,D3 ;temporarily store into D3 to check the remainder SWAP D3 ;swap the high and low words of D3. Now bottom 2 bytes contain remainder. CMP.W #0,D3 ;is the bottom word equal to 0? BNE D2_Wasnt_A_Divisor ;if not, D2 was not a factor of D1.
ADDQ.L #1,D4 ;increment the count of proper divisors. CMP.L #2,D4 ;is D4 two or greater? BCC notPrime ;if so, it's not prime! (Ends early. We'll need to check again if we made it to the end.)
D2_Wasnt_A_Divisor: MOVE.L D0,D1 ;restore D1. ADDQ.L #1,D2 ;increment D2
CMP.L D2,D1 ;is D2 now greater than D1?
BLS computeFactors ;if not, loop again
CMP.L #1,D4 ;was there only one factor? BEQ wasPrime ;if so, D0 was prime.
notPrime: MOVE.L #0,D0 ;return false MOVEM.L (SP)+,D1-D4 ;pop D1-D4 RTS
wasPrime: MOVE.L #1,D0 ;return true MOVEM.L (SP)+,D1-D4 ;pop D1-D4 RTS
- end of program</lang>
AArch64 Assembly
<lang AArch64 Assembly> /* ARM assembly AARCH64 Raspberry PI 3B */ /* program testPrime64.s */
/*******************************************/ /* Constantes file */ /*******************************************/ /* for this file see task include a file in language AArch64 assembly*/ .include "../includeConstantesARM64.inc"
/*******************************************/ /* Initialized data */ /*******************************************/ .data szMessStartPgm: .asciz "Program start \n" szMessEndPgm: .asciz "Program normal end.\n" szMessNotPrime: .asciz "Not prime.\n" szMessPrime: .asciz "Prime\n" szCarriageReturn: .asciz "\n"
/*******************************************/ /* UnInitialized data */ /*******************************************/ .bss .align 4 /*******************************************/ /* code section */ /*******************************************/ .text .global main main: // program start
ldr x0,qAdrszMessStartPgm // display start message bl affichageMess ldr x0,qVal bl isPrime // test prime ? cmp x0,#0 beq 1f ldr x0,qAdrszMessPrime // yes bl affichageMess b 2f
1:
ldr x0,qAdrszMessNotPrime // no bl affichageMess
2:
ldr x0,qAdrszMessEndPgm // display end message bl affichageMess
100: // standard end of the program
mov x0,0 // return code mov x8,EXIT // request to exit program svc 0 // perform system call
qAdrszMessStartPgm: .quad szMessStartPgm qAdrszMessEndPgm: .quad szMessEndPgm qAdrszCarriageReturn: .quad szCarriageReturn qAdrszMessNotPrime: .quad szMessNotPrime qAdrszMessPrime: .quad szMessPrime //qVal: .quad 1042441 // test not prime //qVal: .quad 1046527 // test prime //qVal: .quad 37811 // test prime //qVal: .quad 1429671721 // test not prime (37811 * 37811) qVal: .quad 100000004437 // test prime /******************************************************************/ /* test if number is prime */ /******************************************************************/ /* x0 contains the number */ /* x0 return 1 if prime else return 0 */ isPrime:
stp x1,lr,[sp,-16]! // save registers cmp x0,1 // <= 1 ? ble 98f cmp x0,3 // 2 and 3 prime ble 97f tst x0,1 // even ? beq 98f mov x11,3 // first divisor
1:
udiv x12,x0,x11 msub x13,x12,x11,x0 // compute remainder cbz x13,98f // end if zero add x11,x11,#2 // increment divisor cmp x11,x12 // divisors<=quotient ? ble 1b // loop
97:
mov x0,1 // return prime b 100f
98:
mov x0,0 // not prime b 100f
100:
ldp x1,lr,[sp],16 // restaur 2 registers ret // return to address lr x30
/********************************************************/ /* File Include fonctions */ /********************************************************/ /* for this file see task include a file in language AArch64 assembly */ .include "../includeARM64.inc" </lang>
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>
Action!
<lang Action!>BYTE FUNC IsPrime(CARD a)
CARD i
IF a<=1 THEN
RETURN (0)
FI
FOR i=2 TO a/2
DO
IF a MOD i=0 THEN
RETURN (0)
FI
OD
RETURN (1)
PROC Test(CARD a)
IF IsPrime(a) THEN
PrintF("%I is prime%E",a)
ELSE
PrintF("%I is not prime%E",a)
FI
RETURN
PROC Main()
Test(13) Test(997) Test(1) Test(6) Test(120) Test(0)
RETURN</lang>
- Output:
Screenshot from Atari 8-bit computer
13 is prime 997 is prime 1 is not prime 6 is not prime 120 is not prime 0 is not prime
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
Test : Natural;
begin
if Item = 1 then
return False;
elsif Item = 2 then
return True;
elsif Item mod 2 = 0 then
return False;
else
Test := 3;
while Test <= Integer(Sqrt(Float(Item))) loop
if Item mod Test = 0 then
return False;
end if;
Test := Test + 2;
end loop;
end if;
return True;
end Is_Prime;</lang>
Sqrt is made visible by a with / use clause on Ada.Numerics.Elementary_Functions.
With Ada 2012, the function can be made more compact as an expression function (but without loop optimized by skipping even numbers) : <lang ada>function Is_Prime(Item : Positive) return Boolean is
(Item /= 1 and then (for all Test in 2..Integer(Sqrt(Float(Item))) => Item mod Test /= 0));</lang>
As an alternative, one can use the generic function Prime_Numbers.Is_Prime, as specified in Prime decomposition#Ada, which also implements trial division. <lang Ada>with Prime_Numbers;
procedure Test_Prime is
package Integer_Numbers is new
Prime_Numbers (Natural, 0, 1, 2);
use Integer_Numbers;
begin
if Is_Prime(12) or (not Is_Prime(13)) then
raise Program_Error with "Test_Prime failed!";
end if;
end Test_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
ALGOL-M
<lang algol> BEGIN
% RETURN P MOD Q % INTEGER FUNCTION MOD(P, Q); INTEGER P, Q; BEGIN
MOD := P - Q * (P / Q);
END;
% RETURN INTEGER SQUARE ROOT OF N % INTEGER FUNCTION ISQRT(N); INTEGER N; BEGIN
INTEGER R1, R2;
R1 := N;
R2 := 1;
WHILE R1 > R2 DO
BEGIN
R1 := (R1+R2) / 2;
R2 := N / R1;
END;
ISQRT := R1;
END;
% RETURN 1 IF N IS PRIME, OTHERWISE 0 % INTEGER FUNCTION ISPRIME(N); INTEGER N; BEGIN
IF N = 2 THEN
ISPRIME := 1
ELSE IF (N < 2) OR (MOD(N,2) = 0) THEN
ISPRIME := 0
ELSE % TEST ODD NUMBERS UP TO SQRT OF N %
BEGIN
INTEGER I, LIMIT;
LIMIT := ISQRT(N);
I := 3;
WHILE I <= LIMIT AND MOD(N,I) <> 0 DO
I := I + 2;
ISPRIME := (IF I <= LIMIT THEN 0 ELSE 1);
END;
END;
% TEST FOR PRIMES IN RANGE 1 TO 50 % INTEGER I; WRITE(""); FOR I := 1 STEP 1 UNTIL 50 DO
BEGIN IF ISPRIME(I)=1 THEN WRITEON(I," "); % WORKS FOR 80 COL SCREEN % END;
END </lang>
- Output:
2 3 5 7 11 13 17 19 23 29
31 37 41 43 47
ALGOL W
<lang algolw>% returns true if n is prime, false otherwise % % uses trial division % logical procedure isPrime ( integer value n ) ;
if n < 3 or not odd( n ) then n = 2
else begin
% odd number > 2 %
integer f, rootN;
logical havePrime;
f := 3;
rootN := entier( sqrt( n ) );
havePrime := true;
while f <= rootN and havePrime do begin
havePrime := ( n rem f ) not = 0;
f := f + 2
end;
havePrime
end isPrime ;</lang>
Test program: <lang algolw>begin
logical procedure isPrime ( integer value n ) ; algol "isPrime" ; for i := 0 until 32 do if isPrime( i ) then writeon( i_w := 1,s_w := 1, i )
end.</lang>
- Output:
2 3 5 7 11 13 17 19 23 29 31
AppleScript
<lang applescript>on isPrime(n)
if (n < 3) then return (n is 2)
if (n mod 2 is 0) then return false
repeat with i from 3 to (n ^ 0.5) div 1 by 2
if (n mod i is 0) then return false
end repeat
return true
end isPrime
-- Test code: set output to {} repeat with n from -7 to 100
if (isPrime(n)) then set end of output to n
end repeat return output</lang>
Or eliminating multiples of 3 at the start as well as those of 2:
<lang applescript>on isPrime(n)
if (n < 4) then return (n > 1)
if ((n mod 2 is 0) or (n mod 3 is 0)) then return false
repeat with i from 5 to (n ^ 0.5) div 1 by 6
if ((n mod i is 0) or (n mod (i + 2) is 0)) then return false
end repeat
return true
end isPrime
-- Test code: set output to {} repeat with n from -7 to 100
if (isPrime(n)) then set end of output to n
end repeat return output</lang>
- Output:
<lang applescript>{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>
Arturo
<lang rebol>isPrime?: function [n][
if n=2 -> return true if n=3 -> return true if or? n=<1 0=n%2 -> return false high: to :integer sqrt n loop high..2 .step: 3 'i [ if 0=n%i -> return false ]
return true
]
loop 1..20 'i [
print ["isPrime?" i "=" isPrime? i ]
]</lang>
- Output:
isPrime? 1 = false isPrime? 2 = true isPrime? 3 = true isPrime? 4 = false isPrime? 5 = true isPrime? 6 = false isPrime? 7 = true isPrime? 8 = false isPrime? 9 = false isPrime? 10 = false isPrime? 11 = true isPrime? 12 = false isPrime? 13 = true isPrime? 14 = false isPrime? 15 = false isPrime? 16 = false isPrime? 17 = true isPrime? 18 = false isPrime? 19 = true isPrime? 20 = false
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>
B
B as on PDP7/UNIX0
<lang B>isprime(n) {
auto p;
if(n<2) return(0);
if(!(n%2)) return(n==2);
p=3;
while(n/p>p) {
if(!(n%p)) return(0);
p=p+2;
}
return(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
IS-BASIC
<lang IS-BASIC>100 PROGRAM "Prime.bas" 110 FOR X=0 TO 100 120 IF PRIME(X) THEN PRINT X; 130 NEXT 140 DEF PRIME(N) 150 IF N=2 THEN 160 LET PRIME=-1 170 ELSE IF N<=1 OR MOD(N,2)=0 THEN 180 LET PRIME=0 190 ELSE 200 LET PRIME=-1 210 FOR I=3 TO CEIL(SQR(N)) STEP 2 220 IF MOD(N,I)=0 THEN LET PRIME=0:EXIT FOR 230 NEXT 240 END IF 250 END DEF</lang>
True BASIC
<lang qbasic>FUNCTION isPrime (n)
IF n = 2 THEN
LET isPrime = 1
ELSEIF n <= 1 OR REMAINDER(n, 2) = 0 THEN
LET isPrime = 0
ELSE
LET isPrime = 1
FOR i = 3 TO INT(SQR(n)) STEP 2
IF REMAINDER(n, i) = 0 THEN
LET isPrime = 0
EXIT FUNCTION
END IF
NEXT i
END IF
END FUNCTION
FOR n = 1 TO 50
IF isPrime(n) = 1 THEN PRINT n;
NEXT n END</lang>
ZX Spectrum Basic
<lang ZXBasic>10 LET n=0: LET p=0 20 INPUT "Enter number: ";n 30 IF n>1 THEN 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!
BASIC256
<lang freebasic>for i = 1 to 99
if isPrime(i) then print string(i); " ";
next i end
function isPrime(v)
if v < 2 then return False
if v mod 2 = 0 then return v = 2
if v mod 3 = 0 then return v = 3
d = 5
while d * d <= v
if v mod d = 0 then return False else d += 2
end while
return True
end function</lang>
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>
BCPL
<lang bcpl>get "libhdr"
let sqrt(s) =
s <= 1 -> 1, valof
$( let x0 = s >> 1
let x1 = (x0 + s/x0) >> 1
while x1 < x0
$( x0 := x1
x1 := (x0 + s/x0) >> 1
$)
resultis x0
$)
let isprime(n) =
n < 2 -> false, (n & 1) = 0 -> n = 2, valof
$( for i = 3 to sqrt(n) by 2
if n rem i = 0 resultis false resultis true
$)
let start() be $( for i=1 to 100
if isprime(i) then writef("%N ",i)
wrch('*N')
$)</lang>
- Output:
2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 73 79 83 89 97
Befunge
Reads the value to test from stdin and outputs Yes if prime and No if not.
To avoid dealing with Befunge's limited data cells, the implementation is entirely stack-based. However, this requires compressing multiple values into a single stack cell, which imposes an upper limit of 1,046,529 (10232), thus a maximum testable prime of 1,046,527.
<lang befunge>&>:48*:** \1`!#^_2v v_v#`\*:%*:*84\/*:*84::+< v >::48*:*/\48*:*%%!#v_1^ >0"seY" >:#,_@#: "No">#0<</lang>
- Output:
(multiple runs)
0 No 17 Yes 49 No 97 Yes 1042441 No 1046527 Yes
Bracmat
<lang bracmat> ( prime
= incs n I inc
. 4 2 4 2 4 6 2 6:?incs
& 2:?n
& 1 2 2 !incs:?I
& whl
' ( !n*!n:~>!arg
& div$(!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
Brainf***
<lang bf>>->,[.>,]>-<++++++[-<+[---------<+]->+[->+]-<]>+<-<+[-<+]>>+[-<[->++++++++++<]>> +]++++[->++++++++<]>.<+++++++[->++++++++++<]>+++.++++++++++.<+++++++++[->------- --<]>--.[-]<<<->[->+>+<<]>>-[+<[[->>+>>+<<<<]>>[-<<+>>]<]>>[->-[>+>>]>[+[-<+>]>> >]<<<<<]>[-]>[>+>]<<[-]+[-<+]->>>--]<[->+>+<<]>>>>>>>[-<<<<<->>>>>]<<<<<--[>++++ ++++++[->+++++++++++<]>.+.+++++.>++++[->++++++++<]>.>]++++++++++[->+++++++++++<] >++.++.---------.++++.--------.>++++++++++.</lang> Explanation: <lang bf>> ->,[.>,]>-<++++++[-<+[---------<+]->+[->+]-<]>+<-<+[-<+]>>+[-<[->++++++++++<]>>+]< takes input >++++[->++++++++<]>.<+++++++[->++++++++++<]>+++.++++++++++.<+++++++++[->---------<]>--.[-]<< " is " <-> [->+>+<<]>>-[+<[[->>+>>+<<<<]>>[-<<+>>]<]>>[->-[>+>>]>[+[-<+>]>>>]<<<<<]>[-]>[>+>]<<[-]+[-<+]->>>--] finds # of divisors from 1 to n <[->+>+<<]>>>>>>>[-<<<<<->>>>>]<<<<<-- [>++++++++++[->+++++++++++<]>.+.+++++.>++++[->++++++++<]>.>] "not " ++++++++++[->+++++++++++<]>++.++.---------.++++.--------.>++++++++++. "prime" new line</lang> Will format as "# is/is not prime", naturally limited by cell size.
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 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>
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>
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 function used in both versions: <lang clojure>(defn divides? [k n] (zero? (mod k n)))</lang>
Testing divisors are in range from 3 to √ n with step 2: <lang clojure>(defn prime? [x]
(or (= 2 x)
(and (< 1 x)
(odd? x)
(not-any? (partial divides? x)
(range 3 (inc (Math/sqrt x)) 2)))))
</lang>
Testing only prime divisors: <lang clojure>(declare prime?)
(def primes (filter prime? (range)))
(defn prime? [x]
(and (integer? x)
(< 1 x)
(not-any? (partial divides? x)
(take-while (partial >= (Math/sqrt x)) primes))))
</lang>
CLU
<lang clu>isqrt = proc (s: int) returns (int)
x0: int := s/2
if x0=0 then return(s) end
x1: int := (x0 + s/x0)/2
while x1 < x0 do
x0 := x1
x1 := (x0 + s/x0)/2
end
return(x0)
end isqrt
prime = proc (n: int) returns (bool)
if n<=2 then return(n=2) end
if n//2=0 then return(false) end
for d: int in int$from_to_by(3,isqrt(n),2) do
if n//d=0 then return(false) end
end
return(true)
end prime
start_up = proc ()
po: stream := stream$primary_input()
for i: int in int$from_to(1,100) do
if prime(i) then
stream$puts(po, int$unparse(i) || " ")
end
end
end start_up</lang>
- Output:
2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 73 79 83 89 97
CMake
<lang cmake># Prime predicate: does n be a prime number? Sets var to true or false. function(primep var n)
if(n GREATER 2)
math(EXPR odd "${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 (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>
Alternate solution
I use Allegro CL 10.1
<lang lisp>;; Project : Primality by trial division
(defun prime(n)
(setq flag 0)
(loop for i from 2 to (- n 1) do
(if (= (mod n i) 0)
(setq flag 1)))
(if (= flag 0)
(format t "~d is a prime number" n)
(format t "~d is not a prime number" n)))
(prime 7) (prime 8)</lang> Output:
7 is a prime number 8 is not a prime number
Crystal
Mathematicaly basis of Prime Generators https://www.academia.edu/19786419/PRIMES-UTILS_HANDBOOK https://www.academia.edu/42734410/Improved_Primality_Testing_and_Factorization_in_Ruby_revised
<lang ruby>require "big" require "benchmark"
- the simplest PG primality test using the P3 prime generator
- reduces the number space for primes to 2/6 (33.33%) of all integers
def primep3?(n) # P3 Prime Generator primality test
# P3 = 6*k + {5, 7} # P3 primes candidates (pc) sequence
n = n.to_big_i
return n | 1 == 3 if n < 5 # n: 0,1,4|false, 2,3|true
return false if n.gcd(6) != 1 # 1/3 (2/6) of integers are P3 pc
p = typeof(n).new(5) # first P3 sequence value
until p > isqrt(n)
return false if n % p == 0 || n % (p + 2) == 0 # if n is composite
p += 6 # first prime candidate for next kth residues group
end
true
end
- PG primality test using the P5 prime generator
- reduces the number space for primes to 8/30 (26.67%) of all integers
def primep5?(n) # P5 Prime Generator primality test
# P5 = 30*k + {7,11,13,17,19,23,29,31} # P5 primes candidates sequence
n = n.to_big_i
return [2, 3, 5].includes?(n) if n < 7 # for small and negative values
return false if n.gcd(30) != 1 # 4/15 (8/30) of integers are P5 pc
p = typeof(n).new(7) # first P5 sequence value
until p > isqrt(n)
return false if # if n is composite
n % (p) == 0 || n % (p+4) == 0 || n % (p+6) == 0 || n % (p+10) == 0 ||
n % (p+12) == 0 || n % (p+16) == 0 || n % (p+22) == 0 || n % (p+24) == 0
p += 30 # first prime candidate for next kth residues group
end
true
end
- PG primality test using the P7 prime generator
- reduces the number space for primes to 48/210 (22.86%) of all integers
def primep7?(n)
# P7 = 210*k + {11,13,17,19,23,29,31,37,41,43,47,53,59,61,67,71,73,79,83,
# 89,97,101,103,107,109,113,121,127,131,137,139,143,149,151,157,163,
# 167,169,173,179,181,187,191,193,197,199,209,211}
n = n.to_big_i
return [2, 3, 5, 7].includes?(n) if n < 11
return false if n.gcd(210) != 1 # 8/35 (48/210) of integers are P7 pc
p = typeof(n).new(11) # first P7 sequence value
until p > isqrt(n)
return false if
n % (p) == 0 || n % (p+2) == 0 || n % (p+6) == 0 || n % (p+8) == 0 ||
n % (p+12) == 0 || n % (p+18) == 0 || n % (p+20) == 0 || n % (p+26) == 0 ||
n % (p+30) == 0 || n % (p+32) == 0 || n % (p+36) == 0 || n % (p+42) == 0 ||
n % (p+48) == 0 || n % (p+50) == 0 || n % (p+56) == 0 || n % (p+60) == 0 ||
n % (p+62) == 0 || n % (p+68) == 0 || n % (p+72) == 0 || n % (p+78) == 0 ||
n % (p+86) == 0 || n % (p+90) == 0 || n % (p+92) == 0 || n % (p+96) == 0 ||
n % (p+98) == 0 || n % (p+102) == 0 || n % (p+110) == 0 || n % (p+116) == 0 ||
n % (p+120) == 0 || n % (p+126) == 0 || n % (p+128) == 0 || n % (p+132) == 0 ||
n % (p+138) == 0 || n % (p+140) == 0 || n % (p+146) == 0 || n % (p+152) == 0 ||
n % (p+156) == 0 || n % (p+158) == 0 || n % (p+162) == 0 || n % (p+168) == 0 ||
n % (p+170) == 0 || n % (p+176) == 0 || n % (p+180) == 0 || n % (p+182) == 0 ||
n % (p+186) == 0 || n % (p+188) == 0 || n % (p+198) == 0 || n % (p+200) == 0
p += 210 # first prime candidate for next kth residues group
end
true
end
- Newton's method for integer squareroot
def isqrt(n)
raise ArgumentError.new "Input must be non-negative integer" if n < 0 return n if n < 2 bits = n.bit_length one = typeof(n).new(1) # value 1 of type self x = one << ((bits - 1) >> 1) | n >> ((bits >> 1) + 1) while (t = n // x) < x; x = (x + t) >> 1 end x # output is same integer class as input
end
- Benchmarks to test for various size primes
p = 541 Benchmark.ips do |b|
print "\np = #{p}"
b.report("primep3?") { primep3?(p) }
b.report("primep5?") { primep5?(p) }
b.report("primep7?") { primep7?(p) }
puts
end
p = 1000003 Benchmark.ips do |b|
print "\np = #{p}"
b.report("primep3?") { primep3?(p) }
b.report("primep5?") { primep5?(p) }
b.report("primep7?") { primep7?(p) }
puts
end
p = 2147483647i32 # largest I32 prime Benchmark.ips do |b|
print "\np = #{p}"
b.report("primep3?") { primep3?(p) }
b.report("primep5?") { primep5?(p) }
b.report("primep7?") { primep7?(p) }
puts
end
p = 4294967291u32 # largest U32 prime Benchmark.ips do |b|
print "\np = #{p}"
b.report("primep3?") { primep3?(p) }
b.report("primep5?") { primep5?(p) }
b.report("primep7?") { primep7?(p) }
puts
end
p = 4294967311 # first prime > 2**32 Benchmark.ips do |b|
print "\np = #{p}"
b.report("primep3?") { primep3?(p) }
b.report("primep5?") { primep5?(p) }
b.report("primep7?") { primep7?(p) }
puts
end</lang>
- Output:
p = 541 primep3? 290.17k ( 3.45µs) (± 2.76%) 1.35kB/op 1.64× slower primep5? 476.47k ( 2.10µs) (± 1.75%) 802B/op fastest primep7? 128.44k ( 7.79µs) (± 2.82%) 2.66kB/op 3.71× slower p = 1000003 primep3? 11.24k ( 88.97µs) (± 2.34%) 33.9kB/op 2.48× slower primep5? 21.91k ( 45.64µs) (± 2.88%) 16.6kB/op 1.27× slower primep7? 27.83k ( 35.94µs) (± 2.68%) 11.9kB/op fastest p = 2147483647 primep3? 105.11 ( 9.51ms) (± 3.25%) 3.89MB/op 5.56× slower primep5? 317.49 ( 3.15ms) (± 2.40%) 1.2MB/op 1.84× slower primep7? 584.92 ( 1.71ms) (± 3.09%) 591kB/op fastest p = 4294967291 primep3? 168.56 ( 5.93ms) (± 2.39%) 2.17MB/op 2.69× slower primep5? 349.24 ( 2.86ms) (± 2.86%) 1.03MB/op 1.30× slower primep7? 454.08 ( 2.20ms) (± 2.62%) 739kB/op fastest p = 4294967311 primep3? 84.61 ( 11.82ms) (± 2.35%) 4.68MB/op 5.02× slower primep5? 248.62 ( 4.02ms) (± 2.21%) 1.54MB/op 1.71× slower primep7? 424.61 ( 2.36ms) (± 2.73%) 813kB/op fastest
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>
EchoLisp
<lang scheme>(lib 'sequences)
- Try divisors iff n = 2k + 1
(define (is-prime? p) (cond [(< p 2) #f] [(zero? (modulo p 2)) (= p 2)] [else (for/and ((d [3 5 .. (1+ (sqrt p))] )) (!zero? (modulo p d)))]))
(filter is-prime? (range 1 100))
→ (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)
- Improve performance , try divisors iff n = 6k+1 or n = 6k+5
(define (is-prime? p) (cond [(< p 2) #f] [(zero? (modulo p 2)) (= p 2)] [(zero? (modulo p 3)) (= p 3)] [(zero? (modulo p 5)) (= p 5)] [else ;; step 6 : try divisors 6n+1 or 6n+5 (for ((d [7 13 .. (1+ (sqrt p))] )) #:break (zero? (modulo p d)) => #f #:break (zero? (modulo p (+ d 4))) => #f #t )]))
(filter is-prime? (range 1 100))
→ (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>
Eiffel
<lang Eiffel>class APPLICATION
create make
feature
make
-- Tests the feature is_prime.
do io.put_boolean (is_prime (1)) io.new_line io.put_boolean (is_prime (2)) io.new_line io.put_boolean (is_prime (3)) io.new_line io.put_boolean (is_prime (4)) io.new_line io.put_boolean (is_prime (97)) io.new_line io.put_boolean (is_prime (15589)) io.new_line end
is_prime (n: INTEGER): BOOLEAN
-- Is 'n' a prime number?
require positiv_input: n > 0 local i: INTEGER max: REAL_64 math: DOUBLE_MATH do create math if n = 2 then Result := True elseif n <= 1 or n \\ 2 = 0 then Result := False else Result := True max := math.sqrt (n) from i := 3 until i > max loop if n \\ i = 0 then Result := False end i := i + 2 end end end
end</lang>
False True True False True False
Elixir
<lang elixir>defmodule RC do
def is_prime(2), do: true def is_prime(n) when n<2 or rem(n,2)==0, do: false def is_prime(n), do: is_prime(n,3) def is_prime(n,k) when n<k*k, do: true def is_prime(n,k) when rem(n,k)==0, do: false def is_prime(n,k), do: is_prime(n,k+2)
end
IO.inspect for n <- 1..50, RC.is_prime(n), do: n</lang>
- Output:
[2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47]
Emacs Lisp
<lang lisp>(defun prime (a)
(not (or (< a 2)
(cl-loop for x from 2 to (sqrt a)
when (zerop (% a x))
return t))))</lang>
More concise, a little bit faster: <lang lisp>(defun prime2 (a)
(and (> a 1)
(cl-loop for x from 2 to (sqrt a)
never (zerop (% a x)))))</lang>
A little bit faster: <lang lisp>(defun prime3 (a)
(and (> a 1)
(or (= a 2) (cl-oddp a))
(cl-loop for x from 3 to (sqrt a) by 2
never (zerop (% a x)))))</lang>
More than 2 times faster, than the previous, doesn't use loop macro: <lang lisp>(defun prime4 (a)
(not (or (< a 2)
(cl-some (lambda (x) (zerop (% a x))) (number-sequence 2 (sqrt a))))))</lang>
Almost 2 times faster, than the previous: <lang lisp>(defun prime5 (a)
(not (or (< a 2)
(and (/= a 2) (cl-evenp a))
(cl-some (lambda (x) (zerop (% a x))) (number-sequence 3 (sqrt a) 2)))))</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>
ERRE
<lang ERRE>PROGRAM PRIME_TRIAL
PROCEDURE ISPRIME(N%->OK%)
LOCAL T%
IF N%<=1 THEN OK%=FALSE EXIT PROCEDURE END IF
IF N%<=3 THEN OK%=TRUE EXIT PROCEDURE END IF
IF (N% AND 1)=0 THEN OK%=FALSE EXIT PROCEDURE END IF
FOR T%=3 TO SQR(N%) STEP 2 DO
IF N% MOD T%=0 THEN OK%=FALSE EXIT PROCEDURE END IF
END FOR
OK%=TRUE
END PROCEDURE
BEGIN
FOR I%=1 TO 100 DO
ISPRIME(I%->OK%)
IF OK% THEN PRINT(i%;"is prime") END IF
END FOR
END PROGRAM</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
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>
F#
<lang fsharp>open NUnit.Framework open FsUnit let inline isPrime n = not ({uint64 2..uint64 (sqrt (double n))} |> Seq.exists (fun (i:uint64) -> uint64 n % i = uint64 0)) [<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>
- Output:
> isPrime 1111111111111111111UL;; val it : bool = true
and if you want to test really big numbers, use System.Numerics.BigInteger, but it's slower: <lang fsharp> let isPrimeI x =
if x < 2I then false else
if x = 2I then true else
if x % 2I = 0I then false else
let rec test n =
if n * n > x then true else
if x % n = 0I then false else test (n + 2I) in test 3I</lang>
If you have a lot of prime numbers to test, caching a sequence of primes can speed things up considerably, so you only have to do the divisions against prime numbers. <lang fsharp>let rec primes = Seq.cache(Seq.append (seq[ 2; 3; 5 ]) (Seq.unfold (fun state -> Some(state, state + 2)) 7 |> Seq.filter (fun x -> IsPrime x)))
and IsPrime number =
let rec IsPrimeCore number current limit =
let cprime = primes |> Seq.nth current
if cprime >= limit then
true
else if number % cprime = 0 then
false
else
IsPrimeCore number (current + 1) (number/cprime)
if number = 2 then
true
else if number < 2 then
false
else
IsPrimeCore number 0 number</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 -- f )
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>
FreeBASIC
<lang freebasic>' FB 1.05.0 Win64
Function isPrime(n As Integer) As Boolean
If n < 2 Then Return False If n = 2 Then Return True If n Mod 2 = 0 Then Return False Dim limit As Integer = Sqr(n) For i As Integer = 3 To limit Step 2 If n Mod i = 0 Then Return False Next Return True
End Function
' To test this works, print all primes under 100 For i As Integer = 1 To 99
If isPrime(i) Then Print Str(i); " "; End If
Next
Print : Print Print "Press any key to quit" Sleep</lang>
- Output:
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
Frink
It is unnecessary to write this function because Frink has an efficient isPrime[x] function to test primality of arbitrarily-large integers. Here is a version that works for arbitrarily-large integers. Beware some of these solutions that calculate up to sqrt[x] but because of floating-point error the square root is slightly smaller than the true square root!
<lang frink>isPrimeByTrialDivision[x] :=
{
for p = primes[]
{
if p*p > x
return true
if x mod p == 0
return false
}
return true
}</lang>
FunL
<lang funl>import math.sqrt
def
isPrime( 2 ) = true isPrime( n ) | n < 3 or 2|n = false | otherwise = (3..int(sqrt(n)) by 2).forall( (/|n) )
(10^10..10^10+50).filter( isPrime ).foreach( println )</lang>
- Output:
10000000019 10000000033
FutureBasic
<lang futurebasic>include "ConsoleWindow"
def tab 6
local fn isPrime( n as long ) as Boolean dim as long i dim as Boolean result
if n < 2 then result = _false : exit fn if n = 2 then result = _true : exit fn if n mod 2 == 0 then result = _false : exit fn result = _true for i = 3 to int( n^.5 ) step 2
if n mod i == 0 then result = _false : exit fn
next i end fn = result
dim as long i, j
print "Prime numbers between 0 and 1000:" for i = 0 to 1000
if ( fn isPrime(i) != _false ) print i, : j++ if j mod 10 == 0 then print end if
next</lang> Output:
Prime numbers between 0 and 1000: 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 73 79 83 89 97 101 103 107 109 113 127 131 137 139 149 151 157 163 167 173 179 181 191 193 197 199 211 223 227 229 233 239 241 251 257 263 269 271 277 281 283 293 307 311 313 317 331 337 347 349 353 359 367 373 379 383 389 397 401 409 419 421 431 433 439 443 449 457 461 463 467 479 487 491 499 503 509 521 523 541 547 557 563 569 571 577 587 593 599 601 607 613 617 619 631 641 643 647 653 659 661 673 677 683 691 701 709 719 727 733 739 743 751 757 761 769 773 787 797 809 811 821 823 827 829 839 853 857 859 863 877 881 883 887 907 911 919 929 937 941 947 953 967 971 977 983 991 997
Gambas
Click this link to run this code <lang gambas>'Reworked from the BBC Basic example
Public Sub Main() Dim iNum As Integer
For iNum = 1 To 100
If FNisprime(iNum) Then Print iNum & " is prime"
Next
End '___________________________________________________ Public Sub FNisprime(iNum As Integer) As Boolean Dim iLoop As Integer Dim bReturn As Boolean = True
If iNum <= 1 Then bReturn = False If iNum <= 3 Then bReturn = True If (iNum And 1) = 0 Then bReturn = False
For iLoop = 3 To Sqr(iNum) Step 2
If iNum Mod iLoop = 0 Then bReturn = False
Next
Return bReturn
End</lang> Output:
1 is prime 3 is prime 5 is prime 7 is prime 11 is prime ...... 73 is prime 79 is prime 83 is prime 89 is prime 97 is prime
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(n int) bool { if n < 0 { n = -n } switch {
case n == 2:
return true case n < 2 || n % 2 == 0: return false
default: for i = 3; i*i <= n; i += 2 { if n % i == 0 { return false } } } return true }</lang>
Or, using recursion:
<lang go>func IsPrime(n int) bool { if n < 0 { n = -n } if n <= 2 { return n == 2 } return n % 2 != 0 && isPrime_r(n, 3) }
func isPrime_r(n, i int) bool { if i*i <= n { return n % i != 0 && isPrime_r(n, i+2) } 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
(used here and here). 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, and/or primes are generated efficiently enough: <lang haskell>noDivsBy factors n = foldr (\f r-> f*f>n || ((rem n f /= 0) && r)) True factors
-- primeNums = filter (noDivsBy [2..]) [2..] -- = 2 : filter (noDivsBy [3,5..]) [3,5..] primeNums = 2 : 3 : filter (noDivsBy $ tail primeNums) [5,7..]
isPrime n = n > 1 && noDivsBy primeNums n</lang>
Any increasing unbounded sequence of numbers that includes all primes (above the first few, perhaps) could be used with the testing function noDivsBy to define the isPrime function -- but using just primes is better, produced e.g. by Sieve of Eratosthenes, or noDivsBy itself can be used to define primeNums as shown above, because it stops when the square root is reached (so there's no infinite recursion, no "vicious circle").
A less efficient, more basic variant: <lang haskell>isPrime n = n > 1 && []==[i | i <- [2..n-1], rem n i == 0]</lang>
The following is an attempt at improving it, resulting in absolutely worst performing prime testing code I've ever seen, ever. A curious oddity. <lang haskell>isPrime n = n > 1 && []==[i | i <- [2..n-1], isPrime i && rem n i == 0]</lang>
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 || n == 3 || n == 5 || n == 7) {
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>
jq
def is_prime:
if . == 2 then true
else 2 < . and . % 2 == 1 and
. as $in
| (($in + 1) | sqrt) as $m
| (((($m - 1) / 2) | floor) + 1) as $max
| all( range(1; $max) ; $in % ((2 * .) + 1) > 0 )
end;
Example -- the command line is followed by alternating lines of input and output: $ jq -f is_prime.jq
-2 false 1 false 2 true 100 false
Note: if your jq does not have all, the following will suffice:
def all(generator; condition): reduce generator as $i (true; . and ($i|condition));
Julia
Julia already has an isprime function, so this function has the verbose name isprime_trialdivision to avoid overriding the built-in function. Note this function relies on the fact that Julia skips for-loops having invalid ranges. Otherwise the function would have to include additional logic to check the odd numbers less than 9. <lang Julia>function isprime_trialdivision{T<:Integer}(n::T)
1 < n || return false
n != 2 || return true
isodd(n) || return false
for i in 3:isqrt(n)
n%i != 0 || return false
end
return true
end
n = 100 a = filter(isprime_trialdivision, [1:n])
if all(a .== primes(n))
println("The primes <= ", n, " are:\n ", a)
else
println("The function does not accurately calculate primes.")
end</lang>
- Output:
The primes <= 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]
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>
Kotlin
<lang scala>// version 1.1.2 fun isPrime(n: Int): Boolean {
if (n < 2) return false
if (n % 2 == 0) return n == 2
val limit = Math.sqrt(n.toDouble()).toInt()
return (3..limit step 2).none { n % it == 0 }
}
fun main(args: Array<String>) {
// test by printing all primes below 100 say
(2..99).filter { isPrime(it) }.forEach { print("$it ") }
}</lang>
- Output:
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
langur
Recursive
<lang langur>val .isPrime = f(.i) {
val .n = abs(.i) if .n <= 2: return .n == 2
val .chkdiv = f(.n, .i) {
if .i x .i <= .n {
return .n ndiv .i and self(.n, .i+2)
}
return true
}
return .n ndiv 2 and .chkdiv(.n, 3)
}
writeln where .isPrime, series 100</lang>
Functional
following the Raku example, which states, "Integer $i is prime if it is greater than one and is divisible by none of 2, 3, up to the square root of $i" (plus an adjustment for the prime number 2)
Below, we use an implied parameter (.i) on the .isPrime function.
<lang langur>val .isPrime = f .i == 2 or
.i > 2 and not any f(.x) .i div .x, pseries 2 to .i ^/ 2
writeln where .isPrime, series 100</lang>
- Output:
[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]
Liberty BASIC
<lang lb>print "Rosetta Code - Primality by trial division" print [start] input "Enter an integer: "; x if x=0 then print "Program complete.": end if isPrime(x) then print x; " is prime" else print x; " is not prime" goto [start]
function isPrime(p)
p=int(abs(p))
if p=2 or then isPrime=1: exit function 'prime
if p=0 or p=1 or (p mod 2)=0 then exit function 'not prime
for i=3 to sqr(p) step 2
if (p mod i)=0 then exit function 'not prime
next i
isPrime=1
end function</lang>
- Output:
Rosetta Code - Primality by trial division Enter an integer: 1 1 is not prime Enter an integer: 2 2 is prime Enter an integer: Program complete.
Lingo
<lang Lingo>on isPrime (n)
if n<=1 or (n>2 and n mod 2=0) then return FALSE
sq = sqrt(n)
repeat with i = 3 to sq
if n mod i = 0 then return FALSE
end repeat
return TRUE
end</lang> <lang Lingo>primes = [] repeat with i = 0 to 100
if isPrime(i) then primes.add(i)
end repeat put primes</lang>
- Output:
-- [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]
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> Type of number Decimal.
M2000 Interpreter
<lang M2000 Interpreter>
Inventory Known1=2@, 3@
IsPrime=lambda Known1 (x as decimal) -> {
=0=1
if exist(Known1, x) then =1=1 : exit
if x<=5 OR frac(x) then {if x == 2 OR x == 3 OR x == 5 then Append Known1, x : =1=1
Break}
if frac(x/2) else exit
if frac(x/3) else exit
x1=sqrt(x):d = 5@
{if frac(x/d ) else exit
d += 2: if d>x1 then Append Known1, x : =1=1 : exit
if frac(x/d) else exit
d += 4: if d<= x1 else Append Known1, x : =1=1: exit
loop}
}
i=2
While Len(Known1)<20 {
dummy=Isprime(i)
i++
}
Print "first ";len(known1);" primes"
Print Known1
Print "From 110 to 130"
count=0
For i=110 to 130 {
If isPrime(i) Then Print i, : count++
}
Print
Print "Found ";count;" primes"
</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/Wolfram Language
<lang mathematica>IsPrime[n_Integer] := Block[{},
If[n <= 1, Return[False]]; If[n == 2, Return[True]]; If[Mod[n, 2] == 0, Return[False]]; For[k = 3, k <= Sqrt[n], k += 2, If[Mod[n, k] == 0, Return[False]]]; 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>
min
<lang min>(
:n 3 :i n sqrt :m true :p (i m <=) ( (n i mod 0 ==) (m @i false @p) when i 2 + @i ) while p
) :_prime? ; helper function
(
( ((2 <) (false)) ((dup even?) (2 ==)) ((true) (_prime?)) ) case
) :prime?</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,R SET R=N#2 IF R FOR I=3:2:(N**.5) SET R=N#I Q:'R KILL I QUIT R</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 $$ISPRIME^ROSETTA(97) 7 USER>W $$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>
newLISP
Short-circuit evaluation ensures that the many Boolean expressions are calculated in the right order so as not to waste time. <lang newlisp>; Here are some simpler functions to help us:
(define (divisible? larger-number smaller-number) (zero? (% larger-number smaller-number)))
(define (int-root number) (floor (sqrt number)))
(define (even-prime? number) (= number 2))
- Trial division for odd numbers
(define (find-odd-factor? odd-number) (catch (for (possible-factor 3 (int-root odd-number) 2) (if (divisible? odd-number possible-factor) (throw true)))))
(define (odd-prime? number) (and (odd? number) (or (= number 3) (and (> number 3) (not (find-odd-factor? number))))))
- Now for the final overall Boolean function.
(define (is-prime? possible-prime) (or (even-prime? possible-prime) (odd-prime? possible-prime)))
- Let's use this to actually find some prime numbers.
(println (filter is-prime? (sequence 1 100))) (exit)</lang>
- Output:
(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)
Nim
Here are three ways to test primality using trial division. <lang nim>import sequtils, math
proc prime(a: int): bool =
if a == 2: return true
if a < 2 or a mod 2 == 0: return false
for i in countup(3, sqrt(a.float).int, 2):
if a mod i == 0:
return false
return true
proc prime2(a: int): bool =
result = not (a < 2 or any(toSeq(2 .. sqrt(a.float).int), a mod it == 0))
proc prime3(a: int): bool =
if a == 2: return true if a < 2 or a mod 2 == 0: return false return not any(toSeq countup(3, sqrt(a.float).int, 2), a mod it == 0)
for i in 2..30:
echo i, " ", prime(i)</lang>
- Output:
2 true 3 true 4 false 5 true 6 false 7 true 8 false 9 false 10 false 11 true 12 false 13 true 14 false 15 false 16 false 17 true 18 false 19 true 20 false 21 false 22 false 23 true 24 false 25 false 26 false 27 false 28 false 29 true 30 false
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>
Oforth
<lang Oforth>Integer method: isPrime | i |
self 1 <= ifTrue: [ false return ] self 3 <= ifTrue: [ true return ] self isEven ifTrue: [ false return ] 3 self sqrt asInteger for: i [ self i mod ifzero: [ false return ] ] true ;</lang>
- Output:
#isPrime 1000 seq filter println [2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 8 9, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181 , 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271, 277, 281 , 283, 293, 307, 311, 313, 317, 331, 337, 347, 349, 353, 359, 367, 373, 379, 383, 389, 397 , 401, 409, 419, 421, 431, 433, 439, 443, 449, 457, 461, 463, 467, 479, 487, 491, 499, 503 , 509, 521, 523, 541, 547, 557, 563, 569, 571, 577, 587, 593, 599, 601, 607, 613, 617, 619 , 631, 641, 643, 647, 653, 659, 661, 673, 677, 683, 691, 701, 709, 719, 727, 733, 739, 743 , 751, 757, 761, 769, 773, 787, 797, 809, 811, 821, 823, 827, 829, 839, 853, 857, 859, 863 , 877, 881, 883, 887, 907, 911, 919, 929, 937, 941, 947, 953, 967, 971, 977, 983, 991, 997 ]
Ol
<lang scheme> (define (prime? number)
(define max (sqrt number))
(define (loop divisor)
(or (> divisor max)
(and (> (modulo number divisor) 0)
(loop (+ divisor 2)))))
(or (= number 1)
(= number 2)
(and
(> (modulo number 2) 0)
(loop 3))))
</lang> Testing: <lang scheme>
- first prime numbers less than 100
(for-each (lambda (n)
(if (prime? n)
(display n))
(display " "))
(iota 100))
(print)
- few more sintetic tests
(for-each (lambda (n)
(print n " - prime? " (prime? n)))
'(
1234567654321 ; 1111111 * 1111111
679390005787 ; really prime, I know that
679390008337 ; same
666810024403 ; 680633 * 979691 (multiplication of two prime numbers)
12345676543211234567654321
12345676543211234567654321123456765432112345676543211234567654321123456765432112345676543211234567654321
))
</lang>
- Output:
1 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 1234567654321 - prime? #false 679390005787 - prime? #true 679390008337 - prime? #true 666810024403 - prime? #false 12345676543211234567654321 - prime? #false 12345676543211234567654321123456765432112345676543211234567654321123456765432112345676543211234567654321 - prime? #false
Oz
<lang oz> fun {Prime N}
local IPrime in
fun {IPrime N Acc} if N < Acc*Acc then true elseif (N mod Acc) == 0 then false else {IPrime N Acc+1} end end if N < 2 then false else {IPrime N 2} end
end end</lang>
Panda
In Panda you write a boolean function by making it filter, either returning it's input or nothing. <lang panda>fun prime(p) type integer->integer
p.gt(1) where q=p.sqrt NO(p.mod(2..q)==0)
1..100.prime</lang>
PARI/GP
<lang parigp>trial(n)={
if(n < 4, return(n > 1)); /* Handle negatives */ forprime(p=2,sqrtint(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
improved using number wheel
<lang pascal>program TestTrialDiv; {$IFDEF FPC}
{$MODE DELPHI}{$Smartlink ON}
{$ELSE}
{$APPLICATION CONSOLE}// for Delphi
{$ENDIF} uses
primtrial;
{ Test and display primes 0 .. 50 } begin
repeat write(actPrime,' '); until nextPrime > 50;
end.</lang>
- Output
2 3 5 7 11 13 17 19 23 29 31 37 41 43 47
Perl
A simple idiomatic solution: <lang perl>sub prime { my $n = shift || $_;
$n % $_ or return for 2 .. sqrt $n; $n > 1
}
print join(', ' => grep prime, 1..100), "\n";</lang>
Excluding multiples of 2 and 3
One of many ways of writing trial division using a mod-6 wheel. Almost 2x faster than the simple method shown earlier. <lang perl>sub isprime {
my $n = shift;
return ($n >= 2) if $n < 4;
return unless $n % 2 && $n % 3;
my $sqrtn = int(sqrt($n));
for (my $i = 5; $i <= $sqrtn; $i += 6) {
return unless $n % $i && $n % ($i+2);
}
1;
} my $s = 0; $s += !!isprime($_) for 1..100000; print "Pi(100,000) = $s\n";</lang>
By Regular Expression
JAPH by Abigail 1999 [1] in conference slides 2000 [2].
While this is extremely clever and often used for Code golf, it should never be used for real work (it is extremely slow and uses lots of memory). <lang perl>sub isprime {
('1' x shift) !~ /^1?$|^(11+?)\1+$/
} print join(', ', grep(isprime($_), 0..39)), "\n";</lang>
Phix
function is_prime_by_trial_division(integer n) if n<2 then return 0 end if if n=2 then return 1 end if if remainder(n,2)=0 then return 0 end if for i=3 to floor(sqrt(n)) by 2 do if remainder(n,i)=0 then return 0 end if end for return 1 end function ?filter(tagset(32),is_prime_by_trial_division)
- Output:
{2,3,5,7,11,13,17,19,23,29,31}
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>
Picat
Here are four different versions.
Iterative
<lang Picat>is_prime1(N) =>
if N == 2 then
true
elseif N <= 1 ; N mod 2 == 0 then
false
else
foreach(I in 3..2..ceiling(sqrt(N)))
N mod I > 0
end
end.</lang>
Recursive
<lang Picat>is_prime2(N) =>
(N == 2 ; is_prime2b(N,3)).
is_prime2b(N,_Div), N <= 1 => false. is_prime2b(2,_Div) => true. is_prime2b(N,_Div), N mod 2 == 0 => false. is_prime2b(N,Div), Div > ceiling(sqrt(N)) => true. is_prime2b(N,Div), Div > 2 =>
(N mod Div == 0 -> false ; is_prime2b(N,Div+2) ).</lang>
Functional
<lang Picat>is_prime3(2) => true. is_prime3(3) => true. is_prime3(P) => P > 3, P mod 2 =\= 0, not has_factor3(P,3).
has_factor3(N,L), N mod L == 0 => true. has_factor3(N,L) => L * L < N, L2 = L + 2, has_factor3(N,L2).</lang>
Generator approach
prime2(N) can be used to generate primes until memory is exhausted.
Difference from Prolog implementation: Picat does not support between/3 with
"inf" as upper bound, so a high number (here 2**156+1) must be used.
<lang Picat>prime2(2).
prime2(N) :-
between(3, 2**156+1, N), N mod 2 = 1, % odd M is floor(sqrt(N+1)), % round-off paranoia Max is (M-1) // 2, % integer division foreach(I in 1..Max) N mod (2*I+1) > 0 end.</lang>
Test
<lang Picat>go =>
println([I : I in 1..100, is_prime1(I)]),
nl,
foreach(P in 1..6)
Primes = [ I : I in 1..10**P, is_prime1(I)],
println([10**P,Primes.len])
end,
nl.</lang>
- Output:
[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] [10,4] [100,25] [1000,168] [10000,1229] [100000,9592] [1000000,78498]
Benchmark
Times for calculating the number of primes below 10, 100,1_000,10_000, 100_000, and 1_000_000 respectively.
- imperative:
is_prime1/1(0.971) - recursive:
is_prime2/1(3.258s) - functional:
is_prime3/1(0.938s) - test/generate
prime2/1(2.129s)
PicoLisp
<lang PicoLisp>(de prime? (N)
(or
(= N 2)
(and
(> N 1)
(bit? 1 N)
(let S (sqrt N)
(for (D 3 T (+ D 2))
(T (> D S) T)
(T (=0 (% N D)) NIL) ) ) ) ) )</lang>
PL/I
<lang pli>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>
PL/M
This can be compiled with the original 8080 PL/M compiler and run under CP/M or an emulator or clone.
Note that all integers in 8080 PL/M are unsigned.
<lang pli>100H: /* TEST FOR PRIMALITY BY TRIAL DIVISION */
DECLARE FALSE LITERALLY '0', TRUE LITERALLY '0FFH';
/* CP/M BDOS SYSTEM CALL */
BDOS: PROCEDURE( FN, ARG ); DECLARE FN BYTE, ARG ADDRESS; GOTO 5; END;
PR$CHAR: PROCEDURE( C ); DECLARE C BYTE; CALL BDOS( 2, C ); END;
PR$STRING: PROCEDURE( S ); DECLARE S ADDRESS; CALL BDOS( 9, S ); END;
PR$NL: PROCEDURE; CALL PR$STRING( .( 0DH, 0AH, '$' ) ); END;
PR$NUMBER: PROCEDURE( N );
DECLARE N ADDRESS;
DECLARE V ADDRESS, N$STR( 6 ) BYTE, W BYTE;
N$STR( W := LAST( N$STR ) ) = '$';
N$STR( W := W - 1 ) = '0' + ( ( V := N ) MOD 10 );
DO WHILE( ( V := V / 10 ) > 0 );
N$STR( W := W - 1 ) = '0' + ( V MOD 10 );
END;
CALL PR$STRING( .N$STR( W ) );
END PR$NUMBER;
/* INTEGER SUARE ROOT: BASED ON THE ONE IN THE PL/M FOR FROBENIUS NUMBERS */
SQRT: PROCEDURE( N )ADDRESS;
DECLARE ( N, X0, X1 ) ADDRESS;
IF N <= 3 THEN DO;
IF N = 0 THEN X0 = 0; ELSE X0 = 1;
END;
ELSE DO;
X0 = SHR( N, 1 );
DO WHILE( ( X1 := SHR( X0 + ( N / X0 ), 1 ) ) < X0 );
X0 = X1;
END;
END;
RETURN X0;
END SQRT;
IS$PRIME: PROCEDURE( N )BYTE; /* RETURNS TRUE IF N IS PRIME, FALSE IF NOT */
DECLARE N ADDRESS;
IF N < 2 THEN RETURN FALSE;
ELSE IF ( N AND 1 ) = 0 THEN RETURN N = 2;
ELSE DO;
/* ODD NUMBER > 2 */
DECLARE I ADDRESS;
DO I = 3 TO SQRT( N ) BY 2;
IF N MOD I = 0 THEN RETURN FALSE;
END;
RETURN TRUE;
END;
END IS$PRIME;
/* TEST THE IS$PRIME PROCEDURE */
DECLARE I ADDRESS;
DO I = 0 TO 100;
IF IS$PRIME( I ) THEN DO;
CALL PR$CHAR( ' ' );
CALL PR$NUMBER( I );
END;
END;
CALL PR$NL;
EOF</lang>
- Output:
2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 73 79 83 89 97
PowerShell
<lang PowerShell> function isPrime ($n) {
if ($n -eq 1) {$false}
elseif ($n -eq 2) {$true}
elseif ($n -eq 3) {$true}
else{
$m = [Math]::Floor([Math]::Sqrt($n))
(@(2..$m | where {($_ -lt $n) -and ($n % $_ -eq 0) }).Count -eq 0)
}
} 1..15 | foreach{"isPrime $_ : $(isPrime $_)"}</lang> Output:
isPrime 1 : False isPrime 2 : True isPrime 3 : True isPrime 4 : False isPrime 5 : True isPrime 6 : False isPrime 7 : True isPrime 8 : False isPrime 9 : False isPrime 10 : False isPrime 11 : True isPrime 12 : False isPrime 13 : True isPrime 14 : False isPrime 15 : False
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>
Quackery
sqrt is defined at Isqrt (integer square root) of X#Quackery.
<lang Quackery> [ dup 4 < iff [ 1 > ] done
dup 1 & not iff [ drop false ] done
true swap dup sqrt
0 = iff [ 2drop not ] done
1 >> times
[ dup i^ 1 << 3 + mod 0 = if
[ dip not conclude ] ]
drop ] is isprime ( n --> b )
</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>
Raku
(formerly 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 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, up to the square root of $i." <lang perl6>sub prime (Int $i --> Bool) {
$i > 1 and so $i %% none 2..$i.sqrt;
}</lang>
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 avoid recalculating the square root each time. Note the mutual recursion that depends on the implicit laziness of list evaluation: <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) {
my \limit = $i.sqrt.floor;
@isprime[$i] //= so ($i %% none @primes ...^ { $_ > limit })
}
say "$_ is{ "n't" x !prime($_) } prime." for 1 .. 100;</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 ).
Programming note: all the REXX programs below show all primes up and including the number specified.
- If the number is negative, the absolute value of it is used for the upper limit, but no primes are shown.
- The number of primes found is always shown.
Also, it was determined that computing the (integer) square root of the number to be tested in the isPrime
function slowed up the function (for all three REXX versions), however, for larger numbers of N, it would
be faster.
<lang rexx>/*REXX program tests for primality by using (kinda smartish) trial division. */
parse arg n .; if n== then n=10000 /*let the user choose the upper limit. */
tell=(n>0); n=abs(n) /*display the primes only if N > 0. */
p=0 /*a count of the primes found (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 tell then say right(j,20) 'is prime.' /*maybe display prime to the terminal. */
end /*j*/
say say "There are " p " primes up to " n ' (inclusive).' exit /*stick a fork in it, we're all done. */ /*──────────────────────────────────────────────────────────────────────────────────────*/ isPrime: procedure; parse arg x /*get the number to be tested. */
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 /*a pair of divides. ___ */
end /*k*/ /*divide up through the √ x */
/*Note: // is ÷ remainder.*/
return 1 /*done dividing, it's prime. */</lang>
- output when using the default input of: 100
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 are 25 primes up to 100 (inclusive).
optimized version
This version separates multiple-clause if statements, and also tests for low primes,
making it about 8% faster.
<lang rexx>/*REXX program tests for primality by using (kinda smartish) trial division. */
parse arg n .; if n== then n=10000 /*let the user choose the upper limit. */
tell=(n>0); n=abs(n) /*display the primes only if N > 0. */
p=0 /*a count of the primes found (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 tell then say right(j,20) 'is prime.' /*maybe display prime to the terminal. */
end /*j*/
say say "There are " p " primes up to " n ' (inclusive).' exit /*stick a fork in it, we're all done. */ /*──────────────────────────────────────────────────────────────────────────────────────*/ isPrime: procedure; parse arg x /*get integer to be investigated.*/
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 all the even numbers.*/
if x//3==0 then return 0 /* ··· and eliminate the triples.*/
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: REXX // is ÷ remainder.*/
return 1 /*did all divisions, it's prime. */</lang>
- output is identical to the first version when the same input is used.
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 is used, making it about 22% faster.
Note that the do ... until ... was changed to do ... while ....
<lang rexx>/*REXX program tests for primality by using (kinda smartish) trial division. */
parse arg n .; if n== then n=10000 /*let the user choose the upper limit. */
tell=(n>0); n=abs(n) /*display the primes only if N > 0. */
p=0 /*a count of the primes found (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 tell then say right(j,20) 'is prime.' /*maybe display prime to the terminal. */
end /*j*/
say say "There are " p " primes up to " n ' (inclusive).' exit /*stick a fork in it, we're all done. */ /*──────────────────────────────────────────────────────────────────────────────────────*/ isPrime: procedure; parse arg x /*get the integer to be investigated. */
if x<107 then return wordpos(x, '2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53',
'59 61 67 71 73 79 83 89 97 101 103')\==0 /*some low primes.*/
if x// 2 ==0 then return 0 /*eliminate all the even numbers. */
if x// 3 ==0 then return 0 /* ··· and eliminate the triples. */
parse var x -1 _ /* obtain the rightmost digit.*/
if _ ==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: REXX // is ÷ remainder. */
do k=107 by 6 while k*k<=x /*this skips odd multiples of three. */
if x//k ==0 then return 0 /*perform a divide (modulus), */
if x//(k+2)==0 then return 0 /* ··· and the next also. ___ */
end /*k*/ /*divide up through the √ x */
return 1 /*after all that, ··· it's a prime. */</lang>
- output is identical to the first version when the same input is used.
Ring
<lang ring>give n flag = isPrime(n) if flag = 1 see n + " is a prime number" else see n + " is not a prime number" ok
func isPrime num
if (num <= 1) return 0 ok
if (num % 2 = 0 and num != 2) return 0 ok
for i = 3 to floor(num / 2) -1 step 2
if (num % i = 0) return 0 ok
next
return 1</lang>
Ruby
<lang ruby>def prime(a)
if a == 2
true
elsif a <= 1 || a % 2 == 0
false
else
divisors = (3..Math.sqrt(a)).step(2)
divisors.none? { |d| a % d == 0 }
end
end p (1..50).select{|i| prime(i)}</lang>
The prime package in the stdlib for Ruby contains this compact Prime#prime? method:
<lang ruby>require "prime"
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 p (1..50).select{|i| prime?(i)}</lang>
Without any fancy stuff: <lang ruby>def primes(limit)
(enclose = lambda { |primes|
primes.last.succ.upto(limit) do |trial_pri|
if primes.none? { |pri| (trial_pri % pri).zero? }
return enclose.call(primes << trial_pri)
end
end
primes
}).call([2])
end p primes(50)</lang>
- Output:
[2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47]
By Regular Expression
<lang ruby>def isprime(n)
'1'*n !~ /^1?$|^(11+?)\1+$/
end</lang>
Prime Generators Tests
Mathematicaly basis of Prime Generators https://www.academia.edu/19786419/PRIMES-UTILS_HANDBOOK https://www.academia.edu/42734410/Improved_Primality_Testing_and_Factorization_in_Ruby_revised
<lang ruby>require "benchmark/ips"
- the simplest PG primality test using the P3 prime generator
- reduces the number space for primes to 2/6 (33.33%) of all integers
def primep3?(n) # P3 Prime Generator primality test
# P3 = 6*k + {5, 7} # P3 primes candidates (pc) sequence
return n | 1 == 3 if n < 5 # n: 0,1,4|false, 2,3|true
return false if n.gcd(6) != 1 # 1/3 (2/6) of integers are P3 pc
p, sqrtn = 5, Integer.sqrt(n) # first P3 sequence value
until p > sqrtn
return false if n % p == 0 || n % (p + 2) == 0 # if n is composite
p += 6 # first prime candidate for next kth residues group
end
true
end
- PG primality test using the P5 prime generator
- reduces the number space for primes to 8/30 (26.67%) of all integers
def primep5?(n) # P5 Prime Generator primality test
# P5 = 30*k + {7,11,13,17,19,23,29,31} # P5 primes candidates sequence
return [2, 3, 5].include?(n) if n < 7 # for small and negative values
return false if n.gcd(30) != 1 # 4/15 (8/30) of integers are P5 pc
p, sqrtn = 7, Integer.sqrt(n) # first P5 sequence value
until p > sqrtn
return false if # if n is composite
n % (p) == 0 || n % (p+4) == 0 || n % (p+6) == 0 || n % (p+10) == 0 ||
n % (p+12) == 0 || n % (p+16) == 0 || n % (p+22) == 0 || n % (p+24) == 0
p += 30 # first prime candidate for next kth residues group
end
true
end
- PG primality test using the P7 prime generator
- reduces the number space for primes to 48/210 (22.86%) of all integers
def primep7?(n)
# P7 = 210*k + {11,13,17,19,23,29,31,37,41,43,47,53,59,61,67,71,73,79,83,
# 89,97,101,103,107,109,113,121,127,131,137,139,143,149,151,157,163,
# 167,169,173,179,181,187,191,193,197,199,209,211}
return [2, 3, 5, 7].include?(n) if n < 11
return false if n.gcd(210) != 1 # 8/35 (48/210) of integers are P7 pc
p, sqrtn = 11, Integer.sqrt(n) # first P7 sequence value
until p > sqrtn
return false if
n % (p) == 0 || n % (p+2) == 0 || n % (p+6) == 0 || n % (p+8) == 0 ||
n % (p+12) == 0 || n % (p+18) == 0 || n % (p+20) == 0 || n % (p+26) == 0 ||
n % (p+30) == 0 || n % (p+32) == 0 || n % (p+36) == 0 || n % (p+42) == 0 ||
n % (p+48) == 0 || n % (p+50) == 0 || n % (p+56) == 0 || n % (p+60) == 0 ||
n % (p+62) == 0 || n % (p+68) == 0 || n % (p+72) == 0 || n % (p+78) == 0 ||
n % (p+86) == 0 || n % (p+90) == 0 || n % (p+92) == 0 || n % (p+96) == 0 ||
n % (p+98) == 0 || n % (p+102) == 0 || n % (p+110) == 0 || n % (p+116) == 0 ||
n % (p+120) == 0 || n % (p+126) == 0 || n % (p+128) == 0 || n % (p+132) == 0 ||
n % (p+138) == 0 || n % (p+140) == 0 || n % (p+146) == 0 || n % (p+152) == 0 ||
n % (p+156) == 0 || n % (p+158) == 0 || n % (p+162) == 0 || n % (p+168) == 0 ||
n % (p+170) == 0 || n % (p+176) == 0 || n % (p+180) == 0 || n % (p+182) == 0 ||
n % (p+186) == 0 || n % (p+188) == 0 || n % (p+198) == 0 || n % (p+200) == 0
p += 210 # first prime candidate for next kth residues group
end
true
end
- Benchmarks to test for various size primes
p = 541 Benchmark.ips do |b|
print "\np = #{p}"
b.report("primep3?") { primep3?(p) }
b.report("primep5?") { primep5?(p) }
b.report("primep7?") { primep7?(p) }
b.compare!
puts
end
p = 1000003 Benchmark.ips do |b|
print "\np = #{p}"
b.report("primep3?") { primep3?(p) }
b.report("primep5?") { primep5?(p) }
b.report("primep7?") { primep7?(p) }
b.compare!
puts
end
p = 4294967291 # largest prime < 2**32 Benchmark.ips do |b|
print "\np = #{p}"
b.report("primep3?") { primep3?(p) }
b.report("primep5?") { primep5?(p) }
b.report("primep7?") { primep7?(p) }
b.compare!
puts
end
p = (10 ** 16) * 2 + 3 Benchmark.ips do |b|
print "\np = #{p}"
b.report("primep3?") { primep3?(p) }
b.report("primep5?") { primep5?(p) }
b.report("primep7?") { primep7?(p) }
b.compare!
puts
end</lang>
- Output:
p = 541
Warming up --------------------------------------
primep3? 109.893k i/100ms
primep5? 123.949k i/100ms
primep7? 44.216k i/100ms
Calculating -------------------------------------
primep3? 1.598M (± 3.4%) i/s - 8.022M in 5.025036s
primep5? 1.872M (± 6.3%) i/s - 9.420M in 5.059896s
primep7? 502.040k (± 1.2%) i/s - 2.520M in 5.020919s
Comparison:
primep5?: 1871959.0 i/s
primep3?: 1598489.8 i/s - 1.17x slower
primep7?: 502039.8 i/s - 3.73x slower
p = 1000003
Warming up --------------------------------------
primep3? 5.850k i/100ms
primep5? 9.013k i/100ms
primep7? 10.889k i/100ms
Calculating -------------------------------------
primep3? 59.965k (± 1.1%) i/s - 304.200k in 5.073641s
primep5? 92.561k (± 2.1%) i/s - 468.676k in 5.065709s
primep7? 109.335k (± 0.8%) i/s - 555.339k in 5.079549s
Comparison:
primep7?: 109334.7 i/s
primep5?: 92561.4 i/s - 1.18x slower
primep3?: 59964.5 i/s - 1.82x slower
p = 4294967291
Warming up --------------------------------------
primep3? 92.000 i/100ms
primep5? 148.000 i/100ms
primep7? 184.000 i/100ms
Calculating -------------------------------------
primep3? 926.647 (± 1.1%) i/s - 4.692k in 5.064067s
primep5? 1.480k (± 1.7%) i/s - 7.400k in 5.001399s
primep7? 1.804k (± 1.0%) i/s - 9.200k in 5.099110s
Comparison:
primep7?: 1804.4 i/s
primep5?: 1480.0 i/s - 1.22x slower
primep3?: 926.6 i/s - 1.95x slower
p = 20000000000000003
Warming up --------------------------------------
primep3? 1.000 i/100ms
primep5? 1.000 i/100ms
primep7? 1.000 i/100ms
Calculating -------------------------------------
primep3? 0.422 (± 0.0%) i/s - 3.000 in 7.115929s
primep5? 0.671 (± 0.0%) i/s - 4.000 in 5.957077s
primep7? 0.832 (± 0.0%) i/s - 5.000 in 6.007834s
Comparison:
primep7?: 0.8 i/s
primep5?: 0.7 i/s - 1.24x slower
primep3?: 0.4 i/s - 1.97x slower
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
Rust
<lang rust>fn is_prime(n: u64) -> bool {
match n {
0 | 1 => false,
2 => true,
_even if n % 2 == 0 => false,
_ => {
let sqrt_limit = (n as f64).sqrt() as u64;
(3..=sqrt_limit).step_by(2).find(|i| n % i == 0).is_none()
}
}
}
fn main() {
for i in (1..30).filter(|i| is_prime(*i)) {
println!("{} ", i);
}
}</lang>
- Output:
2 3 5 7 11 13 17 19 23 29
S-BASIC
<lang S-BASIC> $lines
$constant FALSE = 0 $constant TRUE = 0FFFFH
rem - return p mod q function mod(p, q = integer) = integer end = p - q * (p / q)
rem - return true (-1) if n is prime, otherwise false (0) function isprime(n = integer) = integer
var i, limit, result = integer
if n = 2 then
result = TRUE
else if (n < 2) or (mod(n,2) = 0) then
result = FALSE
else
begin
limit = int(sqr(n))
i = 3
while (i <= limit) and (mod(n, i) <> 0) do
i = i + 2
result = not (i <= limit)
end
end = result
rem - test by looking for primes in range 1 to 50 var i = integer for i = 1 to 50
if isprime(i) then print using "#####";i;
next i
end </lang>
- Output:
2 3 5 7 11 13 17 19 23 29 31 37 41 43 47
S-lang
<lang S-lang>define is_prime(n) {
if (n == 2) return(1); if (n <= 1) return(0); if ((n & 1) == 0) return(0);
variable mx = int(sqrt(n)), i;
_for i (3, mx, 1) {
if ((n mod i) == 0)
return(0);
}
return(1);
}
define ptest() {
variable lst = {};
_for $1 (1, 64, 1)
if (is_prime($1))
list_append(lst, string($1));
print(strjoin(list_to_array(lst), ", "));
} ptest();</lang>
- Output:
"2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61"
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
<lang Scala> def isPrime(n: Int) =
n > 1 && (Iterator.from(2) takeWhile (d => d * d <= n) forall (n % _ != 0))</lang>
Accelerated version FP and parallel runabled
//
- Output:
Best seen running in your browser Scastie (remote JVM).
<lang Scala>object IsPrimeTrialDivision extends App {
def isPrime(n: Long) =
n > 1 && ((n & 1) != 0 || n == 2) && (n % 3 != 0 || n == 3) &&
(5 to math.sqrt(n).toInt by 6).par.forall(d => n % d != 0 && n % (d + 2) != 0)
assert(!isPrime(-1))
assert(!isPrime(1))
assert(isPrime(2))
assert(isPrime(100000000003L))
assert(isPrime(100000000019L))
assert(isPrime(100000000057L))
assert(isPrime(100000000063L))
assert(isPrime(100000000069L))
assert(isPrime(100000000073L))
assert(isPrime(100000000091L))
println("10 Numbers tested. A moment please …\nNumber crunching biggest 63-bit prime …")
assert(isPrime(9223372036854775783L)) // Biggest 63-bit prime
println("All done")
}</lang>
Accelerated version FP, tail recursion
Tests 1.3 M numbers against OEIS prime numbers. <lang Scala>import scala.annotation.tailrec import scala.io.Source
object PrimesTestery extends App {
val rawText = Source.fromURL("https://oeis.org/A000040/a000040.txt")
val oeisPrimes = rawText.getLines().take(100000).map(_.split(" ")(1)).toVector
def isPrime(n: Long) = {
@tailrec
def inner(d: Int, end: Int): Boolean = {
if (d > end) true
else if (n % d != 0 && n % (d + 2) != 0) inner(d + 6, end) else false
}
n > 1 && ((n & 1) != 0 || n == 2) &&
(n % 3 != 0 || n == 3) && inner(5, math.sqrt(n).toInt)
}
println(oeisPrimes.size) for (i <- 0 to 1299709) assert(isPrime(i) == oeisPrimes.contains(i.toString), s"Wrong $i")
}</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: isPrime (in integer: number) is func
result
var boolean: prime is FALSE;
local
var integer: upTo is 0;
var integer: testNum is 3;
begin
if number = 2 then
prime := TRUE;
elsif odd(number) and number > 2 then
upTo := sqrt(number);
while number rem testNum <> 0 and testNum <= upTo do
testNum +:= 2;
end while;
prime := testNum > upTo;
end if;
end func;</lang>
Original source: [3]
Sidef
<lang ruby>func is_prime(a) {
given (a) {
when (2) { true }
case (a <= 1 || a.is_even) { false }
default { 3 .. a.isqrt -> any { .divides(a) } -> not }
}
}</lang>
Alternative version, excluding multiples of 2 and 3: <lang ruby>func is_prime(n) {
return (n >= 2) if (n < 4)
return false if (n%%2 || n%%3)
for k in (5 .. n.isqrt -> by(6)) {
return false if (n%%k || n%%(k+2))
}
return true
}</lang>
Smalltalk
<lang smalltalk>| isPrime | isPrime := [:n |
n even ifTrue: [ ^n=2 ]
ifFalse: [
3 to: n sqrt do: [:i |
(n \\ i = 0) ifTrue: [ ^false ]
].
^true
]
]</lang>
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>
Swift
<lang swift>import Foundation
extension Int {
func isPrime() -> Bool {
switch self {
case let x where x < 2:
return false
case 2:
return true
default:
return
self % 2 != 0 &&
!stride(from: 3, through: Int(sqrt(Double(self))), by: 2).contains {self % $0 == 0}
}
}
}</lang>
Tcl
<lang tcl>proc is_prime n {
if {$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
Tiny BASIC
<lang tinybasic> PRINT "ENTER A NUMBER "
INPUT P GOSUB 100 IF Z = 1 THEN PRINT "It is prime." IF Z = 0 THEN PRINT "It isn't prime." END
100 REM PRIMALITY OF THE NUMBER P BY TRIAL DIVISION
IF P < 2 THEN RETURN LET Z = 1 IF P < 4 THEN RETURN LET I = 2
110 IF (P/I)*I = P THEN LET Z = 0
IF Z = 0 THEN RETURN LET I = I + 1 IF I*I <= P THEN GOTO 110 RETURN</lang>
uBasic/4tH
<lang>10 LET n=0: LET p=0 20 INPUT "Enter number: ";n 30 LET p=0 : IF n>1 THEN GOSUB 1000 40 IF p=0 THEN PRINT n;" is not prime!" 50 IF p#0 THEN PRINT n;" is prime!" 60 GOTO 10 1000 REM *************** 1001 REM * PRIME CHECK * 1002 REM *************** 1010 LET p=0 1020 IF (n%2)=0 THEN RETURN 1030 LET p=1 : PUSH n,0 : GOSUB 9030 1040 FOR i=3 TO POP() STEP 2 1050 IF (n%i)=0 THEN LET p=0: PUSH n,0 : GOSUB 9030 : LET i=POP() 1060 NEXT i 1070 RETURN 9030 Push ((10^(Pop()*2))*Pop()) : @(255)=Tos() 9040 Push (@(255) + (Tos()/@(255)))/2
If Abs(@(255)-Tos())<2 Then @(255)=Pop() : If Pop() Then Push @(255) : Return
@(255) = Pop() : Goto 9040
REM ** This is an integer SQR subroutine. Output is scaled by 10^(TOS()).</lang>
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>
VBA
<lang vb>Option Explicit
Sub FirstTwentyPrimes() Dim count As Integer, i As Long, t(19) As String
Do
i = i + 1
If IsPrime(i) Then
t(count) = i
count = count + 1
End If
Loop While count <= UBound(t)
Debug.Print Join(t, ", ")
End Sub
Function IsPrime(Nb As Long) As Boolean
If Nb = 2 Then
IsPrime = True
ElseIf Nb < 2 Or Nb Mod 2 = 0 Then
Exit Function
Else
Dim i As Long
For i = 3 To Sqr(Nb) Step 2
If Nb Mod i = 0 Then Exit Function
Next
IsPrime = True
End If
End Function</lang>
- Output:
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71
VBScript
<lang vb>Function IsPrime(n) If n = 2 Then IsPrime = True ElseIf n <= 1 Or n Mod 2 = 0 Then IsPrime = False Else IsPrime = True For i = 3 To Int(Sqr(n)) Step 2 If n Mod i = 0 Then IsPrime = False Exit For End If Next End If End Function
For n = 1 To 50 If IsPrime(n) Then WScript.StdOut.Write n & " " End If Next</lang>
- Output:
2 3 5 7 11 13 17 19 23 29 31 37 41 43 47
Wren
<lang ecmascript>import "/fmt" for Fmt
var isPrime = Fn.new { |n|
if (n < 2) return false
if (n%2 == 0) return n == 2
var p = 3
while (p * p <= n) {
if (n%p == 0) return false
p = p + 2
}
return true
}
var tests = [2, 5, 12, 19, 57, 61, 97] System.print("Are the following prime?") for (test in tests) {
System.print("%(Fmt.d(2, test)) -> %(isPrime.call(test) ? "yes" : "no")")
}</lang>
- Output:
Are the following prime? 2 -> yes 5 -> yes 12 -> no 19 -> yes 57 -> no 61 -> yes 97 -> yes
XPL0
<lang XPL0>func Prime(N); \Return 'true' if N is a prime number int N; int I; [if N <= 1 then return false; for I:= 2 to sqrt(N) do
if rem(N/I) = 0 then return false;
return true; ];
int Num; repeat Num:= IntIn(0);
Text(0, if Prime(Num) then "is " else "not ");
Text(0, "prime^M^J");
until Num = 0</lang>
- Output:
777777777 not prime 777777773 is prime 0 not prime
zkl
The Method filter1 stops at the first non False result, which, if there is one, is the first found diviser, thus short cutting the rest of the test <lang zkl>fcn isPrime(n){
if(n.isEven or n<2) return(n==2);
(not [3..n.toFloat().sqrt().toInt(),2].filter1('wrap(m){n%m==0}))
}</lang>
- Output:
zkl: [1..].filter(20,isPrime) L(2,3,5,7,11,13,17,19,23,29,31,37,41,43,47,53,59,61,67,71) zkl: isPrime(777777773) True zkl: isPrime(777777777) False
- Programming Tasks
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