Primality by trial division: Difference between revisions
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=={{header|Mathematica}}== |
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<lang>IsPrime[n_Integer] := |
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Module[{k = 2}, |
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If[n <= 1, Return False]; |
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If[n == 2, Return True]; |
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While[k <= Sqrt[n], |
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If[Mod[n, k] == 0, Return[False], k++] |
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]; |
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Return[True] |
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]</lang> |
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=={{header|MAXScript}}== |
=={{header|MAXScript}}== |
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Revision as of 03:58, 15 July 2010
You are encouraged to solve this task according to the task description, using any language you may know.
Write a boolean function that tells whether a given integer is prime. Remember that 1 and all non-positive numbers are not prime.
Use trial division. Even numbers may be eliminated right away. A loop from 3 to √(n) will suffice, but other loops are allowed.
ActionScript
<lang ActionScript>function isPrime(n:int):Boolean { if(n < 2) return false; if(n == 2) return true; if((n & 1) == 0) return false; for(var i:int = 3; i <= Math.sqrt(n); i+= 2) if(n % i == 0) return false; return true; }</lang>
Ada
<lang ada>function Is_Prime(Item : Positive) return Boolean is
Result : Boolean := True; Test : Natural;
begin
if Item /= 2 and Item mod 2 = 0 then
Result := False;
else
Test := 3;
while Test < Integer(Sqrt(Float(Item))) loop
if Item mod Test = 0 then
Result := False;
exit;
end if;
Test := Test + 2;
end loop;
end if;
return Result;
end Is_Prime;</lang>
ALGOL 68
<lang algol68>main:(</lang>
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> 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: <lang algol68>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</lang>
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>
AWK
$ awk 'func prime(n){for(d=2;d<=sqrt(n);d++)if(!(n%d)){return 0};return 1}{print prime($1)}'
BASIC
Going with the classic 1 for "true" and 0 for "false":
FUNCTION prime% (n!)
IF n = 2 THEN prime = 1
IF n <= 1 OR n MOD 2 = 0 THEN prime = 0
FOR a = 3 TO INT(SQR(n)) STEP 2
IF n MOD a = 0 THEN prime = 0
NEXT a
prime = 1
END FUNCTION
C
<lang c>#include <math.h>
- define FALSE 0
- define TRUE 1
int isPrime( unsigned int n ) {
unsigned int i;
if ( n == 2 )
return TRUE;
if ( n <= 1 || ( n & 1 ) == 0 )
return FALSE;
for ( i = 3 ; i <= sqrt( n ) ; i += 2 )
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>
C#
<lang csharp>static bool isPrime(int n)
{
if (n <= 1) return false;
for (int i = 2; i <= Math.Sqrt(n); i++)
if (n % i == 0) return false;
return true;
}</lang>
Clojure
The symbol # is a shortcut for creating lambda functions; the arguments in such a function are %1, %2, %3... (or simply % if there is only one argument). Thus, #(< (* % %) n) is equivalent to (fn [x] (< (* x x) n)) or more mathematically f(x) = x * x < n.
(defn divides? [k n] (= (rem n k) 0))
(defn prime? [n]
(if (< n 2)
false
(empty? (filter #(divides? % n) (take-while #(<= (* % %) n) (range 2 n))))))
Common Lisp
<lang Lisp>
(defun primep (a)
(cond ((= a 2) T)
((or (<= a 1) (= (mod a 2) 0)) nil)
((loop for i from 3 to (sqrt a) by 2 do
(if (= (mod a i) 0)
(return nil))) nil)
(T T)))
</lang> <lang Lisp> (defun primep (n)
"Is N prime?"
(and (> n 1)
(or (= n 2) (oddp n))
(loop for i from 3 to (isqrt n) by 2
never (zerop (rem n i))))) </lang>
D
<lang d>import std.math: sqrt; bool isPrime(int n) {
if (n == 2)
return true;
if (n <= 1 || (n & 1) == 0)
return false;
for(int i = 3; i <= sqrt(cast(float)n); i += 2)
if (n % i == 0)
return false;
return true;
}</lang>
Version with excluded multiplies of 2 and 3
<lang d>/**
* to compile: * $ dmd -run prime_trial.d * to optimize: * $ dmd -O -inline -release prime_trial.d */
module prime_trial;
import std.conv : to; import std.stdio : writefln;
/// Adapted from: http://www.devx.com/vb2themax/Tip/19051 bool isprime(Integer)(in Integer number) {
/* manually test 1, 2, 3 and multiples of 2 and 3 */
if (number == 2 || number == 3)
return true;
else if (number < 2 || number % 2 == 0 || number % 3 == 0)
return false;
/* we can now avoid to consider multiples
* of 2 and 3. This can be done really simply
* by starting at 5 and incrementing by 2 and 4
* alternatively, that is:
* 5, 7, 11, 13, 17, 19, 23, 25, 29, 31, 35, 37, ...
* we don't need to go higher than the square root of the number */
for (Integer divisor = 5, increment = 2; divisor*divisor <= number;
divisor += increment, increment = 6 - increment)
if (number % divisor == 0)
return false;
return true; // if we get here, the number is prime
}
/// print all prime numbers less then a given limit void main(char[][] args) {
const limit = (args.length == 2) ? to!(uint)(args[1]) : 100;
for (uint i = 0; i < limit; ++i)
if (isprime(i))
writefln(i);
}</lang>
E
<lang e> def isPrime(n :int) {
if (n == 2) {
return true
} else if (n <= 1 || n %% 2 == 0) {
return false
} else {
def limit := (n :float64).sqrt().ceil()
var divisor := 1
while ((divisor += 2) <= limit) {
if (n %% divisor == 0) {
return false
}
}
return true
}
}</lang>
Erlang
<lang erlang>is_prime(N) when N == 2 -> true; is_prime(N) when N < 2 orelse N rem 2 == 0 -> false; is_prime(N) -> is_prime(N,3).
is_prime(N,K) when K*K > N -> true; is_prime(N,K) when N rem K == 0 -> false; is_prime(N,K) -> is_prime(N,K+2). </lang>
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
[0\$2=$[@~@@]?~[$$2>\1&&[\~\ 3[\$@$@1+\$*>][\$@$@$@$@\/*=[%\~\$]?2+]#% ]?]?%]p:
Forth
: prime? ( n -- ? )
dup 2 < if drop false
else dup 2 = if drop true
else dup 1 and 0= if drop false
else 3
begin 2dup dup * >=
while 2dup mod 0=
if 2drop false exit
then 2 +
repeat 2drop true
then then then ;
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>
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>
Sample output:
[2, 3, 5, 7, 11, 13, 17, 19]
Haskell
Without square roots:
divides k n = n `mod` k == 0
isPrime :: Integer -> Bool
isPrime n | n < 2 = False
| otherwise = not $ any (`divides` n) $ takeWhile (\k -> k*k <= n) (2:[3,5..])
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>
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;
}
for(long n= 3; n <= (long)Math.sqrt(a); 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>
Joy
From http://www.latrobe.edu.au/philosophy/phimvt/joy/jp-imper.html <lang joy> DEFINE prime ==
2
[ [dup * >] nullary [rem 0 >] dip and ]
[ succ ]
while
dup * < .
</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
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
M4
<lang M4> define(`testnext',
`ifelse(eval($2*$2>$1),1,
1,
`ifelse(eval($1%$2==0),1,
0,
`testnext($1,eval($2+2))')')')
define(`isprime',
`ifelse($1,2,
1,
`ifelse(eval($1<=1 || $1%2==0),1,
0,
`testnext($1,3)')')')
isprime(9) isprime(11) </lang>
Output:
0 1
Mathematica
<lang>IsPrime[n_Integer] :=
Module[{k = 2},
If[n <= 1, Return False];
If[n == 2, Return True];
While[k <= Sqrt[n],
If[Mod[n, k] == 0, Return[False], k++]
];
Return[True]
]</lang>
MAXScript
fn isPrime n =
(
if n == 2 then
(
return true
)
else if (n <= 1) OR (mod n 2 == 0) then
(
return false
)
for i in 3 to (sqrt n) by 2 do
(
if mod n i == 0 then return false
)
true
)
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>
Perl
A more idiomatic solution: <lang perl>sub prime { my $n = shift || $_;
$n % $_ or return for 2 .. sqrt $n; $n > 1
}
print join(', ' => grep prime, 1..100), "\n";</lang>
By Regular Expression
<lang perl>sub isprime {
('1' x shift) !~ /^1?$|^(11+?)\1+$/
}
- A quick test
print join(', ', grep(isprime($_), 0..39)), "\n";</lang>
Perl 6
Most simply:
<lang perl6>sub prime (Int $n) { $n > 1 and $n !% none 2 .. sqrt $n }</lang>
Skipping even numbers when possible:
<lang perl6>sub prime (Int $n where (* > 0)) returns Bool {
when $n == 1 { False }
when $n == 2|3 { True }
when $n < 25 { $n % 2 and $n % 3 }
default { $n !% none 2, 3 ... * + 2, sqrt $n }
}</lang>
Testing:
<lang perl6>say "$_ is{ prime($_) ?? !! 'n\'t' } prime." for 1 .. 100;</lang>
PHP
<lang php><?php function prime($a) {
if ($a < 2)
return false;
$limit = sqrt($a);
for ($i = 2; $i <= $limit; $i++)
if ($a % $i == 0)
return false;
return true;
}
foreach (range(1, 100) as $x)
if (prime($x)) echo "$x\n";
?></lang>
By Regular Expression
<lang php><?php function prime($a) {
return !preg_match('/^1?$|^(11+?)\1+$/', str_repeat('1', $a));
} ?></lang>
PicoLisp
<lang PicoLisp>(de prime? (N)
(or
(= N 2)
(and
(> N 1)
(bit? 1 N)
(for (D 3 T (+ D 2))
(T (> D (sqrt N)) T)
(T (=0 (% N D)) NIL) ) ) ) )</lang>
PowerShell
<lang powershell>function isPrime ($n) {
if ($n -eq 1) {
return $false
} else {
return (@(2..[Math]::Sqrt($n) | Where-Object { $n % $_ -eq 0 }).Length -eq 0)
}
}</lang>
PureBasic
<lang PureBasic>Procedure.i IsPrime(n)
Protected k
If n = 2
ProcedureReturn #True
EndIf
If n <= 1 Or n % 2 = 0
ProcedureReturn #False
EndIf
For k = 3 To Int(Sqr(n)) Step 2
If n % k = 0
ProcedureReturn #False
EndIf
Next
ProcedureReturn #True
EndProcedure</lang>
Python
The simplest primality test, using trial division:
<lang python>def prime(a):
return not (a < 2 or any(a % x == 0 for x in xrange(2, int(a**0.5) + 1)))</lang>
Another test. Exclude even numbers first:
<lang python>def prime2(a):
if a == 2: return True if a < 2 or a % 2 == 0: return False return not any(a % x == 0 for x in xrange(3, int(a**0.5) + 1, 2))</lang>
Yet another test. Exclude multiples of 2 and 3, see http://www.devx.com/vb2themax/Tip/19051:
<lang python>def prime3(a):
if a < 2: return False if a == 2 or a == 3: return True # manually test 2 and 3 if a % 2 == 0 or a % 3 == 0: return False # exclude multiples of 2 and 3
maxDivisor = a**0.5
d, i = 5, 2
while d <= maxDivisor:
if a % d == 0: return False
d += i
i = 6 - i # this modifies 2 into 4 and viceversa
return True</lang>
By Regular Expression
Regular expression by "Abigail".
(An explanation is given in "The Story of the Regexp and the Primes").
<lang python>>>> import re
>>> def isprime(n):
return not re.match(r'^1?$|^(11+?)\1+$', '1' * n)
>>> # A quick test >>> [i for i in range(40) if isprime(i)] [2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37] </lang>
R
<lang R>isPrime <- function(n) {
if (n == 2) return(TRUE)
if ( (n <= 1) || ( n %% 2 == 0 ) ) return(FALSE)
for( i in 3:sqrt(n) ) {
if ( n %% i == 0 ) return(FALSE)
}
TRUE
}</lang>
<lang R>print(lapply(1:100, isPrime))</lang>
Ruby
<lang ruby>def prime(a)
if a == 2
true
elsif a <= 1 || a % 2 == 0
false
else
divisors = Enumerable::Enumerator.new(3..Math.sqrt(a), :step, 2)
# this also creates an enumerable object: divisors = (3..Math.sqrt(a)).step(2)
!divisors.any? { |d| a % d == 0 }
end
end</lang> The mathn package in the stdlib for Ruby 1.9.2 contains this compact Prime#prime? method: <lang ruby> def prime?(value, generator = Prime::Generator23.new)
return false if value < 2
for num in generator
q,r = value.divmod num
return true if q < num
return false if r == 0
end
end</lang>
Without any fancy stuff: <lang ruby>def primes(limit)
(enclose = lambda { |primes|
primes.last.succ.upto(limit) do |trial_pri|
if primes.find { |pri| (trial_pri % pri).zero? }.nil?
return enclose.call(primes << trial_pri)
end
end
primes
}).call([2])
end</lang>
By Regular Expression
<lang ruby>def isprime(n)
'1'*n !~ /^1?$|^(11+?)\1+$/
end</lang>
Scala
<lang scala>def isPrime(n: Int) = n > 1 && (Iterator.from(2) takeWhile (d => d * d <= n) forall (n % _ != 0))</lang>
Scheme
<lang scheme>(define (prime? number)
(define (*prime? divisor)
(or (> (* divisor divisor) number)
(and (> (modulo number divisor) 0)
(*prime? (+ divisor 1)))))
(and (> number 1)
(*prime? 2)))</lang>
Standard ML
<lang sml>fun is_prime n =
if n = 2 then true
else if n < 2 orelse n mod 2 = 0 then false
else let
fun loop k =
if k * k > n then true
else if n mod k = 0 then false
else loop (k+2)
in loop 3
end</lang>
Tcl
<lang tcl>proc is_prime n {
if {$n <= 1} {return false}
if {$n == 2} {return true}
if {$n % 2 == 0} {return false}
for {set i 3} {$i <= sqrt($n)} {incr i 2} {
if {$n % $i == 0} {return false}
}
return true
}</lang>
TI-83 BASIC
Prompt A If A=2:Then Disp "PRIME" Stop End If (fPart(A/2)=0 and A>0) or A<2:Then Disp "NOT PRIME" Stop End 1→P For(B,3,int(√(A))) If FPart(A/B)=0:Then 0→P √(A)→B End B+1→B End If P=1:Then Disp "PRIME" Else Disp "NOT PRIME" End
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
like joy
[prime?
2
[[dup * >] [true] [false] ifte [% 0 >] dip and]
[succ]
while
dup * <].
Using it
|11 prime? =true |4 prime? =false