# Is this number a prime?

Believe it or not, we do not yet have a code golf challenge for a simple primality test. While it may not be the most interesting challenge, particularly for "usual" languages, it can be nontrivial in many languages.

Rosetta code features lists by language of idiomatic approaches to primality testing, one using the Miller-Rabin test specifically and another using trial division. However, "most idiomatic" often does not coincide with "shortest." In an effort to make Programming Puzzles and Code Golf the go-to site for code golf, this challenge seeks to compile a catalog of the shortest approach in every language, similar to "Hello, World!" and Golf you a quine for great good!.

Furthermore, the capability of implementing a primality test is part of our definition of programming language, so this challenge will also serve as a directory of proven programming languages.

Write a full program that, given a strictly positive integer n as input, determines whether n is prime and prints a truthy or falsy value accordingly.

For the purpose of this challenge, an integer is prime if it has exactly two strictly positive divisors. Note that this excludes 1, who is its only strictly positive divisor.

Your algorithm must be deterministic (i.e., produce the correct output with probability 1) and should, in theory, work for arbitrarily large integers. In practice, you may assume that the input can be stored in your data type, as long as the program works for integers from 1 to 255.

### Input

• If your language is able to read from STDIN, accept command-line arguments or any other alternative form of user input, you can read the integer as its decimal representation, unary representation (using a character of your choice), byte array (big or little endian) or single byte (if this is your languages largest data type).

• If (and only if) your language is unable to accept any kind of user input, you may hardcode the input in your program.

In this case, the hardcoded integer must be easily exchangeable. In particular, it may appear only in a single place in the entire program.

For scoring purposes, submit the program that corresponds to the input 1.

### Output

Output has to be written to STDOUT or closest alternative.

If possible, output should consist solely of a truthy or falsy value (or a string representation thereof), optionally followed by a single newline.

The only exception to this rule is constant output of your language's interpreter that cannot be suppressed, such as a greeting, ANSI color codes or indentation.

• This is not about finding the language with the shortest approach for prime testing, this is about finding the shortest approach in every language. Therefore, no answer will be marked as accepted.

• Submissions in most languages will be scored in bytes in an appropriate preexisting encoding, usually (but not necessarily) UTF-8.

The language Piet, for example, will be scored in codels, which is the natural choice for this language.

Some languages, like Folders, are a bit tricky to score. If in doubt, please ask on Meta.

• Unlike our usual rules, feel free to use a language (or language version) even if it's newer than this challenge. If anyone wants to abuse this by creating a language where the empty program performs a primality test, then congrats for paving the way for a very boring answer.

Note that there must be an interpreter so the submission can be tested. It is allowed (and even encouraged) to write this interpreter yourself for a previously unimplemented language.

• If your language of choice is a trivial variant of another (potentially more popular) language which already has an answer (think BASIC or SQL dialects, Unix shells or trivial Brainfuck derivatives like Headsecks or Unary), consider adding a note to the existing answer that the same or a very similar solution is also the shortest in the other language.

• Built-in functions for testing primality are allowed. This challenge is meant to catalog the shortest possible solution in each language, so if it's shorter to use a built-in in your language, go for it.

• Unless they have been overruled earlier, all standard rules apply, including the http://meta.codegolf.stackexchange.com/q/1061.

As a side note, please don't downvote boring (but valid) answers in languages where there is not much to golf; these are still useful to this question as it tries to compile a catalog as complete as possible. However, do primarily upvote answers in languages where the author actually had to put effort into golfing the code.

### Catalog

The Stack Snippet at the bottom of this post generates the catalog from the answers a) as a list of shortest solution per language and b) as an overall leaderboard.

## Language Name, N bytes


where N is the size of your submission. If you improve your score, you can keep old scores in the headline, by striking them through. For instance:

## Ruby, <s>104</s> <s>101</s> 96 bytes


If there you want to include multiple numbers in your header (e.g. because your score is the sum of two files or you want to list interpreter flag penalties separately), make sure that the actual score is the last number in the header:

## Perl, 43 + 2 (-p flag) = 45 bytes


You can also make the language name a link which will then show up in the snippet:

## [><>](http://esolangs.org/wiki/Fish), 121 bytes


<style>body { text-align: left !important} #answer-list { padding: 10px; width: 290px; float: left; } #language-list { padding: 10px; width: 290px; float: left; } table thead { font-weight: bold; } table td { padding: 5px; }</style><script src="https://ajax.googleapis.com/ajax/libs/jquery/2.1.1/jquery.min.js"></script> <link rel="stylesheet" type="text/css" href="//cdn.sstatic.net/codegolf/all.css?v=83c949450c8b"> <div id="language-list"> <h2>Shortest Solution by Language</h2> <table class="language-list"> <thead> <tr><td>Language</td><td>User</td><td>Score</td></tr> </thead> <tbody id="languages"> </tbody> </table> </div> <div id="answer-list"> <h2>Leaderboard</h2> <table class="answer-list"> <thead> <tr><td></td><td>Author</td><td>Language</td><td>Size</td></tr> </thead> <tbody id="answers"> </tbody> </table> </div> <table style="display: none"> <tbody id="answer-template"> <tr><td>{{PLACE}}</td><td>{{NAME}}</td><td>{{LANGUAGE}}</td><td>{{SIZE}}</td><td><a href="{{LINK}}">Link</a></td></tr> </tbody> </table> <table style="display: none"> <tbody id="language-template"> <tr><td>{{LANGUAGE}}</td><td>{{NAME}}</td><td>{{SIZE}}</td><td><a href="{{LINK}}">Link</a></td></tr> </tbody> </table><script>var QUESTION_ID = 57617; var ANSWER_FILTER = "!t)IWYnsLAZle2tQ3KqrVveCRJfxcRLe"; var COMMENT_FILTER = "!)Q2B_A2kjfAiU78X(md6BoYk"; var OVERRIDE_USER = 12012; var answers = [], answers_hash, answer_ids, answer_page = 1, more_answers = true, comment_page; function answersUrl(index) { return "https://api.stackexchange.com/2.2/questions/" + QUESTION_ID + "/answers?page=" + index + "&pagesize=100&order=desc&sort=creation&site=codegolf&filter=" + ANSWER_FILTER; } function commentUrl(index, answers) { return "https://api.stackexchange.com/2.2/answers/" + answers.join(';') + "/comments?page=" + index + "&pagesize=100&order=desc&sort=creation&site=codegolf&filter=" + COMMENT_FILTER; } function getAnswers() { jQuery.ajax({ url: answersUrl(answer_page++), method: "get", dataType: "jsonp", crossDomain: true, success: function (data) { answers.push.apply(answers, data.items); answers_hash = []; answer_ids = []; data.items.forEach(function(a) { a.comments = []; var id = +a.share_link.match(/\d+/); answer_ids.push(id); answers_hash[id] = a; }); if (!data.has_more) more_answers = false; comment_page = 1; getComments(); } }); } function getComments() { jQuery.ajax({ url: commentUrl(comment_page++, answer_ids), method: "get", dataType: "jsonp", crossDomain: true, success: function (data) { data.items.forEach(function(c) { if (c.owner.user_id === OVERRIDE_USER) answers_hash[c.post_id].comments.push(c); }); if (data.has_more) getComments(); else if (more_answers) getAnswers(); else process(); } }); } getAnswers(); var SCORE_REG = /<h\d>\s*([^\n,<]*(?:<(?:[^\n>]*>[^\n<]*<\/[^\n>]*>)[^\n,<]*)*),.*?(\d+)(?=[^\n\d<>]*(?:<(?:s>[^\n<>]*<\/s>|[^\n<>]+>)[^\n\d<>]*)*<\/h\d>)/; var OVERRIDE_REG = /^Override\s*header:\s*/i; function getAuthorName(a) { return a.owner.display_name; } function process() { var valid = []; answers.forEach(function(a) { var body = a.body; a.comments.forEach(function(c) { if(OVERRIDE_REG.test(c.body)) body = '<h1>' + c.body.replace(OVERRIDE_REG, '') + '</h1>'; }); var match = body.match(SCORE_REG); if (match) valid.push({ user: getAuthorName(a), size: +match[2], language: match[1], link: a.share_link, }); else console.log(body); }); valid.sort(function (a, b) { var aB = a.size, bB = b.size; return aB - bB }); var languages = {}; var place = 1; var lastSize = null; var lastPlace = 1; valid.forEach(function (a) { if (a.size != lastSize) lastPlace = place; lastSize = a.size; ++place; var answer = jQuery("#answer-template").html(); answer = answer.replace("{{PLACE}}", lastPlace + ".") .replace("{{NAME}}", a.user) .replace("{{LANGUAGE}}", a.language) .replace("{{SIZE}}", a.size) .replace("{{LINK}}", a.link); answer = jQuery(answer); jQuery("#answers").append(answer); var lang = a.language; lang = jQuery('<a>'+lang+'</a>').text(); languages[lang] = languages[lang] || {lang: a.language, lang_raw: lang.toLowerCase(), user: a.user, size: a.size, link: a.link}; }); var langs = []; for (var lang in languages) if (languages.hasOwnProperty(lang)) langs.push(languages[lang]); langs.sort(function (a, b) { if (a.lang_raw > b.lang_raw) return 1; if (a.lang_raw < b.lang_raw) return -1; return 0; }); for (var i = 0; i < langs.length; ++i) { var language = jQuery("#language-template").html(); var lang = langs[i]; language = language.replace("{{LANGUAGE}}", lang.lang) .replace("{{NAME}}", lang.user) .replace("{{SIZE}}", lang.size) .replace("{{LINK}}", lang.link); language = jQuery(language); jQuery("#languages").append(language); } }</script>

• Is there a reason for the full program requirement, rather than allowing the full range of default input types? E.g. answering with a function that takes its input as an argument, is currently disallowed? codegolf.meta.stackexchange.com/questions/2447/… Commented Dec 12, 2017 at 6:21
• @LyndonWhite This was intended as a catalog (like “Hello, World!”) of primality tests, so a unified submission format seemed preferable. It's one of two decisions about this challenge that I regret, the other being only allowing deterministic primality tests. Commented Dec 12, 2017 at 12:51
• Could a case be made for locking this challenge and posting a new, less restrictive one? Commented Jun 25, 2018 at 12:59
• @Shaggy Seems like a question for meta. Commented Jun 25, 2018 at 13:44
• Yeah, that's what I was thinking. I'll let you do the honours, seeing as it's your challenge. Commented Jun 25, 2018 at 13:45

# Pharo, 7 bytes

isPrime


Pharo is an open source flavour of Smalltalk, a fork of Squeak. All integer instances understand the message isPrime and will respond with a boolean value. To quickly test this, simply select the text containing the number and the message (e.g. 1 isPrime) and "print-it" (or "inspect-it") from the context menu or via keyboard shortcut.

For anyone interested, here's the implementation of that method on the class Integer:

isPrime
"Answer true if the receiver is a prime number. See isProbablyPrime for a probabilistic
implementation that is much faster for large integers, and that is correct to an extremely
high statistical level of confidence (effectively deterministic)."

self <= 1 ifTrue: [ ^false ].
self even ifTrue: [ ^self = 2].
3 to: self sqrtFloor by: 2 do: [ :each |
self \\ each = 0 ifTrue: [ ^false ] ].
^true


So the receiver of this message first covers the base cases (numbers <= 1 aren't prime, even numbers are only prime if they're 2), then checks each odd number from 3 to the floor of its squareroot to see whether dividing itself by each of those numbers results in a zero remainder. If such a divisor is found, false is returned, otherwise true. As the method comment states, there is also a faster (but probabilistic) implementation.

And just for fun, because Smalltalk is so simple with collections, here's how to get a collection of prime numbers up to 100:

(1 to: 100) select: [:each | each isPrime]


The output would be:

#(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)

• I'm pretty sure (but unable to confirm) that the same method exists in Squeak, and probably several other flavours of Smalltalk. If so, please feel free to update this answer. Commented Sep 15, 2015 at 7:57

# JavaScript, 50 47 bytes

n=+prompt();for(i=2;n%i&&i*i<n;i++);n<3?n-1:n%i


This is a basic translation of @steveverrill's C answer :)

• "Write a full program that..." Commented Sep 11, 2015 at 15:10
• It is now a full program, and I have attributed you at the bottom of my answer :)
– Sam
Commented Sep 11, 2015 at 15:15
• I believe you need an alert or something to output from a full program in JS? Commented Sep 11, 2015 at 15:17
• @FryAmTheEggman I'd say that a web browser's console counts as a standard JS interpreter, and pretty much all consoles output the result. This outputs 0 (which is falsy) if the number is not prime. Commented Sep 11, 2015 at 16:14
• You could save three bytes by getting rid of the 'n' in prompt('n'). I believe moving the i++ into the condition, like so: for(i=2;n%i&&i*i++<n;); would also save a byte. Commented Sep 11, 2015 at 16:35

## Perl 6, 16 bytes

Using built-ins is allowed (even if makes for a boring answer), and this is about shortest solutions, so...

say is-prime get


get obtains a line from STDIN, is-prime when used on string converts input to integer, say outputs the boolean.

# Common Lisp, 111 88 bytes

(defun p(n c)(cond((= n 1)0)((= n c)1)((eq(rem n c)0)0)(t(p n(1+ c)))))(print(p(read)2))


Try it online!

I'm kinda new to Lisp, so this could probably be improved. It's a very straightforward algorithm.

Ungolfed and commented:

(defun primep (n c)                ; define a function p with args n and c
(cond                          ; conditional statement similar to switch-case
((= 1 n) 0)                ; if n is 1, return 0. Difficult to handle this.
((= n c) 1)                ; if c = n, then we have gotten all the way through
; without finding a single divisor. It's prime; return 1.
((eq (rem n c) 0) 0)       ; if n is evenly divisible by c, return 0
(t (primep n (1+ c)))      ; if all else fails, increment c and recurse
)
)
(print (primep (read) 2))          ; eval one line from stdin, call primep with an initial
; count of 2, and print the result


# C,59 bytes

main(i,j){scanf("%d",&i);for(j=2;i%j++;);printf("%d",j>i);}


It outputs 1 if the input is a prime and 0 otherwise.

• +1 This very nearly fails for an input of 1, where the for() loop will repeat almost forever. But eventually j will overflow and become negative, and then count back up again towards zero. When j==-1, i%j==0, so the loop exits and the program outputs 0 because 1>-1. This is fortunate because the next iteration would raise a divide-by-zero error. Commented Oct 13, 2015 at 19:04

# TeaScript, 5 bytes

$P(x)  Returns true if input is a prime and false of it is a composite number. Try it online here Does NOT work in Chrome. Input is given in the first field. # Carrot (version ^3), 3 bytes #^P  Basically takes the input (#) and checks if it is a prime number (P) or not. Note that this program takes the input as a string and,without converting it into a integer, it checks if the number is a prime or not. Try it online here. Although my programming language has been created after the challenge, it was not created to "abuse" this challenge. • This is scary good. +1 Commented Nov 4, 2015 at 20:29 ## Stack, 132 bytes: '' '' input num n set 2 i set { n i % 0 = n 1 = or { 0 print } { i n 2 - gt { 1 print } { i 1 + i set p } ifelse } ifelse } p def p  Your basic test every number from 2 to n primality test. • Unnecessary whitespace is unnecessary? '' input num n set 2 i set { n i % 0 = n 1 = or { 0 print } { i n 2 - gt { 1 print } { i 1 + i set p } ifelse } ifelse } p def p saves 10 bytes Commented Nov 6, 2015 at 17:00 • @towerofnix thanks Commented Nov 7, 2015 at 15:49 ## Minkolang, 29 5 bytes I have since added a built-in for primality testing (it uses a parallelized version of the Sieve of Sundaram). n2MN.  ### Explanation n Take integer from input 2M Push 1 if prime, 0 otherwise N. Output as integer and stop.  ### Old version (does not use a built-in): This was surprisingly long. I'll probably implement a built-in for this (since primality checking will probably be useful elsewhere). nd24&1-N.d2-[0ci2+%3&0N.]1N.  ### Explanation n Take integer as input. d24&1-N. Output 0 if 1, 1 if 2. d2-[ ] Trial division loop. 0ci2+% Check to see if it's divisible by <loop counter>. 3&0N. If so, output 0 and stop. 1N. It's prime! Output 1 and stop.  # pl, 1 byte ≡  Try it online. At the moment, pl uses a lazy trial division builtin. I'll replace it with something that doesn't suck later. • Is that really one byte? I count three but I'm not sure what the rules are. Commented Dec 21, 2015 at 16:56 • pl uses CP437 as its encoding. This char is one byte (0xF0) in it. Commented Dec 21, 2015 at 16:57 • And now I know a thing. Thank you! Commented Dec 21, 2015 at 17:08 ## ROOP, 17 bytes I w#H P w O  The w operator reads a number from the keyboard because it has an input object above (I). The input object moves to the right and the number created falls down. The P operator checks whether the number is prime and places a 1 or a 0 on the right (eliminating the number). Then the input object is moved to the right, the number created can not move anywhere. The operator H is activated because it has a object above and terminates the program at the end of all operators. The operator w, puts the number 1 or 0 in the display. # jq, 38 bytes . as$n|[range(2;.)]|all($n%.>0)and$n>1


Sample run:

bash-4.3$jq '. as$n|[range(2;.)]|all($n%.>0)and$n>1' <<< 2015
false


On-line test:

• [.>1,.%range(2;.)>0]|all works for 24: tio.run/##yyr8/z9az85QR0@1KDEvPVXDyFpP084gtiYxJ@f/…
– ovs
Commented Sep 5, 2021 at 9:52
• Wow! That's great @ovs. Sure you not want to post it as standalone solution? It's far better than mine to be mentioned in the same answer. Commented Sep 5, 2021 at 15:40

## F#, 99 88 bytes

[<EntryPoint>]
let f a=
let i=int a.[0]
Seq.forall((%)i>>(<)0)[2..i-1]|>printfn"%b";0


Explanation of the interesting part:

                      [2..i-1]              // Generate a list of possible divisors (from 2 to i-1) - for 2, this is an empty list.
Seq.forall(          )                      // Check if none of them are actual divisors, that is
(%)i>>(<)0                           // That i % it is greater than 0. This is equivalent to (fun d -> i % d > 0)
|>printfn"%b" // And print the answer as a boolean


Update: Turns out, the entry point doesn't have to be called "main", and the argument array doesn't have to be called "argv"! =)

# Oracle SQL 11.2, 73 bytes

SELECT SUM(DECODE(MOD(:1,LEVEL),0,1,0))-2 FROM DUAL CONNECT BY LEVEL<=:1;


Return 0 if prime, any other value is false

• A slightly smaller version SELECT MIN(MOD(:1,LEVEL+1))FROM DUAL CONNECT BY LEVEL<:1-1; that will return 0 or 1 Commented Apr 10, 2016 at 21:37

# Reng v.1, 36 39 bytes

This tests if the input is a prime. (+3 bytes because I forgot to check if input = 1)

i:11#x   eqv0n~
x1+#x:x,?v$\ ~nex$/


All because I forgot to implement member switching. I am rather proud of this, however, because it uses an interesting feature of Reng: you can redefine any character's meaning. In this case, we see constant redefinition.

i takes input, and 1 pushes 1. We also define x as 1 (1#x) and print zero and terminate if the input is 1 (eqv0n~). Then, the pointer is directed to the next line, x1+#x:x,?v$\. (The last character is a NOP for this instance.) For the first iteration, 1+ increments x, yielding 2. This is where our trial division starts. #x defines x to be the top element of the stack, in this case, 2. : duplicates the TOS, the input element, and x puts down its value, 2 in this case. , is the modulus operator, and yields i % x. If this is zero, we are directed downwards by v. Otherwise, we drop the modulus and redo the line again. When we are directed downward by v, we meet / which executes the line ~nex$, equivalent in a left-to-right form to $xen~. $ drops a value, x lays down x, and e checks for equality. If they are equal (in the case that x is the input), this is 1. Otherwise, this is 0.

## Test cases

The programs pictured are still valid, but fail to handle 1 correctly, unlike the program above.

Yes!

No!

# VHDL, 236 bytes

entity e is
port(n:natural;b:out bit);end;architecture a of e is
function p(n:natural)return bit is
begin
if n=1 then
return'0';end if;for i in 2 to n-1 loop
if n mod i=0 then
return'0';end if;end loop;return'1';end;begin
b<=p(n);end a;


The input n is an input port of the entity e; a natural number (starts at 0 for VHDL). b is an output port of entity e; a bit (meaning '1' or '0'). This works by the ever-famous method of trial division, with a special case for 1. This is what it looks like formatted nicely:

entity e is
port(   n : in natural;
b : out bit);
end;

architecture a of e is
function p(n:natural) return bit is
begin
if n=1 then
return '0';
end if;
for i in 2 to n-1 loop
if n mod i=0 then
return '0';
end if;
end loop;
return'1';
end;
begin
b<=p(n);
end a;


Here's the testbench I used for verification:

entity m is
end;

architecture a of m is
signal i_n : natural := 2;
signal i_b : bit;

type int_vector is array(natural range<>) of natural;

constant primes : int_vector := (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);

function is_prime(n : natural) return boolean is
begin
for i in primes'range loop
if n=primes(i) then
return true;
end if;
end loop;
return false;
end;
begin
x:entity work.e port map(n=>i_n, b=>i_b);
process
begin
for i in 1 to primes'right loop
i_n <= i;
wait for 10 ns;
if is_prime(i) then
assert i_b='1';
else
assert i_b='0';
end if;
end loop;
report "finished" severity error; -- error to stop simulation
end process;
end a;


This is the result.

• @ais523 except the question specifies that it must be a full program Commented May 11, 2017 at 16:30

## Labyrinth, 21 bytes

?
:
}()"{
{: *}{:*{%!


Terminates with an error, with the error message going to STDERR.

Try it online!

### Explanation

This uses the same approach based on Wilson's theorem as Sp3000's answer, but I managed to decompose his 3x3 loop into two 2x2 loops which is generally better for golfing Labyrinth.

Labyrinth primer:

• Labyrinth has two stacks of arbitrary-precision integers, main and aux(iliary), which are initially filled with an (implicit) infinite amount of zeros.
• The source code resembles a maze, where the instruction pointer (IP) follows corridors when it can (even around corners). The code starts at the first valid character in reading order, i.e. in the top left corner in this case. When the IP comes to any form of junction (i.e. several adjacent cells in addition to the one it came from), it will pick a direction based on the top of the main stack. The basic rules are: turn left when negative, keep going ahead when zero, turn right when positive. And when one of these is not possible because there's a wall, then the IP will take the opposite direction. The IP also turns around when hitting dead ends.

Now we can look at the code. The program starts with a short linear (vertical) bit:

?   Read input N as an integer and push onto main.
:   Duplicate N.
}   Move one copy over to aux.


The IP is now at a junction, and in fact the small 2x2 block acts as a loop which is traversed in clockwise order:

(   Decrement N. If it hits zero, we exit the loop.
:   Duplicate N.
{}  Move a value over from aux and push it back. Together this does nothing.


So this loop leaves all the numbers from N-1 down to 0 on the main stack. Now there's the single ), which increments that 0 back up to 1, such that all the non-zero numbers on top of the stack will multiply together to (N-1)! (including the special case of 0! == 1).

The next 2x2 block is a loop which performs this multiplication and its traversed in counter-clockwise order. The tricky part is that we need to check the value below the product to see if we're done, otherwise we'd end up losing the product by multiplying it with an implicit 0 below.

"   No-op, does nothing.
*   Multiply the top two values on main.
}   Move the product over to aux, exposing the value below. If that's zero,
we exit the loop.
{   Pull the product back onto main for the next iteration.


Now the main stack is empty and the aux stack contains (N-1)! on top of N. Time to wrap things up:

{   Pull (N-1)! over from aux.
:*  Duplicate and multiply, squaring the factorial.
{   Pull N over from aux.
%   Take modulo, computing (N-1)!^2 % N.
!   Output the result.


Now the IP hits a dead end and has to turn around. The next thing it tries to execute is % but that would now attempt to compute 0 % 0 which terminates with a division-by-zero error.

# Fuzzy Octo Guacamole, 21 bytes

^-!.1C[d2ss.p*.+]s.Zm


Outputs 0 for false, 1 for true. Thanks to @xnor for the algorithm.

Explanation:

^-!.1C[d2ss.p*.+]s.Zm
^                      # Get Input
-                     # Decrement
!                    # Set for-loop counter
.                   # Swap stacks
1C                 # Set both stacks to [1].  (push 1 and copy to other stack)
[         ]      # For loop for duration of input - 1
d2              # Duplicate the top of stack and push 2 after
ss            # Put them on the other stack, in reverse order.
.           # Switches stacks
p          # Power function. x^y (ToS^2)
*         # Multiply
.+       # Move back to the other stack and increment
]      # (repeat)
s.    # Move the ToS over and go with it
Z   # Reverse the stack.
m  # Modulus, x%y.
# (implicit output)


We start with each stack having 1. Let the bottom of one stack be n, the other bottom be P. We multiply P by n^2, and then increment n. Do this n-1 times. Then take P%n.

# WistfulC, 381 bytes

if only int n were 0...
wish for "%d", &n upon a star
if n < 2 were true...
wish "not prime" upon a star
if wishes were horses...
*sigh*
if only int i were 2...
someday i will be n...
if n % i were 0...
wish "not prime" upon a star
if wishes were horses...
*sigh*
if only i were i + 1...
*sigh*
wish "prime" upon a star
if wishes were horses...


Outputs prime if prime, not prime if not prime.

I went for entertainment value more than small size. Here's a golfed, (arguably) less funny version (286 bytes):

if only int n were 0...wish for "%d",&n upon a star
if n<2...wish "0" upon a star
if wishes were horses...*sigh*if only int i were 2...someday i==n...if !(n%i)...wish "0" upon a star
if wishes were horses...*sigh*if only i were i+1...*sigh*wish "1" upon a star
if wishes were horses...

• You can save some bytes by return truthy/falsy values? Commented Jul 21, 2016 at 17:55

# Sesos, 40 39 bytes

0000000: 16f0be afcf9c 37fcfe 8c19d7 c671d7 668ee3 f57b33  ......7......q.f...{3
0000015: 877bc6 662edb b961ba 8763bc 666e3c 66ec01         .{.f...a..c.fn<f..


Try it online! Check Debug to see the generated binary code.

### Background

To identify primes, we use a corollary of Wilson's theorem:

### How it works (WIP)

The binary file above has been generated by assembling the following SASM code.

set numin
set numout

get
jmp
jmp, fwd 1, add 1, fwd 1, add 1, fwd 1, add 1, rwd 3, sub 1, jnz
fwd 1, sub 1, fwd 2
jmp, rwd 3, add 1, fwd 3, sub 1, jnz
rwd 1, sub 1
jnz
rwd 1, add 1, rwd 2
jmp
fwd 1
jmp, fwd 2, add 1, rwd 2, sub 1, jnz
fwd 1
jmp
fwd 1
jmp, fwd 1, add 1, rwd 3, add 1, fwd 2, sub 1, jnz
fwd 1
jmp, rwd 1, add 1, fwd 1, sub 1, jnz
rwd 2
sub 1
jnz
fwd 1
get
rwd 4
jnz
fwd 2
jmp
rwd 1, sub 1, rwd 1, add 1, fwd 1
jmp, rwd 1, jnz
rwd 1
jmp, fwd 1, add 1, rwd 1, sub 1, jnz
fwd 2
jmp, fwd 1, jnz
rwd 1, sub 1
jnz
rwd 2
put


# Idris, 78 bytes

main:IO()
main=let S n=length!getLine in print$0<(pow(product[1..n])2modS n)  This takes input as a unary string (it uses the string length as input). Compile this with idris -O3 or you’ll spend forever in Nat-computey hell. # Python 3.5, 5852 58 bytes import math;n=int(input());print(math.factorial(n-1)**2%n)  I unassumed. Thanks Dennis. I assumed something about the way the input would be given. Thanks daHugLenny. import math;lambda n:print(math.factorial(n-1)**2%n) • You can save 13 bytes by using a lambda: import math;lambda n:math.factorial(n-1)**2%n Commented Aug 7, 2016 at 2:21 • In this particular challenge, lambdas are not allowed. The spec says Write a full program that... (CC @daHugLenny) Commented Aug 10, 2016 at 2:57 # PHP, 51 bytes An alternative PHP answer, but required the GMP extension to be installed: <?=gmp_strval(gmp_nextprime($argv[1]-1))==$argv[1];  Simply subtracts 1 from the input and compares the nextprime result against the input • gmp_cmp can save 3 bytes. But unfortunately, that algorithm is only probably correct, not always. +1 anyway for showing it off. :) Commented Oct 16, 2016 at 14:14 ## Clojure, 98 bytes (let[n(Integer/parseInt(read-line))](println(and(not= 1 n)(not(some #(=(rem n %)0)(range 2 n))))))  Uses some to check if the number has any divisors, then negates the result to indicate whether or not it's prime. A simple function would have only been 59 bytes :( (fn[n](and(not= 1 n)(not(some #(=(rem n %)0)(range 2 n)))))  Ungolfed: (let [n (Integer/parseInt (read-line))] (println (and (not= 1 n) ; 1 is an unfortunate special case (not ; Negate to indicate primality (some #(= (rem n %) 0) ; Check if n has any divsors... (range 2 n)))))) ; in the range of 2 to (n-1)  There's a previous Clojure answer, but this beats it by a little over 60 bytes • You can also turn Integer/parseInt to read-string to save a further 5 bytes. Commented Jul 9, 2017 at 11:38 • 1: You can replace (not= 1 n) with (> n 1) to handle the edgecase of 1. 2: You can replace (not (some ...)) with (every? ...) and change the condition around a bit. 3: You can swap the fn[n] and the shorthand around to remove a space, like so: #(and(> % 1)(every?(fn[a](>(rem % a)0))(range 2 %))). These three save 7 bytes. Commented Jul 9, 2017 at 11:46 • @Qwerp-Derp Thanks. I'll try to remember to update my answer later today. Commented Jul 9, 2017 at 13:09 • @Carcigenicate Here's a quick reminder. :P Commented Nov 17, 2017 at 0:30 • @totallyhuman LOL. JULY. I clearly dropped that ball. I hate to delay this again, but I'm not quite on the mental state to deal with this right now. I think I'll set a reminder on my phone Commented Nov 17, 2017 at 0:31 # Clojure, 124 Full program, reading standard input and writing to standard output as required by the terms. (let[n(read-string(first *command-line-args*))](print(and(> n 1)(=(reduce #(* %1(if(=(mod n %2)0)0 1))1(range 2 n))1))))  Obviously inefficient to test all the way up to n but trying to be clever with (range 2 (inc (int (Math/sqrt n)))) adds length. Ungolfed version: (let [n (read-string (first *command-line-args*))] (print (and (> n 1) (= (reduce #(* %1 (if (= (mod n %2) 0) 0 1) ) 1 (range 2 n) ) 1 ) ) ) )  Put in file prime.clj, execute as: java -cp location-of-clojure.jar-in-your-system clojure.main prime.clj 1  E.g. in my system I get: $ java -cp ../../repo-wide-libs/clojure.jar clojure.main prime.clj  1
false
$java -cp ../../repo-wide-libs/clojure.jar clojure.main prime.clj 7 true$ java -cp ../../repo-wide-libs/clojure.jar clojure.main prime.clj  8
false

• Is there a way to remove any of those spaces? Since this is Lisp-like, you could probably take out the ones before parentheses. Commented Sep 28, 2015 at 13:37
• yeah so this overflows easily, just uploaded version that doesn't overflow (3 chars longer) Commented Jan 6, 2017 at 19:26

# APL (NARS2000), 6 bytes

Function: 0∘π

Program: 0π⎕

# Taxi, 1519 1309 bytes

Go to Post Office:w 1 l 1 r 1 l.Pickup a passenger going to The Babelfishery.Go to The Babelfishery:s 1 l 1 r.Pickup a passenger going to Cyclone.Go to Cyclone:n 1 l 1 l 2 r.Pickup a passenger going to Cyclone.Pickup a passenger going to The Underground.Go to Zoom Zoom:n.[a]Go to Cyclone:w.Pickup a passenger going to Sunny Skies Park.Pickup a passenger going to Divide and Conquer.Go to Sunny Skies Park:n 1 r.Go to The Underground:s 1 l 1 r 2 l.Switch to plan "1" if no one is waiting.Pickup a passenger going to Cyclone.Go to Cyclone:n 3 l 2 l.Pickup a passenger going to Divide and Conquer.Pickup a passenger going to The Underground.Go to Divide and Conquer:n 2 r 2 r 1 r.Pickup a passenger going to Cyclone.Go to Cyclone:e 1 l 1 l 2 l.Pickup a passenger going to Trunkers.Pickup a passenger going to Equal's Corner.Go to Trunkers:s 1 l.Pickup a passenger going to Equal's Corner.Go to Equal's Corner:w 1 l.Switch to plan "b" if no one is waiting.Go to The Underground:n 3 r 1 r 2 l.Switch to plan "z" if no one is waiting.[1]'0' is waiting at Writer's Depot.[z]'1' is waiting at Writer's Depot.Go to Writer's Depot:n 3 l 2 l.Pickup a passenger going to Post Office.Go to Post Office:n 1 r 2 r 1 l.[b]Go to Sunny Skies Park:n.Pickup a passenger going to Cyclone.Go to Zoom Zoom:n 1 r.Switch to plan "a".


Try it online!

[ Test for Primality ]
[ Inspired by: https://codegolf.stackexchange.com/q/57617 ]

[ Pickup stdin and triplicate it]
Go to Post Office: west 1st left 1st right 1st left.
Pickup a passenger going to The Babelfishery.
Go to The Babelfishery: south 1st left 1st right.
Pickup a passenger going to Cyclone.
Go to Cyclone: north 1st left 1st left 2nd right.
Pickup a passenger going to Cyclone.
Pickup a passenger going to The Underground.
Go to Zoom Zoom: north.

[a]
Go to Cyclone: west.
Pickup a passenger going to Sunny Skies Park.
Pickup a passenger going to Divide and Conquer.

[ Store a copy of the original stdin ]
Go to Sunny Skies Park: north 1st right.

[ Iterate down to the next lowest number ]
[ If the input is 1, the switch will immediately output '0' ]
Go to The Underground: south 1st left 1st right 2nd left.
Switch to plan "1" if no one is waiting.
Pickup a passenger going to Cyclone.
Go to Cyclone: north 3rd left 2nd left.
Pickup a passenger going to Divide and Conquer.
Pickup a passenger going to The Underground.

[ Divide the original by the current iteration and check if it's an integer ]
Go to Divide and Conquer: north 2nd right 2nd right 1st right.
Pickup a passenger going to Cyclone.
Go to Cyclone: east 1st left 1st left 2nd left.
Pickup a passenger going to Trunkers.
Pickup a passenger going to Equal's Corner.
Go to Trunkers: south 1st left.
Pickup a passenger going to Equal's Corner.
Go to Equal's Corner: west 1st left.
Switch to plan "b" if no one is waiting.

[ Someone was waiting so it was an integer result ]
[ This is going to eventually happen when we divide by one ]
[ If the current iteration is 1, we want to return 1 as a truthy result ]
[ It it's anything higher than 1, we want to return 0 as a falsey result ]
Go to The Underground: north 3rd right 1st right 2nd left.
Switch to plan "z" if no one is waiting.

[1]
'0' is waiting at Writer's Depot.
[z]
'1' is waiting at Writer's Depot.
Go to Writer's Depot: north 3rd left 2nd left.
Pickup a passenger going to Post Office.
Go to Post Office: north 1st right 2nd right 1st left.
[ No need to return to taxi garage or even switch to the end of the program ]
[ It will output to stderr because it can't drive in the next direction and terminate the program ]

[b]
[ No one was waiting so it was not an integer result ]
[ Continue on to the next iteration ]
Go to Sunny Skies Park: north.
Pickup a passenger going to Cyclone.
Go to Zoom Zoom: north 1st right.
Switch to plan "a".


The only thing that felt golf-y when writing it was allowing it to error out instead of terminate properly.

# Jolf, 2 bytes

Try it here!

m{


Simple. m{ is the isPrime function of the math module, and j is the (implicit) user input as a number.

# Swift 4, 58 bytes

let n=Int(readLine()!)!;print((1..<n).filter{n%\$0<1}==[1])


Try it online!

Saved a few bytes thanks to Dennis.

## Wumpus, 13 bytes

I=&l2-&*=*r%O


Try it online!

### Explanation

Uses Wilson's theorem in the form (n-1)!2isprime(n) (mod n) (where isprime gives 0 or 1, respectively).

The implementation doesn't really work the way it's supposed to for n = 1, but the result still ends up being 0.

I=  Read n and duplicate it.
&l  Push the stack depth n times. Since there's another copy of n on the stack to
begin with, this pushes 1, 2, ..., n-1, n.
2-  Subtract 2 from n.
&*  Multiply the top two numbers that many times. This essentially folds
multiplication over the range [1,n-1], which computes (n-1)!. At
the end of this, the stack will be [n, (n-1)!]. However, in the case
of n = 1, we only ever pushed one element for the range, so there's
nothing left to form the factorial and the stack contains only one
copy of 1.
=*  Square the (n-1)!.
r   Reverse the stack. This is normally just used to swap n and (n-1)!²,
but we're using stack reversal instead of swap (~), so that the stack
remains unchanged for n = 1.
%   Modulo. For n = 1, this computes 0 % 1 = 0 (because there's an implicit
zero underneath the 1 on the stack). For all other n, this computes
(n-1)!² % n, i.e. the result of the primality test.
O   Output the result as a decimal integer.


The IP then bounces off the end of the program and starts moving left again. % tries another modulo, but there's only implicit zeros on the stack, so the program ends due to the attempted division by zero.