Find the Nth pair of twin primes

Question

Two prime numbers are defined as twin primes if they differ by two. For example, 3 and 5 are twin primes as are 29 and 31.

Write a program that finds the nth pair of twin primes (where n comes from STDIN) and prints them on STDOUT, separated by a comma and a space. This is code-golf, so the shortest code wins.

Sample input:

3

Sample output:

11, 13

Answer 1

Haskell 118

main=putStrLn.(!!)[show n++", "++show(n+2)|n<-[2..],all((>0).rem n)[2..n-1],all((>0).rem(n+2))[2..n]].(+)(-1)=<<readLn

Brute-force all twin primes and prints the n^th pair.

Answer 2

CJam, 29 26 bytes

Y4]{{:)_{mp}/&!}g}q~*", "*

Try it online.

Examples

$ for i in {1..10}; do cjam twin-primes.cjam <<< $i; echo; done
3, 5
5, 7
11, 13
17, 19
29, 31
41, 43
59, 61
71, 73
101, 103
107, 109

How it works

Y4]        " Push [ 2 4 ].                                                            ";
{          "                                                                          ";
  {        "                                                                          ";
    :)     " Increment each integer in the array.                                     ";
    _      " Duplicate the array.                                                     ";
    {mp}/  " For each integer in the array, push 1 if it's prime and 0 otherwise.     ";
    &!     " Compute the logical NOT of the bitwise AND of the two previous integers. "; 
  }g       " If the result is non-zero, repeat the loop.                              ";
}q~*       " Do the above “N” times, where “N” is the integer read from STDIN.        ";
", "       " Join the array by comma and space.                                       ";

Answer 3

Perl, 101 87

87 characters, building over aschepler's comment

$n=pop;$r='^1$|^(11+?)\1+$';($t=1x$s)=~$r||"11t"=~$r||--$n||die"$s, ",$s+2,$/while++$s

101 characters, earlier answer

$n=pop;$r=qr/^1$|^(11+?)\1+$/;(1x$s)!~$r&&(1x($s+2))!~$r&&++$i==$n&&say($s,", ",$s+2)&&exit while++$s

Usage:

$ perl ./twin_primes.pl 10
107, 109

Explanation

$n = pop;                 # Pulls twin prime pair counter from @ARGV
$r = qr/^1$|^(11+?)\1+$/; # The money line - a regex that verifies
                          # if a string of 1's has non-prime length

while ( ++$s ) {          # Loop over integers

                          # '&&' short-circuits
    (1 x  $s    ) !~ $r   # Negated regex match evaluates to true if $s is prime
 && (1 x ($s+2) ) !~ $r   # Same for $s + 2
 &&          ++$i == $n   # Counter to control which pair to print
 && say( $s, ", ", $s+2 ) # Print the line
 && exit                  # Terminate program
}

The workings of the non-primality regex is explained in this SO question.

Answer 4

C: 113

n,c,l;main(i){for(scanf("%d",&n),l=2;n;l=c==i?n-=i==l+2,i:l,i+=2)for(c=2;c<i&&i%c++;);printf("%d, %d\n",l-2,l);}

Sample run:

$ for i in $(seq 1 10); do echo $i | ./twinprimes; done
3, 5
5, 7
11, 13
17, 19
29, 31
41, 43
59, 61
71, 73
101, 103
107, 109

Thanks for help from Dennis, bebe, and Alchymist.

Answer 5

CJam - 26

1e4,{mp},_2f-&qi(=_2+", "\

It works for primes smaller than 10000; you can replace 4 with a higher exponent for larger numbers (potentially up to 10²⁰), but the program will get slower and will use more memory.

Try it at http://cjam.aditsu.net/

Explanation:

1e4, creates the array [0 1 2 ... 9999]
{mp}, selects only the prime numbers
_2f- copies the array and subtracts 2 from each item
& intersects the two arrays, thus finding the lower primes from each twin prime pair
qi reads the input and converts to integer
(= adjusts the index and gets the corresponding (lower) twin prime from the array
_2+ copies the prime and adds 2
", "\ puts the comma and space between the two primes

Answer 6

Mathematica - 63 characters

Print[#-2,", ",#]&@Nest[NestWhile[NextPrime,#,#2-#!=2&,2]&,1,n]

Notes

This is in fact a rather straightforward implementation. Shortening resulted in almost no obfuscation.

NextPrime is a builtin that finds the next prime after a number.

NestWhile[NextPrime,#,#2-#1!=2&,2]& is an anonymous function that finds the larger prime of the next twin prime pair after a number.

Nest applies this anonymous function n times.

Print[#-2,", ",#]& is an anonymous function that prints to stdout according to the specifications. Sadly this alone takes up 18 characters of the 63 character solution.

Example

In[1]:= Do[                                                                     
         Print[#-2,", ",#]&@Nest[NestWhile[NextPrime,#,#2-#!=2&,2]&,1,n],
         {n, 1, 10}
        ]
3, 5
5, 7
11, 13
17, 19
29, 31
41, 43
59, 61
71, 73
101, 103
107, 109

---

Update: Two characters could be saved by reimplementing this CJam solution. However, this algorithm limits the maximum value of n. Just replace the Nest... part by Intersection[#,#-2][[5]]&@Prime@Range[999]

Answer 7

Javascript (E6) 92 96

Shorter and compliant - use spidermonkey shell to read stdin/write stdout (with comma and space). It finds the 10000th pair 1260989, 1260991 in under a minute on my PC
Could be shorter using p[n]=o=n instead of p.push(o=n), so that the p array is sparse. But that's quite slower, and I'm not going to win for code length anyway.

m=readline();for(n=3,o=p=[];m;n+=2)p.every(e=>n%e)&&(m-=n-o<3,p.push(o=n));print(o-2+', '+o)

To try in firefox console:

m=prompt();for(n=3,o=p=[];m;n+=2)p.every(e=>n%e)&&(m-=n-o<3,p.push(o=n));alert(o-2+', '+o)

Ungolfed

A function that found all first m twins (returns the largest value):

T=m=>{
  for (o=n=3, p=[2], t=[]; !t[m-1]; n+=2)
    p.every(e => n%e) && (n-o-2 ? 0 : t.push(n), p.push(o=n))
  return t
}

Example: console.log(T(50))

[5, 7, 13, 19, 31, 43, 61, 73, 103, 109, 139, 151, 181, 193, 199, 229, 241, 271, 283, 313, 349, 421, 433, 463, 523, 571, 601, 619, 643, 661, 811, 823, 829, 859, 883, 1021, 1033, 1051, 1063, 1093, 1153, 1231, 1279, 1291, 1303, 1321, 1429, 1453, 1483, 1489]

Just the last:

L=m=>{
  for (o=n=3,p=[2]; m; n+=2)
    p.every(e => n%e) && (m -= n-o==2, p.push(o=n))
  return o
}

Then, take that 2 lines and add IO

m = prompt()
for (o=n=3, p=[2]; m; n+=2)
  p.every(e => n%e) && (m -= n-o==2, p.push(o=n))
alert('o-2+', '+o)

Answer 8

J - 49 60 55 51 bytes

I decided to go with a simple approach. Function t finds the next twin prime given a prime number as input (now this is included in the f function). Function f finds the nth twin prime. This also happens to be the first actual program I've written in J.

f=:[:(":,', ',":@+&2)(4&p:(,{~-=2:)])^:_@>:^:(]`2:)

Examples:

   f 1
3, 5
   f 2
5, 7
   f 3
11, 13
   f 4
17, 19
   f 5
29, 31
   f 100000
18409199, 18409201

Just for some eyebrowraises, have the ungolfed version.

twin =: (4&p:)(($:@[)`(,)@.(=(]+2:)))]
f    =: ((]-2:),])((0:{twin) ^: (]`(2:)))

Explanation:

f=:[:(":,', ',":@+&2)(4&p:(,{~-=2:)])^:_@>:^:(]`2:)
                     (4&p:(,{~-=2:)])^:_@>:^:(]`2:)
                                        @>:^:(]`2:)  main loop
                                           ^:(]`2:)  Repeat n times, starting with value of 2
                                        @>:          Add one to the current value and apply to the following function.
                     (4&p:(,{~-=2:)])^:_             Get the next twin prime
                                     ^:_             Recurse until there's no change
                          (,{~-=2:)                  If next prime - current value == 2, return current value, otherwise the next prime.
                      4&p:                           Get the next prime
     (":,', ',":@+&2)                                Format the output and add 2 to the second value.
   [:                                                Apply the twin prime to the formatter.

Basically, if n is 4, this creates a recursion tree like this:
 let T be the recursion inside t
 and numbers between rows the return values of according function
   (t * n) 3
-> (t * 4) 3
-> t  t  t  t  3
   17 11 5  3
-> (T  T) (T  T) T  T  3
    17 13  11 7  5  3
-> 17

Answer 9

C#, 265

using System.Linq;class P{static void Main(string[] args){var i=int.Parse(args[0]);int f=0,c=0;for(int j=1;;j+=2){var b=(Enumerable.Range(1,j).Count(x=>j%x==0)==2);if(f==0 && b){f=j;continue;}if(b){c++;if(c==i){System.Console.WriteLine(f+","+j);break;}j-=2;}f=0;}}}

Answer 10

Ruby 94

require'mathn'
n=gets.to_i
a=1
(a+=2;a.prime?&&(a+2).prime?&&n-=1)while n>0
$><<"#{a}, #{a+2}"

Online test: http://ideone.com/B2wxnG

Answer 11

Perl, 100 95

$n=<>;$i=3;while($c<$n&&($l=$i++)){$i++until!grep{$i%$_<1}(2..$i-1);$c++if$i-$l<3}print"$l, $i"

Ungolfed:

$n = <>;          # Read from STDIN
$i = 3;           # Tiny hack because I know I don't need the number 2
while ($c<$n && ($l = $i++)) {   # $c counts the pairs, $l is the last prime
  $i++ until ! grep {$i%$_<1} (2..$i-1);   # Increase $i until it's not divisible by anything
  $c++ if $i-$l < 3   # If $i and $l are twin primes, count it
}
print "$l, $i"    # That damned comma added a whole character to my code!

Answer 12

T-SQL (2008+): 344

Brute force a CTE to find primes, window function to count n, followed by a join to find the twin. Works in a second for outputs < 1,000, just under a minute for outputs < 10,000.

Golfed (SQLFiddle here):

WITH x(i) AS(SELECT 99 UNION ALL SELECT i-2
FROM x WHERE i>3),z AS(SELECT RANK()OVER(ORDER BY x.i)n,x.i
FROM x x LEFT JOIN x y ON x.i%y.i=0 AND y.i NOT IN(x.i,1)
WHERE y.i IS NULL)SELECT LTRIM(a)+', '+LTRIM(b)FROM(SELECT RANK()
OVER(ORDER BY x.i)n,x.i a,y.i b FROM z x,z y WHERE x.n=y.n-1
AND x.i=y.i-2) o WHERE n=3
OPTION(MAXRECURSION 0)

Legible:

WITH x(i) AS (
   SELECT 99
    UNION ALL
   SELECT i-2
   FROM x
   WHERE i > 3
)
,z AS (
SELECT RANK()OVER(ORDER BY x.i)n,x.i
FROM x x
WHERE NOT EXISTS
  (SELECT *
   FROM x y
   WHERE x.i%y.i = 0
    AND y.i NOT IN (x.i,1)
  )
)
SELECT LTRIM(a)+', '+LTRIM(b)
FROM (
    SELECT RANK()OVER(ORDER BY x.i)n,x.i a, y.i b
    FROM z x, z y
    WHERE x.n = y.n+1
    AND x.i = y.i+2
) o
WHERE n = 3
OPTION(MAXRECURSION 0)

Answer 13

GolfScript 46

~[1 3]\{\{))}%.{:x,{)x\%!},,2=}/*@\-.}do;', '*

Online test: link

Annotated code:

~                       # parse the input as an int
[1 3]                   # add the array [1, 3] on the stack
\                       # invert the items on the stack
{                       # begin loop
  \                     # bring the array to the top of the stack
  {))}%                 # add 2 to each of the numbers in the array
  .{:x,{)x\%!},,2=}/    # check if numbers are prime (leaves a 0 or 1 for both numbers on the stack)
  *                     # multiply the two 0/1 numbers (will only get 1 if both are 1)
  @\-                   # subtract the result from the inital int
  .                     # copy the new int value on the stack to be consumed by the 'do' loop
}do                     # repeat until the initial int was taken down to 0
                        # at this point the array contains the two numbers we're looking for
;                       # get rid of the 0 from the stack
', '*                   # format the output

Answer 14

PHP 5.4, 223

Not a smaller one, But one try from php.

$n=$argv[1];function i($k){for($i=2;$i<=(int)($k/2);$i++)if($k%$i==0)return 0;return 1;}function t($t){return (i($t) && i($t+2))?1:0;}$g=1;$d=0;do{if(t($g)==1){if($d<$n){$d++;}else{print_r([$g,$g+2]);break;}}$g++;}while(1);

Answer 15

C 309

Keeps getting next primes and store odd and even terms then checks if the difference is two.

int main()
{
int n;
scanf("%d",&n);
int a=2,b=3,k=2,q;
int odd=1;
int p;
if(n>0)
{
while(n)
{
k++;
p=1;
q=ceil(sqrt(k));
for(int i=2;i<=q;i++)
{
if(k%i==0)
{
p=0;
break;
}
}
if(p)
{
if(odd%2==0)a=k;
else b=k;
if(abs(a-b)==2)n--;
odd++;
}
}
}
printf("%d %d\n",a,b);
return 0;
}

Answer 16

R, 91 chars

a=scan();n=1;p=5;while(n!=a){p=p+1;q=p-2;if(sum(!p%%2:p,!q%%2:q)<3)n=n+1};cat(q,p,sep=", ")

Nothing really fancy:

a=scan()
n=1
p=5
while(n!=a){
    p=p+1
    q=p-2
    if(sum(!p%%2:p,!q%%2:q)<3) # Check that p and q are both primes by checking
       n=n+1                   # the number of zeroes resulting from 
}                              # p modulo each integers 2 to p and same for q
cat(q,p,sep=", ")

Usage:

> a=scan();n=1;p=5;while(n!=a){p=p+1;q=p-2;if(sum(!p%%2:p,!q%%2:q)<3)n=n+1};cat(q,p,sep=", ")
1: 10
2: 
Read 1 item
107, 109

Answer 17

Japt, 23 19 bytes

-4 bytes thanks to Shaggy

@µX°j ©°Xj}f
[UU+2]

Run it online

Answer 18

JavaScript (Node.js), 162 chars

Reads from stdin, outputs to stdout, exits "early" for input <= 0.

t=process.argv[2],c=0,l=1;if(t>0){for(i=0;;i++){p=!Array(i+1).join(1).match(/^1?$|^(11+?)\1+$/);if(p){if(i-2==l){if(c>=t-1){console.log(l+", "+i);break}c++}l=i}}}

Usage (script above saved as ntp.js):

>for /l %x in (0, 1, 10) do node ntp.js %x
>node ntp.js 0
>node ntp.js 1
3, 5
>node ntp.js 2
5, 7
>node ntp.js 3
11, 13
>node ntp.js 4
17, 19
>node ntp.js 5
29, 31
>node ntp.js 6
41, 43
>node ntp.js 7
59, 61
>node ntp.js 8
71, 73
>node ntp.js 9
101, 103
>node ntp.js 10
107, 109

Answer 19

AWK - 129

The file fsoe-pairs.awk:

{n=2;N=1
for(;;){if(n in L){p=L[n];del L[n]}else{p=n
if(n-N==2)if(!--$0){print N", "n;exit}N=n}P=p+n++
while(P in L)P+=p;L[P]=p}}

Running it:

$ awk -f fsoe-pairs.awk
1
3, 5
$ awk -f fsoe-pairs.awk
2
5, 7
$ awk -f fsoe-pairs.awk
10
107, 109

(1st line after command is input, the 2nd one is output)

This is based on an own prime generator algorithm I call "floating sieve of erastosthenes" (until I find it described elswhere) which only stores the needed part of the sieve and the already calculated primes.

Answer 20

Python 2 (75)

c=input()
n=3
while c:n+=2;c-=all(n%i&~2for i in range(2,n-2))
print(n-2,n)

So what's going on here?

First, let's look at the expression all(n%i&~2for i in range(2,n-2)), which checks if (n-2,n) are a pair of twin primes.

The simpler expression all(n%i for i in range(2,n)) simply checks if n is prime by trying every divisor i in the range 2<=i<=n-1, and seeing if all remainders are nonzero. This all checks exactly this, since Python treats 0 as False and all other numbers as True.

Now, observe that (n-2)%i==0 exactly when n%i==2 for divisors i>2. So, we can perform the primality check on n and n-2 at the same time by checking the remainders for both 0 and 2. This could be done as all(n%i not in [0,2] for i in range(2,n-2)). We only try divisors in the range 2<=i<=n-3 for the sake of n-2, but this suffices for n as well since n-1 and n-2 can't be divisors unless n<=4. We will only try odd n starting from 5 to avoid this complication and that of the divisor i=2.

We golf the expression n%i not in [0,2] into n%i&~2, remembering that 0 is False and other numbers are True. The operator precedence (n%i)&(~2) is exactly what's needed. The bit-complement ~2 is ...11111101, so its bitwise and with a number zeroes-out the 2's binary place value. This gives 0 (i.e., False) only for 0 and 2, exactly what we want.

Phew! Now we have that the expression all(n%i&~2for i in range(2,n-2)) checks whether n is the upper number of a twin prime pair. What remains is to iterate over them until we see c of them, where c is the inputted number. We start with 5 and counting up by 2 to avoid divisor issues. We decrement c each time we encounter an n that works, stopping when c=0. Finally, we print the twin prime pair that we end with.

Answer 21

T-SQL(2012+),255 characters

A more compact T-SQL twin prime finder that also gets a little bit of a speed up.

with t(n)as(select 2+number from spt_values where type='p')select*from(select concat(b,', ',a),rank()over(order by a)from(select n,lag(n)over(order by n)from t where not exists(select*from t f where f.n<t.n and t.n%f.n=0))z(a,b)where a=b+2)r(s,k)where k=2

Human readable format::

    with t(n)as(
        select 2+number 
        from spt_values 
        where type='p'
    )
    select *
    from(
        select concat(b,', ',a),rank() over (order by a)
        from(
            select n, lag(n) over(order by n)
        

    from t 
        where not exists(
            select 1 from t f 
            where f.n<t.n and t.n%f.n=0)
    ) z(a,b)
    where a=b+2
) r(s,k)
where k=2

The basic gist is we use the built in table of numbers (master..spt_values type='p') and alias that with a CTE as something short. We add 2 to remove the worry of pulling 0 or 1 trivial errors for our set, so now we have candidates of 2,2050.

Z the inner most query gets all primes from 2 to 2050, by filtering out any number n that is divisible by a number less than n. We then use a nifty T-SQL 2012 windowing function lag that lets us pull the previous result, so now Z's results a and b are the primes P[n] and P[n-1] respectively. The R query creates the output string, and filters out non-twin primes and also creates a sequence number for the output we call K. Finally the last query R allows us to filter and get the Kth twin prime by changing its variable.

Answer 22

Mathematica -- 71 bytes

n=Input[];
i=j=0;
While[j<n,i++;If[And@@PrimeQ[x={i,i+2}],j++]];Print@x

Answer 23

Husk, 11 bytes

!fo=2F≠Ẋeİp

Try it online!

non-competing, since it takes input from arguments rather than STDIN.

Returns a 2-element list.

Interesting version using Ψ: Try it online!