# Emirp primes

Emirp primes
You are encouraged to solve this task according to the task description, using any language you may know.

An   emirp   (prime spelled backwards)   are primes that when reversed   (in their decimal representation)   are a different prime.

(This rules out palindromic primes.)

•   show the first   twenty   emirps
•   show all emirps between   7,700   and   8,000
•   show the   10,000th   emirp

In each list, the numbers should be in order.

Invoke the (same) program once per task requirement, this will show what limit is used as the upper bound for calculating surplus (regular) primes.

The specific method of how to determine if a range or if specific values are to be shown will be left to the programmer.

he solution uses the package Miller_Rabin from the Miller-Rabin primality test.

`with Ada.Text_IO, Miller_Rabin; procedure Emirp_Gen is    type Num is range 0 .. 2**63-1; -- maximum for the gnat Ada compiler    MR_Iterations: constant Positive := 25;      -- the probability Pr[Is_Prime(N, MR_Iterations) = Probably_Prime]      -- is 1 for prime N and < 4**(-MR_Iterations) for composed N    function Is_Emirp(E: Num) return Boolean is      package MR is new Miller_Rabin(Num); use MR;       function Rev(E: Num) return Num is	 N: Num := E;	 R: Num := 0;      begin	 while N > 0 loop	    R := 10*R + N mod 10; -- N mod 10 is least significant digit of N	    N := N / 10;          -- delete least significant digit of N	 end loop;	 return R;      end Rev;       R: Num := Rev(E);    begin      return E /= R and then	     (Is_Prime(E, MR_Iterations) = Probably_Prime) and then	     (Is_Prime(R, MR_Iterations) = Probably_Prime);   end Is_Emirp;    function Next(P: Num) return Num is      N: Num := P+1;   begin      while not (Is_Emirp(N)) Loop	 N := N + 1;      end loop;      return N;   end Next;    Current: Num;   Count: Num := 0; begin   -- show the first twenty emirps   Ada.Text_IO.Put("First 20 emirps:");   Current := 1;   for I in 1 .. 20 loop      Current := Next(Current);      Ada.Text_IO.Put(Num'Image(Current));   end loop;   Ada.Text_IO.New_Line;    -- show the emirps between 7700 and 8000   Ada.Text_IO.Put("Emirps between 7700 and 8000:");   Current := 7699;   loop      Current := Next(Current);      exit when Current > 8000;       Ada.Text_IO.Put(Num'Image(Current));   end loop;    -- the 10_000th emirp   Ada.Text_IO.Put("The 10_000'th emirp:");   for I in 1 .. 10_000 loop      Current := Next(Current);   end loop;   Ada.Text_IO.Put_Line(Num'Image(Current));end Emirp_Gen;`
Output:
```First 20 emirps: 13 17 31 37 71 73 79 97 107 113 149 157 167 179 199 311 337 347 359 389
Emirps between 7700 and 8000: 7717 7757 7817 7841 7867 7879 7901 7927 7949 7951 7963
The 10_000'th emirp: 948349```

## ALGOL 68

Works with: ALGOL 68G version Any - tested with release 2.8.3.win32

Uses Algol 68G specific argc and argv procedures to access to command line. Allows the user to specify the from and to range values or ordinals on the command line. The sieve size can also be specified. As suggested by the Fortran sample, from = to is treated as a special case for labeling the output.

`# sieve of Eratosthenes: sets s[i] to TRUE if i is prime, FALSE otherwise     #PROC sieve = ( REF[]BOOL s )VOID:     BEGIN        # start with everything flagged as prime                              #         FOR i TO UPB s DO s[ i ] := TRUE OD;        # sieve out the non-primes                                            #        s[ 1 ] := FALSE;        FOR i FROM 2 TO ENTIER sqrt( UPB s ) DO            IF s[ i ] THEN FOR p FROM i * i BY i TO UPB s DO s[ p ] := FALSE OD FI        OD     END # sieve # ; # parse the command line - ignore errors                                      #INT  emirp from  :=  1;        # lowest emirp required                        #INT  emirp to    := 10;        # highest emirp required                       #BOOL value range := FALSE;     # TRUE if the range is the value of the emirps #                               # FALSE if the range is the ordinal of the     #                               # emirps                                       #INT  max number  := 1 000 000; # sieve size                                   ## returns s converted to an integer - does not check s is a valid integer     #PROC to int = ( STRING s )INT:     BEGIN        INT result := 0;        FOR ch pos FROM LWB s TO UPB s DO            result *:= 10;            result +:= ABS s[ ch pos ] - ABS "0"        OD;        result     END # to int # ;FOR arg pos TO argc DO    IF   argv( arg pos ) = "FROM"    THEN        emirp from  := to int( argv( arg pos + 1 ) )    ELIF argv( arg pos ) = "TO"      THEN        emirp to    := to int( argv( arg pos + 1 ) )    ELIF argv( arg pos ) = "VALUE"   THEN        value range := TRUE    ELIF argv( arg pos ) = "ORDINAL" THEN        value range := FALSE    ELIF argv( arg pos ) = "SIEVE"   THEN        max number  := to int( argv( arg pos + 1 ) )    FIOD; # construct a sieve of primes up to the maximum number required for the task  #[ 1 : max number ]BOOL is prime;sieve( is prime ); # return TRUE if p is an emirp, FALSE otherwise                               #PROC is emirp = ( INT p )BOOL:     IF NOT is prime[ p ] THEN        FALSE     ELSE        # reverse the digits of p, if this is a prime different from p,       #        # p is an emirp                                                       #        INT q    := 0;        INT rest := ABS p;        WHILE rest > 0 DO            q    TIMESAB 10;            q    PLUSAB  rest MOD 10;            rest OVERAB  10        OD;        is prime[ q ] AND q /= p     FI # is emirp # ; # generate the required emirp list                                            #IF value range THEN    # find emirps with values in the specified range                          #    print( ( "emirps between ", whole( emirp from, 0 ), " and ", whole( emirp to, 0 ), ":" ) );    FOR p FROM emirp from TO emirp to DO        IF is emirp( p ) THEN            print( ( " ", whole( p, 0 ) ) )        FI    ODELSE    # find emirps with ordinals in the specified range                        #    INT emirp count := 0;    IF emirp from = emirp to THEN        print( ( "emirp ", whole( emirp from, 0 ), ":" ) )    ELSE        print( ( "emirps ", whole( emirp from, 0 ), " to ", whole( emirp to, 0 ), ":" ) )    FI;    FOR p TO max number WHILE emirp count < emirp to DO        IF is emirp( p ) THEN            # have another emirp                                              #            emirp count +:= 1;            IF emirp count >= emirp from THEN                print( ( " ", whole( p, 0 ) ) )            FI        FI    ODFI;print( ( newline ) )`
Output:

a68g emirpPrimes.a68 - FROM 1 TO 20

```emirps 1 to 20: 13 17 31 37 71 73 79 97 107 113 149 157 167 179 199 311 337 347 359 389
```

a68g emirpPrimes.a68 - FROM 7700 TO 8000 VALUE

```emirps between 7700 and 8000: 7717 7757 7817 7841 7867 7879 7901 7927 7949 7951 7963
```

a68g emirpPrimes.a68 - FROM 10000 TO 10000

```emirp 10000: 948349
```

## AutoHotkey

`SetBatchLines, -1p := 1Loop, 20 {	p := NextEmirp(p)	a .= p " "}p := 7700Loop {	p := NextEmirp(p)	if (p > 8000)		break	b .= p " "}p :=1Loop, 10000	p := NextEmirp(p)MsgBox, % "First twenty emirps: " a	. "`nEmirps between 7,700 and 8,000: " b	. "`n10,000th emirp: " p IsPrime(n) {	if (n < 2)		return, 0	else if (n < 4)		return, 1	else if (!Mod(n, 2))		return, 0	else if (n < 9)		return 1	else if (!Mod(n, 3))		return, 0	else {		r := Floor(Sqrt(n))		f := 5		while (f <= r) {			if (!Mod(n, f))				return, 0			if (!Mod(n, (f + 2)))				return, 0			f += 6		}		return, 1	}} NextEmirp(n) {	Loop		if (IsPrime(++n)) {			rev := Reverse(n)			if (rev = n)				continue			if (IsPrime(rev))				return n		}} Reverse(s) {	Loop, Parse, s		r := A_LoopField r	return r}`
Output:
```First twenty emirps: 13 17 31 37 71 73 79 97 107 113 149 157 167 179 199 311 337 347 359 389
Emirps between 7,700 and 8,000: 7717 7757 7817 7841 7867 7879 7901 7927 7949 7951 7963
10,000th emirp: 948349```

## AWK

Based on C example here :

cat emirp.awk

` function is_prime(n,	p){        if (!(n%2) || !(n%3)) {		return 0 }        p = 1        while(p*p < n)                if (n%(p += 4) == 0 || n%(p += 2) == 0) {                        return 0 }        return 1} function reverse(n,	r){	r = 0        for (r = 0; int(n) != 0; n /= 10)                r = r*10 + int(n%10);        return r} function is_emirp(n,   r){        r = reverse(n)	return ((r != n) && is_prime(n) && is_prime(r)) ? 1 : 0} BEGIN {	c = 0	for (x = 11; c < 20; x += 2) {		if (is_emirp(x)) {			printf(" %i,", x); ++c }	}	printf("\n")	for (x = 7701; x < 8000; x += 2) {		if (is_emirp(x)) {			printf(" %i,", x); ++c }	}	printf("\n")	c = 0	for (x = 11; ; x += 2)                        if (is_emirp(x) && ++c == 10000) {                                printf(" %i", x);                                break;                        }	printf("\n")} `
Output:
```\$ awk -f emirp.awk
13, 17, 31, 37, 71, 73, 79, 97, 107, 113, 149, 157, 167, 179, 199, 311, 337, 347, 359, 389,
7717, 7757, 7817, 7841, 7867, 7879, 7901, 7927, 7949, 7951, 7963,
948349
```

## C

Note the unusual commandline argument parsing to sastisfy the "invoke three times" magic requirement.

`#include <stdio.h> typedef unsigned uint;int is_prime(uint n){        if (!(n%2) || !(n%3)) return 0;        uint p = 1;        while(p*p < n)                if (n%(p += 4) == 0 || n%(p += 2) == 0)                        return 0;        return 1;} uint reverse(uint n){        uint r;        for (r = 0; n; n /= 10)                r = r*10 + (n%10);        return r;} int is_emirp(uint n){        uint r = reverse(n);        return r != n && is_prime(n) && is_prime(r);} int main(int argc, char **argv){        uint x, c = 0;        switch(argc) { // advanced args parsing        case 1: for (x = 11; c < 20; x += 2)                        if (is_emirp(x))                                printf(" %u", x), ++c;                break;         case 2: for (x = 7701; x < 8000; x += 2)                        if (is_emirp(x))                                printf(" %u", x);                break;         default:                for (x = 11; ; x += 2)                        if (is_emirp(x) && ++c == 10000) {                                printf("%u", x);                                break;                        }        }         putchar('\n');        return 0;}`
Output:
```% ./a.out           # no argument: task 1
13 17 31 37 71 73 79 97 107 113 149 157 167 179 199 311 337 347 359 389
% ./a.out a         # one argument: task 2
7717 7757 7817 7841 7867 7879 7901 7927 7949 7951 7963
% ./a.out a b       # you get the idea
948349
```

## C++

`#include <vector>#include <iostream>#include <algorithm>#include <sstream>#include <string>#include <cmath> bool isPrime ( int number ) {   if ( number <= 1 )       return false ;   if ( number == 2 )      return true ;   for ( int i = 2 ; i <= std::sqrt( number ) ; i++ ) {      if ( number % i == 0 )	 return false ;   }   return true ;} int reverseNumber ( int n ) {   std::ostringstream oss ;   oss << n ;   std::string numberstring ( oss.str( ) ) ;   std::reverse ( numberstring.begin( ) , numberstring.end( ) ) ;   return std::stoi ( numberstring ) ;} bool isEmirp ( int n ) {   return isPrime ( n ) && isPrime ( reverseNumber ( n ) )      && n != reverseNumber ( n ) ;} int main( ) {   std::vector<int> emirps ;   int i = 1 ;   while ( emirps.size( ) < 20 ) {      if ( isEmirp( i ) ) {         emirps.push_back( i ) ;      }      i++ ;   }   std::cout << "The first 20 emirps:\n" ;   for ( int i : emirps )       std::cout << i << " " ;   std::cout << '\n' ;   int newstart = 7700 ;   while ( newstart < 8001 ) {      if ( isEmirp ( newstart ) ) 	std::cout << newstart << '\n' ;      newstart++ ;   }   while ( emirps.size( ) < 10000 ) {      if ( isEmirp ( i ) ) {	 emirps.push_back( i ) ;      }      i++ ;   }   std::cout << "the 10000th emirp is " << emirps << " !\n" ;    return 0 ;}`
Output:
```13 17 31 37 71 73 79 97 107 113 149 157 167 179 199 311 337 347 359 389
7717
7757
7817
7841
7867
7879
7901
7927
7949
7951
7963
the 10000th emirp is 948349 !
```

## C#

Works with: C sharp version 7
`using static System.Console;using System;using System.Linq;using System.Collections.Generic; public class Program{    public static void Main() {        const int limit = 1_000_000;        WriteLine("First 20:");        WriteLine(FindEmirpPrimes(limit).Take(20).Delimit());        WriteLine();         WriteLine("Between 7700 and 8000:");        WriteLine(FindEmirpPrimes(limit).SkipWhile(p => p < 7700).TakeWhile(p => p < 8000).Delimit());        WriteLine();         WriteLine("10000th:");        WriteLine(FindEmirpPrimes(limit).ElementAt(9999));    }     private static IEnumerable<int> FindEmirpPrimes(int limit)    {        var primes = Primes(limit).ToHashSet();         foreach (int prime in primes) {            int reverse = prime.Reverse();            if (reverse != prime && primes.Contains(reverse)) yield return prime;	}    }     private static IEnumerable<int> Primes(int bound) {        if (bound < 2) yield break;        yield return 2;         BitArray composite = new BitArray((bound - 1) / 2);        int limit = ((int)(Math.Sqrt(bound)) - 1) / 2;        for (int i = 0; i < limit; i++) {            if (composite[i]) continue;	    int prime = 2 * i + 3;	    yield return prime; 	    for (int j = (prime * prime - 2) / 2; j < composite.Count; j += prime)	        composite[j] = true;        }	for (int i = limit; i < composite.Count; i++)	    if (!composite[i]) yield return 2 * i + 3;    }} public static class Extensions{    public static HashSet<T> ToHashSet<T>(this IEnumerable<T> source) => new HashSet<T>(source);     private const string defaultSeparator = " ";    public static string Delimit<T>(this IEnumerable<T> source, string separator = defaultSeparator) =>        string.Join(separator ?? defaultSeparator, source);     public static int Reverse(this int number)    {	if (number < 0) return -Reverse(-number);	if (number < 10) return number;	int reverse = 0;	while (number > 0) {	    reverse = reverse * 10 + number % 10;	    number /= 10;	}	return reverse;    }}`
Output:
```First 20:
13 17 31 37 71 73 79 97 107 113 149 157 167 179 199 311 337 347 359 389

Between 7700 and 8000:
7717 7757 7817 7841 7867 7879 7901 7927 7949 7951 7963

10000th:
948349
```

## Clojure

### Using biginteger's isProbablePrime()

The isProbablePrime() method performs a Miller-Rabin primality test to within a given certainty.

`(defn emirp? [v]  (let [a (biginteger v)        b (biginteger (clojure.string/reverse (str v)))]    (and (not= a b)         (.isProbablePrime a 16)         (.isProbablePrime b 16)))) ; Generate the output(println "first20:    " (clojure.string/join " " (take 20 (filter emirp? (iterate inc 0)))))(println "7700-8000:  " (clojure.string/join " " (filter emirp? (range 7700 8000))))(println "10,000:     " (nth (filter emirp? (iterate inc 0)) 9999))  `
Output:
```first20:     13 17 31 37 71 73 79 97 107 113 149 157 167 179 199 311 337 347 359 389
7700-8000:   7717 7757 7817 7841 7867 7879 7901 7927 7949 7951 7963
10,000:      948349
```

## Common Lisp

It uses a primitive prime function found in http://www.rosettacode.org/wiki/Primality_by_trial_division, not optimized at all.

`(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))))) (defun reverse-digits (n)  (labels ((next (n v)             (if (zerop n) v                 (multiple-value-bind (q r)                     (truncate n 10)                   (next q (+ (* v 10) r))))))    (next n 0))) (defun emirp (&key (count nil) (start 10) (end nil) (print-all nil))  (do* ((n start (1+ n))        (c count) )       ((or (and count (<= c 0)) (and end (>= n end))))    (when (and (primep n) (not (= n (reverse-digits n))) (primep (reverse-digits n)))      (when print-all (format t "~a " n))      (when count (decf c)) )))  (progn  (format t "First 20 emirps: ") (emirp :count 20 :print-all t)  (format t "~%Emirps between 7700 and 8000: ") (emirp :start 7700 :end 8000 :print-all t)  (format t "~%The 10,000'th emirp: ") (emirp :count 10000 :print-all nil) ) `
Output:
```First 20 emirps: 13 17 31 37 71 73 79 97 107 113 149 157 167 179 199 311 337 347 359 389
Emirps between 7700 and 8000: 7717 7757 7817 7841 7867 7879 7901 7927 7949 7951 7963
The 10,000'th emirp: 948349```

## D

`bool isEmirp(uint n) pure nothrow @nogc {    bool isPrime(in uint n) pure nothrow @nogc {        if (n == 2 || n == 3)            return true;        else if (n < 2 || n % 2 == 0 || n % 3 == 0)            return false;        for (uint div = 5, inc = 2; div ^^ 2 <= n;             div += inc, inc = 6 - inc)            if (n % div == 0)                return false;         return true;    }     uint reverse(uint n) pure nothrow @nogc {        uint r;        for (r = 0; n; n /= 10)            r = r * 10 + (n % 10);        return r;    }     immutable r = reverse(n);    return r != n && isPrime(n) && isPrime(r);} void main() {    import std.stdio, std.algorithm, std.range;     auto uints = uint.max.iota;    writeln("First 20:\n", uints.filter!isEmirp.take(20));    writeln("Between 7700 and 8000:\n",            iota(7_700, 8_001).filter!isEmirp);    writeln("10000th: ", uints.filter!isEmirp.drop(9_999).front);}`
Output:
```First 20:
[13, 17, 31, 37, 71, 73, 79, 97, 107, 113, 149, 157, 167, 179, 199, 311, 337, 347, 359, 389]
Between 7700 and 8000:
[7717, 7757, 7817, 7841, 7867, 7879, 7901, 7927, 7949, 7951, 7963]
10000th: 948349```

This code is not efficient, but the run-time is acceptable, about 0.33 seconds with the ldc2 compiler.

### Sieve-Based Version

`import std.stdio, std.algorithm, std.range, std.bitmanip; /// Not extendible Sieve of Eratosthenes.BitArray sieve(in uint n) pure nothrow /*@safe*/ {    BitArray composites;    composites.init([true, true]);    composites.length = n;    if (n < 2)        return composites;     foreach (immutable uint i; 2 .. cast(uint)(n ^^ 0.5) + 1)        if (!composites[i])            for (uint k = i * i; k < n; k += i)                composites[k] = true;     return composites;} __gshared BitArray composites; bool isEmirp(uint n) nothrow @nogc {    uint reverse(uint n) pure nothrow @safe @nogc {        uint r;        for (r = 0; n; n /= 10)            r = r * 10 + (n % 10);        return r;    }     immutable r = reverse(n);    // BitArray doesn't perform bound tests yet.    assert(n < composites.length && r < composites.length);    return r != n && !composites[n] && !composites[r];} void main() {    composites = 1_000_000.sieve;     auto uints = uint.max.iota;    writeln("First 20:\n", uints.filter!isEmirp.take(20));    writeln("Between 7700 and 8000:\n",            iota(7_700, 8_001).filter!isEmirp);    writeln("10000th: ", uints.filter!isEmirp.drop(9_999).front);}`

The output is the same. With ldc2 compiler the run-time is about 0.06 seconds.

## Elixir

`defmodule Emirp do  defp prime?(2), do: true  defp prime?(n) when n<2 or rem(n,2)==0, do: false  defp prime?(n), do: prime?(n,3)   defp prime?(n,k) when n<k*k, do: true  defp prime?(n,k) when rem(n,k)==0, do: false  defp prime?(n,k), do: prime?(n,k+2)   def emirp?(n) do    if prime?(n) do      reverse = to_string(n) |> String.reverse |> String.to_integer      n != reverse and prime?(reverse)    end  end   def task do    emirps = Stream.iterate(1, &(&1+1)) |> Stream.filter(&emirp?/1)    first = Enum.take(emirps,20) |> Enum.join(" ")    IO.puts "First 20 emirps: #{first}"    between = Enum.reduce_while(emirps, [], fn x,acc ->      cond do        x < 7700        -> {:cont, acc}        x in 7700..8000 -> {:cont, [x | acc]}        true            -> {:halt, Enum.reverse(acc)}      end    end) |> Enum.join(" ")    IO.puts "Emirps between 7,700 and 8,000: #{between}"    IO.puts "10,000th emirp: #{Enum.at(emirps, 9999)}"  endend Emirp.task`
Output:
```First 20 emirps: 13 17 31 37 71 73 79 97 107 113 149 157 167 179 199 311 337 347 359 389
Emirps between 7,700 and 8,000: 7717 7757 7817 7841 7867 7879 7901 7927 7949 7951 7963
10,000th emirp: 948349
```

## F#

### The function

This task uses Extensible Prime Generator (F#)

` // Generate emirps. Nigel Galloway: November 19th., 2017let emirp =  let rec fN n g = match n with |0->g |_->fN (n/10) (g*10+n%10)  let     fG n g = n<>g && isPrime g  primes |> Seq.filter (fun n -> fG n (fN n 0)) `

` emirps |> (Seq.take 20) |> Seq.iter (printf "%d ") `
Output:
```13 17 31 37 71 73 79 97 107 113 149 157 167 179 199 311 337 347 359 389
```
` emirps |> Seq.skipWhile (fun n->n<7700) |> Seq.takeWhile (fun n->n<=8000) |> Seq.iter (printf "%d ")  `
Output:
```7717 7757 7817 7841 7867 7879 7901 7927 7949 7951 7963
```
` printfn "%d" (Seq.item 9999 emirps) `
Output:
```948349
```
` // count # of emirps with n = 2 to 7 digits. Nigel Galloway: August 8th., 2018 let n=emirp |> Seq.takeWhile(fun n->n<10000000) |> Seq.countBy(fun n->match n with |n when n>999999->7                                                                                   |n when n> 99999->6                                                                                   |n when n>  9999->5                                                                                   |n when n>   999->4                                                                                   |n when n>    99->3                                                                                   |_              ->2)for n,g in n do printfn "%d -> %d" n g `
Output:
```2 -> 8
3 -> 28
4 -> 204
5 -> 1406
6 -> 9538
7 -> 70474
Real: 00:07:19.408, CPU: 00:07:23.250, GC gen0: 59744, gen1: 3
```

## Factor

`USING: io kernel lists lists.lazy math.extras math.parser    math.primes sequences ;FROM: prettyprint => . pprint ;IN: rosetta-code.emirp : rev ( n -- n' )    number>string reverse string>number ; : emirp? ( n -- ? )    dup rev [ = not ] [ [ prime? ] [email protected] ] 2bi and and ; : nemirps ( n -- seq )    0 lfrom [ emirp? ] lfilter ltake list>array ; : print-seq ( seq -- )    [ pprint bl ] each nl ; : part1 ( -- )    "First 20 emirps:" print 20 nemirps print-seq ; : part2 ( -- )    "Emirps between 7700 and 8000:" print    7700 ... 8000 [ emirp? ] filter print-seq ; : part3 ( -- )    "10,000th emirp:" print 10,000 nemirps last . ; : main ( -- )    part1 nl part2 nl part3 ; MAIN: main`
Output:
```First 20 emirps:
13 17 31 37 71 73 79 97 107 113 149 157 167 179 199 311 337 347 359 389

Emirps between 7700 and 8000:
7717 7757 7817 7841 7867 7879 7901 7927 7949 7951 7963

10,000th emirp:
948349
```

## Fortran

Fortran has no standard interface arrangements whereby a run can be supplied with parameters from a command line. Some implementations do provide a routine, possibly called something like GETARG and with a variety of parameters and usages. One can of course read a disc file containing suitable parameters, but this is not as specified. So, to meet the three invocations, a subroutine is devised with parameters that allow it to perform the three different tasks. To handle "the first twenty", the parameters are easy 1,20. For "the 10,000'th", they are 10000,10000. Meeting the requirement for an invocation that lists all emirPs between 7,700 and 8,000 involved further bending, with the result that the subroutine has four parameters. Then I thought: why not specify the base for the numerology? So, five.

Now arise such questions as well-based emirPs. For instance, 17 is a base ten emirP, and in (say) base thirteen, 17 is 14 (and not an even number); 41 in base thirteen is 53 and that is a prime also. So, 17 (the number) is an emirP in both base ten and base thirteen. Is there a maximally-based emirP? But, for now, onwards in base ten.

The source would be F77, except for the idea of having the assistant routines GETPRIME(i), NEXTPRIME(n), and ISPRIME(n) all share the responsibility for the updating of a stash of prime numbers as they find the need rather than pre-emptively calculating a table of primes that is large enough for any expected usage, possibly by some high-speed trickery. Since the routines invoke each other back and forth, the dreaded attribute of RECURSIVE must be declared to encourage the compiler and so F90 is required. Otherwise, each routine would have to be careful over its own usage. Each would separately have to be able to proceed past the end of the current stash of prime numbers should its need arise, and augment the table as possible. For use in factoring numbers, the table need not be large as P(4792) = 46337, and the square of this exceeds the capacity of a signed thirty-two bit integer. But in this task, actual prime numbers are required well beyond that. Each run announces the table size, thereby showing the limit of its table of primes; there seemed no point in clearing the table each time to more closely follow the notion of separate runs.

For factoring numbers up to the 32-bit two's complement integer limit, the table need not be large, and it can easily enough be stored as a collection of sixteen and thirty-two bit numbers to save some space. Accessing an array PRIME(i) can be made a function GETPRIME(i) without a change in syntax (as needed in pascal: Prime[i] for an array, GetPrime(i) for a function), at least for reading. So, instead of 4792x4 = 19168 bytes, 12144 are needed, to set against the additional code complexity. These days, this is a difference of small importance. Actually, a further value is needed to hold Prime(4793) = 46349. Function ISPRIME does not determine its stepping point via the near universal usage of SQRT(n). If calculated in double precision this will give acceptable results for a 32-bit integer, but I have been burnt by an ad-hoc calculation nDgits = LOG10(x) + 1 failing for x = 10 because Log10(10) = 0·9999etc. which may well round to one, but truncates to zero. So, a SQRT-free demonstration, needed if the MOD function were unavailable. Actually, if P(i) is the last factor to be checked, this suffices up to the square of P(i + 1), not P(i). But this bound is only useful when successive numbers are being tested; for an individual factorisation it is too messy.

The initial version ran very slowly once past the first run, and this prompted some instrumentation, the addition of counters for the invocations. It transpired that GETPRIME(i) was being invoked thousands of millions of times... Once again, a N2 process is to be avoided, here when NEXTPRIME(n) was stepping linearly along the array of primes (in the hope of knowing the next prime along without having to recalculate it) and being invoked many times to do so. This was fixed by introducing a binary search, the list of primes being of course in order. The early version of NEXTPRIME(n) also did not attempt to save new primes, as it might be invoked with a value well beyond the end of the table and the next value on from n might be past many lesser primes. But by working on from PRIME(NP) up to n they can be found and saved along the way. Saving new primes in NEXTPRIME meant that GETPRIME should no longer itself attempt saving, as it is invoking NEXTPRIME. Mutual recursion is all very well, but organisation is important also.
`      MODULE BAG	!A mixed assortment.       INTEGER MSG	!I/O unit number to share about.       INTEGER PF16LIMIT,PF32LIMIT,NP			!Know that P(3512) = 32749, the last within two's complement 16-bit integers.       PARAMETER (PF16LIMIT = 3512, PF32LIMIT = 4793)	!32749² = 1,072,497,001; the integer limit is 2,147,483,647 in 32-bit integers.       INTEGER*2 PRIME16(PF16LIMIT)			!P(4792) =  46337, next is 46349 and 46337² = 2,147,117,569.       INTEGER*4 PRIME32(PF16LIMIT + 1:PF32LIMIT)	!Let the compiler track the offsets.       DATA NP,PRIME16(1),PRIME16(2)/2,2,3/	!But, start off with this. Note that Prime(NP) is odd...       INTEGER NGP,NNP,NIP	!Invocation counts.       DATA NGP,NNP,NIP/3*0/	!Starting at zero.       CONTAINS		!Some co-operating routines.        RECURSIVE INTEGER FUNCTION GETPRIME(I)	!They are numbered. As if in an array Prime(i).Chooses from amongst two arrays, of sizes known from previous work.         INTEGER I		!The desired index.         INTEGER P		!A potential prime.         INTEGER MP		!Counts beyond NP.          NGP = NGP + 1		!Another try.          IF (I.LE.0) THEN	!A silly question?            GETPRIME = -666		!This should cause trouble!          ELSE IF (I.LE.NP) THEN	!I have a little list.            IF (I.LE.PF16LIMIT) THEN		!Well actually, two little lists.              GETPRIME = PRIME16(I)		!So, direct access from this.             ELSE				!Or, for the larger numbers,              GETPRIME = PRIME32(I)		!This.            END IF			!So much for previous effort.          ELSE IF (I.LE.PF32LIMIT) THEN	!My list may not yet be completely filled.            MP = NP			!This is the last stashed so far.            P = GETPRIME(NP)		!I'll ask me to figure out where this is stashed.   10       P = NEXTPRIME(P)		!Go for the next one along.            MP = MP + 1			!Advance my count.            IF (MP.LT.I) GO TO 10	!Are we there yet?            GETPRIME = P		!Yep.           ELSE			!But, my list may be too short.            WRITE (MSG,*) "Hic!",I	!So, give an indication.            STOP "Too far..."		!And quit.          END IF		!For factoring 32-bit, need only 4792 elements.        END FUNCTION GETPRIME	!This is probably faster than reading from a monster disc file.         SUBROUTINE STASHPRIME(P)	!Saves a value in the stash.         INTEGER P	!The prime to be stashed.          NP = NP + 1		!Count another in.          IF (NP.LE.PF16LIMIT) THEN	!But, where to?            PRIME16(NP) = P			!The short list.          ELSE IF (NP.LE.PF32LIMIT) THEN!Or,            PRIME32(NP) = P			!The long list (which is shorter)          ELSE				!Or,            STOP "Stash overflow!"		!Oh dear.          END IF			!It is stashed.        END SUBROUTINE STASHPRIME	!The checking should be redundant.         INTEGER FUNCTION FINDPRIME(IT)	!Via binary search.         INTEGER IT	!The value to be found.         INTEGER L,R,P	!Assistants.          L = 0		!This is the *exclusive bounds* version.          R = NP + 1	!Thus, L = first - 1; R = Last + 1.    1     P = (R - L)/2		!Probe offset.          IF (P.LE.0) THEN	!No span?            FINDPRIME = -L		!Not found. IT follows Prime(L).            RETURN			!Escape.          END IF		!But otherwise,          P = P + L		!Convert to an index into array PRIME, manifested via GETPRIME.          IF (IT - GETPRIME(P)) 2,4,3	!Compare... Three way result.    2     R = P; GO TO 1	!IT < PRIME(P): move R back.    3     L = P; GO TO 1	!PRIME(P) < IT: move L forward.    4     FINDPRIME = P		!PRIME(P) = IT: Found here!        END FUNCTION FINDPRIME	!Simple and fast.         RECURSIVE INTEGER FUNCTION NEXTPRIME(P)	!Some effort may ensue.Checks the stash in PRIME in the hope of finding the next prime directly, otherwise advances from P.Collates a stash of primes in PRIME16 and PRIME32, advancing NP from 2 to PF32LIMIT as it goes.         INTEGER P	!Not necessarily itself a prime number.         INTEGER PI	!A possibly prime increment.         INTEGER IT	!A finger.          NNP = NNP + 1	!Another try          IF (P.LE.1) THEN	!Dodge annoying effects.	Otherwise, FINDPRIME(P) would be zero.            PI = 2		!The first prime is known.	Because P precedes Prime(1).           ELSE			!The first stashed value is two.            IT = (ABS(FINDPRIME(P)))	!The stash is ordered, and P = 2 will be found at 1.            IF (IT.LT.NP) THEN		!Before my last-known prime? FINDPRIME(4) = -2 as it follows Prime(NP=2).              PI = GETPRIME(IT + 1)	!Yes, so I know the next along already.             ELSE	!Otherwise, it is past Prime(NP). and IT = NP thanks to the ABS.              IF (NP.LT.PF32LIMIT) THEN	!If my stash is not yet filled,                PI = GETPRIME(IT)	!I want to start with its last entry, known to be an odd number.               ELSE			!So that I can stash each next prime along the way.                PI = P			!Otherwise, start with P.                IF (MOD(PI,2).EQ.0) PI = PI - 1	!And some suspicion.              END IF			!So  much for a starting position.              DO WHILE (PI.LE.P)	!Perhaps I must go further.   11           PI = PI + 2			!Advance to a possibility.                IF (.NOT.ISPRIME(PI)) GO TO 11	!Discard it?                IF (IT.EQ.NP .AND. IT.LT.PF32LIMIT) THEN	!Am I one further on from NP?                  CALL STASHPRIME(PI)		!Yes, and there is space to stash it.                  IT = IT + 1			!Ready for the next one along, if it comes.                END IF			!All are candidates for my stash.              END DO		!Perhaps this prime will be big enough.            END IF		!It may be a long way past PRIME(NP).          END IF		!And I may have filled my stash along the way.          NEXTPRIME = PI	!Take that.        END FUNCTION NEXTPRIME	!Messy.         RECURSIVE LOGICAL FUNCTION ISPRIME(N)	!Checks an arbitrary number, though limited by INTEGER size.Crunches up to SQRT(N), and at worst needs to be able to reach Prime(4793) = 46349; greater than SQRT(2147483647) = 46340·95...         INTEGER N	!The number.         INTEGER I,F,Q	!Assistants.          NIP = NIP + 1	!Another try.          IF (N.LT.2) THEN	!Dodge annoyances.            ISPRIME = .FALSE.	!Such as N = 1, and the first F being 2.          ELSE			!Otherwise, some effort.            ISPRIME = .FALSE.	!The usual result.            I = 1		!Start at the start with PRIME(1).   10       F = GETPRIME(I)	!Thus, no special case with F = 2.            Q = N/F		!So, how many times? (Truncation, remember)            IF (Q .GE. F) THEN	!Q < F means F² > N.              IF (Q*F .EQ. N) RETURN	!A factor is found!              I = I + 1		!Very well.              GO TO 10		!Try the next possible factor.            END IF		!And if we get through all that,            ISPRIME = .TRUE.	!It is a prime number.          END IF		!And we're done.        END FUNCTION ISPRIME	!After a lot of divisions.         INTEGER FUNCTION ESREVER(IT,BASE)	!Reversed digits.         INTEGER IT	!The number to be reversed. Presumably positive.         INTEGER BASE	!For the numerology.         INTEGER N,R	!Assistants.          IF (BASE.LE.1) STOP "Base 2 at least!"	!Ah, distrust.          N = IT	!A copy I can damage.          R = 0		!Here we go.          DO WHILE(N.GT.0)	!A digit remains?            R = R*BASE + MOD(N,BASE)	!Yes. Grab the low-order digit of N.            N = N/BASE			!And reduce N by another power of BASE.          END DO		!Test afresh.          ESREVER = R		!That's it.        END FUNCTION ESREVER	!Easy enough.         SUBROUTINE EMIRP(BASE,N1,N2,I1,I2)	!Two-part interface.         INTEGER BASE	!Avoid decimalist chauvinism.         INTEGER N1,N2	!Count span to show those found.         INTEGER I1,I2	!Search span.         INTEGER N	!Counter.         INTEGER P,R	!Assistants.          WRITE (MSG,1) N1,N2,BASE,I1,I2	!Declare the purpose.    1     FORMAT ("Show the first ",I0," to ",I0,	!So as to encompass     &     " emirP numbers (base ",I0,") between ",I0," and ",I0)	!The specified options.          N = 0		!None found so far.          P = I1 - 1	!Syncopation. The starting position might itself be a prime number.Chase another emirP.   10     P = NEXTPRIME(P)		!I want the next prime.          IF (P.LT.I1) GO TO 10		!Up to the starting mark yet?          IF (P.GT.I2) GO TO 900	!Past the finishing mark?          R = ESREVER(P,BASE)		!Righto, a candidate.          IF (P .EQ. R) GO TO 10	!Palindromes are rejected.          IF (.NOT.ISPRIME(R)) GO TO 10	!As are non-primes.          N = N + 1			!Aha, a success!c          if (mod(n,100) .eq. 0) thenc            write (6,66) N,P,R,NP,NGP,NNP,NIPc   66       format ("N=",I5,",p=",I6,",R=",I6,",NP=",I6,3I12)c          end if          IF (N.GE.N1) WRITE (6,*) P,R	!Are we within the count span?          IF (N.LT.N2) GO TO 10		!Past the end?Closedown.  900     WRITE (MSG,901) NP,GETPRIME(NP)	!Might be of interest.  901     FORMAT ("Stashed up to Prime(",I0,") = ",I0,/)        END SUBROUTINE EMIRP	!Well, that was odd.      END MODULE BAG	!Mixed.       PROGRAM POKE	!Now put it all to the test.      USE BAG		!With ease.      MSG = 6		!Standard output.       CALL EMIRP(10,    1,   20,   1,   1000)	!These parameters      CALL EMIRP(10,    1,   28,7700,   8000)	!Meet the specifiction      CALL EMIRP(10,10000,10000,   1,1000000)	!Of three separate invocations.       END	!Whee!`

Output:

```Show the first 1 to 20 emirP numbers (base 10) between 1 and 1000
13          31
17          71
31          13
37          73
71          17
73          37
79          97
97          79
107         701
113         311
149         941
157         751
167         761
179         971
199         991
311         113
337         733
347         743
359         953
389         983
Stashed up to Prime(77) = 389

Show the first 1 to 28 emirP numbers (base 10) between 7700 and 8000
7717        7177
7757        7577
7817        7187
7841        1487
7867        7687
7879        9787
7901        1097
7927        7297
7949        9497
7951        1597
7963        3697
Stashed up to Prime(1008) = 8009

Show the first 10000 to 10000 emirP numbers (base 10) between 1 and 1000000
948349      943849
Stashed up to Prime(4793) = 46349
```

And the invocation counts: GETPRIME 15,200,926; NEXTPRIME 74,799; ISPRIME 548,944. The execution time is small: the run completes even as the new output window stabilises on the screen.

An earlier version used a larger table of primes (size 123,456) as EMIRP advanced via `I = I + 1; P = GETPRIME(I)` thereby only considering successive primes as candidates without having to check factors to find them. By converting to `P = NEXTPRIME(P)` the table could be made smaller, but this meant being clear within NEXTPRIME that if P was greater than the last stashed prime, and the table was filled, then the table no longer offered an advantage and the search should start from P. With larger P, starting from Prime(NP) meant more and more catching up.

Function ISPRIME uses GETPRIME(i) for its successive factor trials, and thus works only up to the table limit unless GETPRIME were to be extended. If NEXTPRIME were used instead the table would be accessed where possible, otherwise a march would begin. If ISPRIME were to be changed to accept say a 64-bit integer the table size limit could be increased, but alas a complete table would require around 139,094,144 entries, and all those trial divisions would take a while. Still, the possible factors go no further than F = SQRT(N), approximately calculated now, and to check that F has no factors requires only tests up to SQRT(F)...

Project Extensible_prime_generator#Fortran offers a scheme supporting such routines as PRIME(i) instead of GETPRIME(i), NEXTPRIME(N), and ISPRIME(N), using a disc file in place of a large array in memory - whose values would be lost when the run finishes. But instead of about a hundred lines of Fortran to provide primes for EMIRP, module PRIMEBAG requires 311 lines.

## FreeBASIC

`' FB 1.05.0 Win64 Function isPrime(n As UInteger) As Boolean  If n < 2 Then Return False  If n Mod 2 = 0 Then Return n = 2  If n Mod 3 = 0 Then Return n = 3  Dim d As Integer = 5  While d * d <= n    If n Mod d = 0 Then Return False    d += 2    If n Mod d = 0 Then Return False    d += 4  Wend  Return TrueEnd Function Function reverseNumber(n As UInteger) As UInteger  If n < 10 Then Return n  Dim As Integer sum = 0  While n > 0    sum = 10 * sum  + (n Mod 10)    n \= 10  Wend  Return sumEnd Function Function isEmirp(n As UInteger) As Boolean  If Not isPrime(n) Then Return False  Dim As UInteger reversed = reverseNumber(n)  Return reversed <> n AndAlso CInt(isPrime(reversed))End Function ' We can immediately rule out all primes from 2 to 11 as these are palindromic' and not therefore Emirp primesPrint "The first 20 Emirp primes are :"Dim As UInteger count = 0, i = 13Do  If isEmirp(i) Then    Print Using "####"; i;    count + = 1  End If  i += 2Loop Until count = 20Print : PrintPrint "The Emirp primes between 7700 and 8000 are:"i = 7701Do  If isEmirp(i) Then Print Using "#####"; i;  i += 2Loop While i < 8000Print : PrintPrint "The 10,000th Emirp prime is : ";i = 13 : count = 0Do  If isEmirp(i) Then count += 1  If count = 10000 Then Exit Do  i += 2LoopPrint iPrintPrint "Press any key to quit"Sleep`
Output:
```The first 20 Emirp primes are :
13  17  31  37  71  73  79  97 107 113 149 157 167 179 199 311 337 347 359 389

The Emirp primes between 7700 and 8000 are:
7717 7757 7817 7841 7867 7879 7901 7927 7949 7951 7963

The 10,000th Emirp prime is : 948349
```

## Go

This has a bit more to it than required but little optimization, other than using a fast Sieve of Atkin implementation for the prime numbers and skipping some tests on ranges of impossible Emirps (thanks to a comment on the discussion page).

As a side note, by using the same API as the prime number generator this also demonstrates how Go interfaces can be used (and note it doesn't require the existing code/package to know anything about the interface being defined).

`package main import (	"flag"	"fmt"	"github.com/jbarham/primegen.go" // Sieve of Atkin implementation	"math") // primeCache is a simple cache of small prime numbers, it very// well might be faster to just regenerate them as needed.type primeCache struct {	gen    *primegen.Primegen	primes []uint64} func NewPrimeCache() primeCache {	g := primegen.New()	return primeCache{gen: g, primes: []uint64{g.Next()}}} // upto returns a slice of primes <= n.// The returned slice is shared with all callers, do not modify it!func (pc *primeCache) upto(n uint64) []uint64 {	if p := pc.primes[len(pc.primes)-1]; p <= n {		for p <= n {			p = pc.gen.Next()			pc.primes = append(pc.primes, p)		}		return pc.primes[:len(pc.primes)-1]	}	for i, p := range pc.primes {		if p > n {			return pc.primes[:i]		}	}	panic("not reached")} var cache = NewPrimeCache() func sqrt(x uint64) uint64 { return uint64(math.Sqrt(float64(x))) } // isprime does a simple test if n is prime.// See also math/big.ProbablyPrime().func isprime(n uint64) bool {	for _, p := range cache.upto(sqrt(n)) {		if n%p == 0 {			return false		}	}	return true} func reverse(n uint64) (r uint64) {	for n > 0 {		r = 10*r + n%10		n /= 10	}	return} // isEmirp does a simple test if n is Emirp, n must be primefunc isEmirp(n uint64) bool {	r := reverse(n)	return r != n && isprime(r)} // EmirpGen is a sequence generator for Emirp primestype EmirpGen struct {	pgen     *primegen.Primegen	nextn    uint64	r1l, r1h uint64	r2l, r2h uint64	r3l, r3h uint64} func NewEmirpGen() *EmirpGen {	e := &EmirpGen{pgen: primegen.New()}	e.Reset()	return e} func (e *EmirpGen) Reset() {	e.pgen.Reset()	e.nextn = 0	// Primes >7 cannot end in 2,4,5,6,8 (leaving 1,3,7)	e.r1l, e.r1h = 20, 30	e.r2l, e.r2h = 40, 70	e.r3l, e.r3h = 80, 90} func (e *EmirpGen) next() (n uint64) {	for n = e.pgen.Next(); !isEmirp(n); n = e.pgen.Next() {		// Skip over inpossible ranges		// Benchmarks show this saves ~20% when generating n upto 1e6		switch {		case e.r1l <= n && n < e.r1h:			e.pgen.SkipTo(e.r1h)		case e.r2l <= n && n < e.r2h:			e.pgen.SkipTo(e.r2h)		case e.r3l <= n && n < e.r3h:			e.pgen.SkipTo(e.r3h)		case n > e.r3h:			e.r1l *= 10			e.r1h *= 10			e.r2l *= 10			e.r2h *= 10			e.r3l *= 10			e.r3h *= 10		}	}	return} func (e *EmirpGen) Next() (n uint64) {	if n = e.nextn; n != 0 {		e.nextn = 0		return	}	return e.next()} func (e *EmirpGen) Peek() uint64 {	if e.nextn == 0 {		e.nextn = e.next()	}	return e.nextn} func (e *EmirpGen) SkipTo(nn uint64) {	e.pgen.SkipTo(nn)	e.nextn = 0	return} // SequenceGen defines an arbitrary sequence generator.// Both *primegen.Primegen and *EmirpGen implement this.type SequenceGen interface {	Next() uint64	Peek() uint64	Reset()	SkipTo(uint64)	//Count(uint64) uint64 // not implemented for *EmirpGen} func main() {	var start, end uint64	var n, skip uint	var oneline, primes bool	flag.UintVar(&n, "n", math.MaxUint64, "number of emirps to print")	flag.UintVar(&skip, "skip", 0, "number of emirps to skip")	flag.Uint64Var(&start, "start", 0, "start at x>=start")	flag.Uint64Var(&end, "end", math.MaxUint64, "stop at x<=end")	flag.BoolVar(&oneline, "oneline", false, "output on a single line")	flag.BoolVar(&primes, "primes", false, "generate primes rather than emirps")	flag.Parse() 	sep := "\n"	if oneline {		sep = " "	} 	// Here's where making SequenceGen an interface comes in handy:	var seq SequenceGen	if primes {		seq = primegen.New()	} else {		seq = NewEmirpGen()	} 	for seq.Peek() < start {		seq.Next()	}	for ; skip > 0; skip-- {		seq.Next()	}	for ; n > 0 && seq.Peek() <= end; n-- {		fmt.Print(seq.Next(), sep)	}	if oneline {		fmt.Println()	}}`
Output:
```\$ ./emirp -h
Usage of ./emirp:
-end=18446744073709551615: stop at x<=end
-n=18446744073709551615: number of emirps to print
-oneline=false: output on a single line
-primes=false: generate primes rather than emirps
-skip=0: number of emirps to skip
-start=0: start at x>=start

\$ ./emirp -oneline -n 20 -primes # not asked for, just demonstrating SequenceGen interface
2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71

\$ ./emirp -oneline -n 20
13 17 31 37 71 73 79 97 107 113 149 157 167 179 199 311 337 347 359 389

\$ ./emirp -oneline -start 7800 -end 8000
7817 7841 7867 7879 7901 7927 7949 7951 7963

\$ ./emirp -skip 9999 -n 1
948349
```

Library: primes
Works with: GHC version 7.8.3
Works with: primes version 0.2.1.0
`#!/usr/bin/env runghc import Data.HashSet (HashSet, fromList, member)import Data.Listimport Data.Numbers.Primesimport System.Environmentimport System.Exitimport System.IO -- optimization mentioned on the talk pagestartDigOK :: Integer -> BoolstartDigOK n = head (show n) `elem` "1379" -- infinite list of primes that have an acceptable first digitfiltPrimes :: [Integer]filtPrimes = filter startDigOK primes -- finite list of primes that have an acceptable first digit and-- are the specified number of digits in lengthnDigsFPr :: Integer -> [Integer]nDigsFPr n =  takeWhile (< hi) \$ dropWhile (< lo) filtPrimes  where lo = 10 ^ (n - 1)        hi = 10 ^ n -- hash set of the filtered primes of the specified number of digitsnDigsFPrHS :: Integer -> HashSet IntegernDigsFPrHS n = fromList \$ nDigsFPr n -- infinite list of hash sets, where each hash set contains primes of-- a specific number of digits, i. e. index 2 contains 2 digit primes,-- index 3 contains 3 digit primes, etc.-- Don't access index 0, because it will return an errorfPrByDigs :: [HashSet Integer]fPrByDigs = map nDigsFPrHS [0 ..] isEmirp :: Integer -> BoolisEmirp n =  let revStr = reverse \$ show n      reversed = read revStr      hs = fPrByDigs !! length revStr  in (startDigOK n) && (reversed /= n) && (reversed `member` hs) emirps :: [Integer]emirps = filter isEmirp primes emirpSlice :: Integer -> Integer -> [Integer]emirpSlice from to =  genericTake numToTake \$ genericDrop numToDrop emirps  where    numToDrop = from - 1    numToTake = 1 + to - from emirpValues :: Integer -> Integer -> [Integer]emirpValues lo hi =  dropWhile (< lo) \$ takeWhile (<= hi) emirps usage = do  name <- getProgName  putStrLn \$ "usage: " ++ name ++ " lo hi [slice | values]"  exitFailure main = do  hSetBuffering stdout NoBuffering  args <- getArgs  fixedArgs <- case length args of    1 -> return \$ args ++ args ++ ["slice"]    2 -> return \$ args ++ ["slice"]    3 -> return args    _ -> usage  let lo = read \$ fixedArgs !! 0      hi = read \$ fixedArgs !! 1  case fixedArgs !! 2 of   "slice" -> print \$ emirpSlice lo hi   "values" -> print \$ emirpValues lo hi   _ -> usage`
Output:

This program uses the same format for command line arguments as the Perl 6 example.

```\$ ./Emirp.hs 1 20
[13,17,31,37,71,73,79,97,107,113,149,157,167,179,199,311,337,347,359,389]
\$ ./Emirp.hs 7700 8000 values
[7717,7757,7817,7841,7867,7879,7901,7927,7949,7951,7963]
\$ ./Emirp.hs 10000

```

### List-based

Using list-based incremental sieve from here and trial division from here,

` λ> let emirp p = let q=(read.reverse.show) p in q /= p && noDivsBy primesW q  λ> take 20 . filter emirp \$ primesW[13,17,31,37,71,73,79,97,107,113,149,157,167,179,199,311,337,347,359,389]   λ> filter emirp . takeWhile (< 8000) . dropWhile (< 7700) \$ primesW[7717,7757,7817,7841,7867,7879,7901,7927,7949,7951,7963]  --  0.02 secs  λ> (!! (10000-1)) . filter emirp \$ primesW948349                                                    -- 0.69 secs`

## J

Solution:
`   emirp =: (] #~ ~: *. 1 p: ]) |.&.:":"0  NB. Input is array of primes`

In other words: select numbers from the argument list whose decimal reverse is both different and prime and return those decimal reversed values as numbers. (For simplicity, we require that our argument be a list of prime numbers.)

Examples
`   /:~ emirp p: 2+i.7513 17 31 37 71 73 79 97 113 311 701 733 743 751 761 941 953 971 983 991    (#~ 7700&< * 8000&>) /:~ emirp i.&.(_1&p:) 99997717 7757 7817 7841 7867 7879 7901 7927 7949 7951 7963    # emirp p: i.74791                          NB. 10,000th emirp is 74,790th prime 10000   p: 74790  948349    NB. alternative approach (first emirp value would be at index 0):   9999 { /:~ emirp p:i.1e5943849`

## Java

This implementation uses a slight optimization discussed in the talk page. It will not actually check the primality (forwards or backwards) for a number that starts or ends with the digits 2, 4, 5, 6, or 8 since no primes greater than 7 end with those digits.

`public class Emirp{ 	//trivial prime algorithm, sub in whatever algorithm you want	public static boolean isPrime(long x){		if(x < 2) return false;		if(x == 2) return true;		if((x & 1) == 0) return false; 		for(long i = 3; i <= Math.sqrt(x);i+=2){			if(x % i == 0) return false;		} 		return true;	} 	public static boolean isEmirp(long x){		String xString = Long.toString(x);		if(xString.length() == 1) return false;		if(xString.matches(".*") || xString.matches(".*")) return false; //eliminate some easy rejects		long xR = Long.parseLong(new StringBuilder(xString).reverse().toString());		if(xR == x) return false;		return isPrime(x) && isPrime(xR);	} 	public static void main(String[] args){		int count = 0;		long x = 1; 		System.out.println("First 20 emirps:");		while(count < 20){			if(isEmirp(x)){				count++;				System.out.print(x + " ");			}			x++;		} 		System.out.println("\nEmirps between 7700 and 8000:");		for(x = 7700; x <= 8000; x++){			if(isEmirp(x)){				System.out.print(x +" ");			}		} 		System.out.println("\n10,000th emirp:");		for(x = 1, count = 0;count < 10000; x++){			if(isEmirp(x)){				count++;			}		}		//--x to fix the last increment from the loop		System.out.println(--x);	}}`
Output:
```First 20 emirps:
13 17 31 37 71 73 79 97 107 113 149 157 167 179 199 311 337 347 359 389
Emirps between 7700 and 8000:
7717 7757 7817 7841 7867 7879 7901 7927 7949 7951 7963
10,000th emirp:
948349```

## jq

Works with: jq version with foreach

The given tasks are simple to implement in jq if unbounded streams can be harnessed, which is possible in versions of jq that support "foreach" and "break". This article accordingly showcases the use of these builtins, which have been available since July 7, 2014.

Infrastructure: prime numbers

`def is_prime:  if  . == 2 then true  else     2 < . and . % 2 == 1 and       (. as \$in       | ((\$in + 1) | sqrt) as \$m       | [false, 3] | until( . or . > \$m; [\$in % . == 0, . + 2])       | .       | not)  end ; def relatively_prime:  . as \$n  | . as \$primes  | (\$n | sqrt) as \$s  | (. | length) as \$length  | [0, true]  | until( . > \$length or (\$primes[.] > \$s) or . == false;          [. + 1, (\$n % \$primes[.] != 0)] )  | . ; def primes:  # The helper function, next, has arity 0 for tail recursion optimization;  # its input must be an array of primes of length at least 2,   # the last also being the greatest.  def next:     . as \$previous     | .[length-1] as \$last     | [(2 + \$last), \$previous]     | until( relatively_prime ; . += 2) as \$nextp     | ( \$previous + [\$nextp] );  2, ([2,3] | recurse( next ) | .[-1]) ;`

Emirps

`def is_emirp:  . as \$n  | tostring | explode | reverse | implode | tonumber | (. != \$n) and is_prime ; # emirps(n) emits [i, p] where p is the i-th emirp, up to and including i == ndef emirps(n):  label \$start  | # state: [count, \$emirp]  foreach primes as \$p ([0, null];    if . >= n then break \$start    else if (\$p | is_emirp) then [. + 1, \$p] else . = null end    end;    if . then . else empty end ) ;`

(0) The three separate subtasks can be accomplished in one step as follows:

`emirps(10000)| select( . <= 20 or (7700 <= . and . <= 8000) or . == 10000)`

The output of the above is shown below.

To accomplish the three subtasks separately:

(1) First twenty:

`emirps(20)`

(2) Selection by value

`label \$top| primes| if (7700 <= .) and (. <= 8000) and is_emirp then .   elif . > 8000 then break \$top   else empty   end`

(3) 10,000th

`last(emirps(10000)) | .`
Output:
`\$ jq -c -n -f Emirp_primes.jq[1,13][2,17][3,31][4,37][5,71][6,73][7,79][8,97][9,107][10,113][11,149][12,157][13,167][14,179][15,199][16,311][17,337][18,347][19,359][20,389][180,7717][181,7757][182,7817][183,7841][184,7867][185,7879][186,7901][187,7927][188,7949][189,7951][190,7963][10000,948349]`

## Julia

Works with: Julia version 0.6
`using Primes function collapse(n::Array{<:Integer})    sum = 0    for (p, d) in enumerate(n)        sum += d * 10 ^ (p - 1)    end    return sumend Base.reverse(n::Integer) = collapse(reverse(digits(n)))isemirp(n::Integer) = isprime(n) && isprime(reverse(n)) function firstnemirps(m::Integer)    rst = Array{typeof(m)}(m)    i, n = 1, 2    @inbounds while i ≤ m        if isemirp(n)            rst[i] = n            i += 1        end        n += 1    end    return rstend emirps = firstnemirps(10000)println("First 20:\n", emirps[1:20])println("Between 7700 and 8000:\n", filter(x -> 7700 ≤ x ≤ 8000, emirps))println("10000th:\n", emirps)`
Output:
```First 20:
[2, 3, 5, 7, 11, 13, 17, 31, 37, 71, 73, 79, 97, 101, 107, 113, 131, 149, 151, 157]
Between 7700 and 8000:
[7717, 7757, 7817, 7841, 7867, 7879, 7901, 7927, 7949, 7951, 7963]
10000th:
942569```

## Kotlin

Translation of: FreeBASIC
`//  version 1.1.4 fun isPrime(n: Int) : Boolean {    if (n < 2) return false    if (n % 2 == 0) return n == 2    if (n % 3 == 0) return n == 3    var d = 5    while (d * d <= n) {        if (n % d == 0) return false        d += 2        if (n % d == 0) return false        d += 4    }    return true} fun reverseNumber(n: Int) : Int {    if (n < 10) return n    var sum = 0    var nn = n    while (nn > 0) {        sum = 10 * sum + nn % 10        nn /= 10    }    return sum} fun isEmirp(n: Int) : Boolean {    if (!isPrime(n)) return false    val reversed = reverseNumber(n)    return reversed != n && isPrime(reversed)} fun main(args: Array<String>) {    println("The first 20 Emirp primes are :")    var count = 0    var i = 13    do {        if (isEmirp(i)) {            print(i.toString() + " ")            count++        }        i += 2    }    while (count < 20)    println()    println()    println("The Emirp primes between 7700 and 8000 are :")    i = 7701    do {        if (isEmirp(i)) print(i.toString() + " ")        i += 2    }    while (i < 8000)    println()    println()    print("The 10,000th Emirp prime is : ")    i = 13    count = 0    do {        if (isEmirp(i)) count++        if (count == 10000) break        i += 2    }    while(true)    print(i)}`
Output:
```The first 20 Emirp primes are :
13 17 31 37 71 73 79 97 107 113 149 157 167 179 199 311 337 347 359 389

The Emirp primes between 7700 and 8000 are :
7717 7757 7817 7841 7867 7879 7901 7927 7949 7951 7963

The 10,000th Emirp prime is : 948349
```

## Lua

` function isPrime (n)    if n < 2 then return false end    if n < 4 then return true end    if n % 2 == 0 then return false end    for d = 3, math.sqrt(n), 2 do        if n % d == 0 then return false end    end    return trueend function isEmirp (n)    if not isPrime(n) then return false end    local rev = tonumber(string.reverse(n))    if rev == n then return false end    return isPrime(rev)end function emirpGen (mode, a, b)    local count, n, eString = 0, 0, ""    if mode == "between" then        for n = a, b do            if isEmirp(n) then eString = eString .. n .. " " end        end        return eString    end    while count < a do        n = n + 1        if isEmirp(n) then            eString = eString .. n .. " "            count = count + 1        end    end    if mode == "first" then return eString end    if mode == "Nth" then return n endend if #arg > 1 and #arg < 4 then    print(emirpGen(arg, tonumber(arg), tonumber(arg)))else    print("Wrong number of arguments")end `

Command prompt session:

```>lua emirp.lua first 20
13 17 31 37 71 73 79 97 107 113 149 157 167 179 199 311 337 347 359 389

>lua emirp.lua between 7700 8000
7717 7757 7817 7841 7867 7879 7901 7927 7949 7951 7963

>lua emirp.lua Nth 10000
948349
```

## Maple

`EmirpPrime := proc(n)    local eprime;    eprime := parse(StringTools:-Reverse(convert(n,string)));    if n <> eprime and isprime(n) and isprime(eprime) then        return n;    end if;end proc:EmirpsList := proc( n )    local i, values;    values := Array([]):    i := 0:    do        i := i + 1;        if EmirpPrime(i) <> NULL then            ArrayTools:-Append(values, i);        end if;    until numelems(values) = n;    return convert(values,list);end proc:EmirpsList(20);EmirpPrime~([seq(7700..8000)]);EmirpsList(10000)[-1]; `
Output:
```[13, 17, 31, 37, 71, 73, 79, 97, 107, 113, 149, 157, 167, 179, 199, 311, 337, 347, 359, 389]
[7717, 7757, 7817, 7841, 7867, 7879, 7901, 7927, 7949, 7951, 7963]
948349```

## Mathematica

First a simple helper function

`reverseDigits[n_Integer] := [email protected]@[email protected]`

A function to test whether n is an emirp prime

`emirpQ[n_Integer] :=  Block[{rev = [email protected]}, And[n != rev, PrimeQ[rev]]]`

Note, this test function assumes n is prime. Adding a check to verify n is prime will have an impact on execution time for finding the mth emirp prime particularly when m is large.

Finally, a function which returns the first emirp prime larger than the supplied argument

`nextEmirp[n_Integer] :=  NestWhile[NextPrime, NextPrime[n], ! emirpQ[#] &]`

With these the first 20 emirp primes are computed as:

`[email protected][nextEmirp, 1, 20]`
Output:
`{13,17,31,37,71,73,79,97,107,113,149,157,167,179,199,311,337,347,359,389}`

The emirp primes betweewn 7700 and 8000 are:

`[email protected][nextEmirp, 7700, # < 8000 &]`
Output:
`{7717,7757,7817,7841,7867,7879,7901,7927,7949,7951,7963,9001}`

The 10,000th emirp prime is:

`Nest[nextEmirp, 1, 10000]`
Output:
`948349`

## MATLAB

` NN=(1:1:1e6); %Natural numbers between 1 and tpns=NN(isprime(NN)); %prime numbersp=fliplr(str2num(fliplr(num2str(pns)))); a=pns(isprime(p)); b=p(isprime(p)); c=a-b;emirps=NN(a(c~=0)); `
Output:
```the first twenty emirps are: emirps(1:20)
ans =

Columns 1 through 14

13    17    31    37    71    73    79    97   107   113   149   157   167   179

Columns 15 through 20

199   311   337   347   359   389
```
```The emirp primes betweewn 7700 and 8000 are:
emirps(emirps>=7700 & emirps<=8000)
ans =

Columns 1 through 7

7717        7757        7817        7841        7867        7879        7901

Columns 8 through 11

7927        7949        7951        7963
```
```The 10,000th emirp prime is: emirps(10000)
ans =

948349
```

## Modula-2

`MODULE Emirp;FROM Conversions IMPORT StrToLong;FROM FormatString IMPORT FormatString;FROM LongMath IMPORT sqrt;FROM Terminal IMPORT WriteString,WriteLn,ReadChar; PROCEDURE IsPrime(x : LONGINT) : BOOLEAN;VAR    i : LONGINT;    u : LONGREAL;    v : LONGINT;BEGIN    IF x<2 THEN RETURN FALSE END;    IF x=2 THEN RETURN TRUE END;    IF x MOD 2 = 0 THEN RETURN FALSE END;     u := sqrt(FLOAT(x));    v := TRUNC(u);     FOR i:=3 TO v BY 2 DO        IF x MOD i = 0 THEN RETURN FALSE END    END;     RETURN TRUEEND IsPrime; PROCEDURE IsEmirp(x : LONGINT) : BOOLEAN;VAR    buf,rev : ARRAY[0..9] OF CHAR;    i,j : INTEGER;    y : LONGINT;BEGIN    (* Terminate early if the number is even *)    IF x MOD 2 = 0 THEN RETURN FALSE END;     (* First convert the input to a string *)    FormatString("%l", buf, x);     (* Create a copy of the string revered *)    j := 0;    WHILE buf[j] # 0C DO INC(j) END;    DEC(j);    i := 0;    WHILE buf[i] # 0C DO        rev[i] := buf[j];        INC(i);        DEC(j)    END;    rev[i] := 0C;     (* Convert the reversed copy to a number *)    StrToLong(rev,y);     (* Terminate early if the number is even *)    IF y MOD 2 = 0 THEN RETURN FALSE END;     (* Discard palindromes *)    IF x=y THEN RETURN FALSE END;     RETURN IsPrime(x) AND IsPrime(y)END IsEmirp; VAR    buf : ARRAY[0..63] OF CHAR;    x,count : LONGINT;BEGIN    count := 0;    x := 1;     WriteString("First 20 emirps:");    WriteLn;    WHILE count<20 DO        IF IsEmirp(x) THEN            INC(count);            FormatString("%l ", buf, x);            WriteString(buf)        END;        INC(x)    END;    WriteLn;     WriteString("Emirps between 7700 and 8000:");    WriteLn;    FOR x:=7700 TO 8000 DO        IF IsEmirp(x) THEN            FormatString("%l ", buf, x);            WriteString(buf)        END    END;    WriteLn;     WriteString("10,000th emirp:");    WriteLn;    count := 0;    x := 1;    WHILE count<10000 DO        IF IsEmirp(x) THEN            INC(count);        END;        INC(x)    END;    FormatString("%l ", buf, x-1);    WriteString(buf);    WriteLn;     ReadCharEND Emirp.`

## Oforth

Using isPrime function of Primality by trial division task :

`: isEmirp(n)   n isPrime ifFalse: [ false return ]   n asString reverse asInteger dup n == ifTrue: [ drop false ] else: [ isPrime ] ; : main(min, max, length)| l |   ListBuffer new ->l   min while(l size length < ) [      dup max > ifTrue: [ break ]       dup isEmirp ifTrue: [ dup l add ] 1 +       ]    drop l ;`
Output:
```>main(2, 9999999, 20) println
[13, 17, 31, 37, 71, 73, 79, 97, 107, 113, 149, 157, 167, 179, 199, 311, 337, 347, 359, 389]

>main(7700, 8000, 300) println
[7717, 7757, 7817, 7841, 7867, 7879, 7901, 7927, 7949, 7951, 7963]

>main(2, 9999999999, 10000) last println
948349
```

## PARI/GP

`rev(n)=subst(Polrev(digits(n)),'x,10);emirp(n)=my(r=rev(n)); isprime(r) && isprime(n) && n!=rselect(emirp, primes(100))[1..20]select(emirp, primes([7700,8000]))s=10000; forprime(p=2,,if(emirp(p) && s--==0, return(p)))`
Output:
```%1 = [13, 17, 31, 37, 71, 73, 79, 97, 107, 113, 149, 157, 167, 179, 199, 311, 337, 347, 359, 389]
%2 = [7717, 7757, 7817, 7841, 7867, 7879, 7901, 7927, 7949, 7951, 7963]
%3 = 948349```

## Pascal

Library: primTrial

using trial division unit , but jumping over number ranges, where the reversed numbers can't be a prime. Compiles with Delphi and Free Pascal.

`program Emirp;//palindrome prime 13 <-> 31{\$IFDEF FPC}  {\$MODE DELPHI}  {\$OPTIMIZATION ON}  {\$OPTIMIZATION REGVAR}  {\$OPTIMIZATION PEEPHOLE}  {\$OPTIMIZATION CSE}  {\$OPTIMIZATION ASMCSE}  {\$Smartlink ON}  {\$CODEALIGN proc=32}{\$ELSE}  {\$APPLICATION CONSOLE}{\$ENDIF}uses  primtrial,sysutils; //IntToStrconst  helptext : array[0..5] of string =     ('  usage ',      '  t     -> test of functions',      '  b l u -> Emirps betwenn l,u       b 7700 8000',      '  c n   -> count of Emirps up to n  c 99999',      '  f n   -> output n first Emirp     f 20',      '  n     -> output the n.th Emirps   10000');   StepToNextPrimeEnd : Array[0..9] of byte =              (1,0,3,0,7,7,7,0,9,0);   base = 10; var  s: AnsiString;  pow,  powLen  : NativeUint; procedure OutputHelp;var  i : NativeUint;Begin  For i := Low(helptext) to High(helptext) do    writeln(helptext[i]);  writeln;end; function GetNumber(const s: string;var n:NativeUint):boolean;var  ErrCode: Word;Begin  val(s,n,Errcode);  result := ErrCode = 0;end; procedure RvsStr(var s: AnsiString);var  i, j: NativeUint;  swapChar : Ansichar;Begin  i := 1;  j := Length(s);  While j>i do Begin    swapChar:= s[i];s[i] := s[j];s[j] := swapChar;    inc(i);dec(j) end;end; function RvsNumL(var n: NativeUint):NativeUint;//reverse and last digitvar  q, c: NativeUint;Begin  result := n;  q := 0;  repeat    c:= result div Base;    q := q*Base+(result-c*Base);    result := c;  until result < Base;  n := q*Base+result; end; procedure InitP(var p: NativeUint);Begin  powLen := 2;  pow := Base;  InitPrime;  repeat p :=NextPrime until p >= 11;end; function isEmirp(p: NativeUint):boolean;var  rvsp: NativeUint;Begin  s := IntToStr(p);  result := StepToNextPrimeEnd[Ord(s)-48] = 0;  IF result then  Begin    RvsStr(s);    rvsp := StrToInt(s);    result := false;    IF rvsp<>p then      result := isPrime(rvsp);  end;end; function NextEmirp:NativeUint;var r,Ldgt: NativeUint;Begin  result:= NextPrime;  repeat    r := result;    //reverse    Ldgt := RvsNumL(r);    Ldgt := StepToNextPrimeEnd[Ldgt];    IF Ldgt = 0 then    Begin      IF r<>result then        IF isPrime(r) then          EXIT;      result:= NextPrime;    end    else    Begin      while actPrime > pow*Base do      Begin        inc(PowLen);        pow := pow*base;      end;      result := Ldgt*pow;      result := PrimeGELimit(result);    end;  until false;end; function GetIthEmirp(i: NativeUint):NativeUint;var  p : NativeUint;Begin  InitP(p);  Repeat    dec(i);    p:= NextEmirp;  until i = 0;  result := p;end; procedure nFirstEmirp(n: NativeUint);var  p : NativeUint; Begin  InitP(p);  Writeln('the first ',n,' Emirp primE: ');  Repeat    dec(n);    p:= NextEmirp;    write(p,' ');  until n = 0;  Writeln;end; function CntToLimit(n: NativeUint):NativeUint;var  p,cnt : NativeUint;Begin  cnt := 0;  InitP(p);  p:= NextEmirp;  While p <= n do  Begin    inc(cnt);    p:= NextEmirp;  end;  result := cnt;end; procedure InRange(l,u:NativeUint);var  p : NativeUint;  b : boolean;Begin  InitP(p);  IF l > u then Begin p:=l;l:=u;u:=p end;  Writeln('Emirp primes between ',l,' and ',u,' : ');  p := PrimeGELimit(l);   b := IsEmirp(p);  if b then    write(p,' ');  p:= NextEmirp;  IF (p> u) AND NOT b  then    Writeln('none')  else  Begin    while p < u do      Begin      write(p,' ');      p:= NextEmirp;    end;    Writeln;  end;end; var  i,u: NativeUint;  select : char;Begin  IF paramcount >= 1 then    select := Lowercase(paramstr(1));  case paramcount of  1: Begin       if select='t' then       Begin         nFirstEmirp(20);         InRange(7700,8000);         Writeln('the ',10000,'.th Emirp prime: ',GetIthEmirp(10000));         writeln(CntToLimit(9999),' Emirp primes up to ',9999);         // as a gag         InRange(400000000,700000000);       end       else         IF GetNumber(paramstr(1),i) then           Writeln('the ',i,'.th Emirp prime: ',GetIthEmirp(i))         else           OutPutHelp;     end;  2: Begin       case select of       'c': If GetNumber(paramstr(2),i) then              writeln(CntToLimit(i),' Eemirp primes up to ',i)            else              OutPutHelp;       'f': If GetNumber(paramstr(2),i) then              nFirstEmirp(i)            else              OutPutHelp;        else          OutPutHelp;        end;     end;  3: IF (select ='b') AND        GetNumber(paramstr(2),i) AND GetNumber(paramstr(3),u) Then       InRange(i,u)     else        OutPutHelp;  else    OutPutHelp;  end;End.`
output
```./Emirp t
the first 20 Emirp primE:
13 17 31 37 71 73 79 97 107 113 149 157 167 179 199 311 337 347 359 389
Emirp primes between 7700 and 8000 :
7717 7757 7817 7841 7867 7879 7901 7927 7949 7951 7963
the 10000.th Emirp prime: 948349
240 Emirp primes up to 9999
Emirp primes between 400000000 and 700000000 :
none

real    0m0.033s
...
a little "stress test"
Emirp primes between 300000000 and 400000000 :
1058667
rumtime for this: 2m 3 secs
```

### Using static sieve

is much faster. Only Counting Emirps. http://rosettacode.org/wiki/Extensible_prime_generator#Pascal It would be nice, if someone could check the results.Like F# today did

output
```Count Emirps
Emirp         Total
Decimals       Count         Count
2           8             8
3          28            36
4         204           240
5        1406          1646
6        9538         11184
7       70474         81658
8      535578        617236
9     4192024       4809260
10    33619380      38428640
11   274890232     313318872```

## Perl

Library: ntheory
`use feature 'say';use ntheory qw(forprimes is_prime); # Return the first \$count emirps using expanding segments.# Can efficiently generate millions of emirps.sub emirp_list {  my \$count = shift;  my(\$i, \$inc, @n) = (13, 100+10*\$count);  while (@n < \$count) {    forprimes {      push @n, \$_ if is_prime(reverse \$_) && \$_ ne reverse(\$_);    } \$i, \$i+\$inc-1;    (\$i, \$inc) = (\$i+\$inc, int(\$inc * 1.03) + 1000);  }  splice @n, \$count;  # Trim off excess emirps  @n;} say "First 20: ", join " ", emirp_list(20);print "Between 7700 and 8000:";forprimes { print " \$_" if is_prime(reverse \$_) && \$_ ne reverse(\$_) } 7700,8000;print "\n";say "The 10_000'th emirp: ", (emirp_list(10000))[-1];`
Output:
```First 20: 13 17 31 37 71 73 79 97 107 113 149 157 167 179 199 311 337 347 359 389
Between 7700 and 8000: 7717 7757 7817 7841 7867 7879 7901 7927 7949 7951 7963
The 10_000'th emirp: 948349```

## Perl 6

Works with: Rakudo version 2018.10

For better performance, build the lazy list using module `Math::Primesieve`, not the built-in, then display results based on parameters passed in. The default is to display an array slice starting and stopping at the given indices. Alternately, ask for all values between two endpoints.

`use Math::Primesieve; sub prime-hash (Int \$max) {    my \$sieve = Math::Primesieve.new;    my @primes = \$sieve.primes(\$max);    @primes.Set;} sub MAIN (\$start, \$stop = Nil, \$display = <slice>) {    my \$end = \$stop // \$start;    my %primes = prime-hash(100*\$end);    my @emirps = lazy gather for 1 .. * -> \$n {        take \$n if %primes{\$n} and %primes{\$n.flip} and \$n != \$n.flip    }     given \$display {        when 'slice'  { return @emirps[\$start-1 .. \$end-1] };        when 'values' {            my @values = gather for @emirps {                .take if \$start < \$_ < \$end;                last if \$_> \$end            }            return @values        }    }}`
Output:

Run with passed parameters: 1 20

('slice' is the default. you could pass it in, but it isn't necessary.)

`13 17 31 37 71 73 79 97 107 113 149 157 167 179 199 311 337 347 359 389`

Run with passed parameters: 7700 8000 values

`7717 7757 7817 7841 7867 7879 7901 7927 7949 7951 7963`

Run with passed parameter: 10000

`948349`

## Phix

Using Extensible_prime_generator#Phix, not that this task makes trial division onerous.
Does not assume anywhere that some pre-guessed value will be enough.

`sequence primes = {2,3,5,7}atom sieved = 10 procedure add_block()integer N = min((sieved-1)*sieved,400000)sequence sieve = repeat(1,N)    -- sieve[i] is really i+sieved    for i=2 to length(primes) do -- (evens filtered on output)        atom p = primes[i], p2 = p*p        if p2>sieved+N then exit end if        if p2<sieved+1 then            p2 += ceil((sieved+1-p2)/p)*p        end if        p2 -= sieved        if and_bits(p2,1)=0 then p2 += p end if        for k=p2 to N by p*2 do            sieve[k] = 0        end for    end for    for i=1 to N by 2 do        if sieve[i] then            primes &= i+sieved        end if    end for    sieved += Nend procedure function is_prime(integer n)    while sieved<n do        add_block()    end while    return binary_search(n,primes)>0end function sequence emirps = {} function rev(integer n)integer res = 0    while n do        res = res*10+remainder(n,10)        n = floor(n/10)    end while    return resend function function emirp(integer n)    if is_prime(n) then        integer r = rev(n)        if r!=n and is_prime(r) then            return 1        end if    end if    return 0end function procedure usage()    printf(1,"use a single command line argument, with no spaces, eg \"1-20\" (first 20), \n")    printf(1,"\"7700..8000\" (between 7700 and 8000), or \"10000\" (the 10,000th).\n")    {} = wait_key()    abort(0)end procedure procedure main(string arg3)sequence argsinteger n,m    if find('-',arg3) then      -- nth to mth emirp range        args = scanf(arg3,"%d-%d")        if length(args)!=1 then usage() end if        {{n,m}} = args        integer k = 1        while length(emirps)<m do            if emirp(k) then emirps &= k end if            k += 1        end while        printf(1,"emirps %d to %d: ",{n,m})        ?emirps[n..m]    elsif match("..",arg3) then -- emirps between n amd m        args = scanf(arg3,"%d..%d")        if length(args)!=1 then usage() end if        {{n,m}} = args        integer k = 1        while length(emirps)=0 or emirps[\$]<m do            if emirp(k) then emirps &= k end if            k += 1        end while        sequence s = {}        for i=1 to length(emirps) do            if emirps[i]>n then                for j=i to length(emirps) do                    if emirps[j]>m then                        printf(1,"emirps between %d and %d: ",{n,m})                        ?emirps[i..j-1]                        exit                    end if                end for                exit            end if        end for    else                        -- nth emirp        args = scanf(arg3,"%d")        if length(args)!=1 then usage() end if        {{n}} = args        integer k = 1        while length(emirps)<n do            if emirp(k) then emirps &= k end if            k += 1        end while        printf(1,"emirp %d: ",{n})        ?emirps[n]    end ifend procedure sequence cl = command_line()    if length(cl)=2 then        main("1-20")        main("7700..8000")        main("10000")    elsif length(cl)!=3 then        usage()    else        main(cl)     end if    {} = wait_key()`
Output:
```emirps 1 to 20: {13,17,31,37,71,73,79,97,107,113,149,157,167,179,199,311,337,347,359,389}
emirps between 7700 and 8000: {7717,7757,7817,7841,7867,7879,7901,7927,7949,7951,7963}
emirp 10000: 938033
```

## PicoLisp

`(de prime? (N)   (and      (bit? 1 N)      (let S (sqrt N)         (for (D 3  T  (+ D 2))            (T (> D S) N)            (T (=0 (% N D)) NIL) ) ) ) )(de palindr? (A)   (and      (<>         (setq A (chop A))         (setq @@ (reverse A)) )      (format @@) ) )(de emirp? (N)   (and (palindr? N) (prime? @) (prime? N)) )(de take1 (N)   (let I 11      (make         (for (X 1 (>= 20 X))            (and               (emirp? (inc 'I 2))               (link @)               (inc 'X) ) ) ) ) )(de take2 (NIL)   (make      (for (I 7701 (> 8000 I) (+ I 2))         (and (emirp? I) (link @)) ) ) )(de take3 (NIL)   (let I 11      (for (X 1 (>= 10000 X))         (and (emirp? (inc 'I 2)) (inc 'X)) )      I ) ) (println (take1 20))(println (take2))(println (take3))`
Output:
```(13 17 31 37 71 73 79 97 107 113 149 157 167 179 199 311 337 347 359 389)
(7717 7757 7817 7841 7867 7879 7901 7927 7949 7951 7963)
948349```

## PL/I

`*process or(!); pt1: Proc(run) Options(main); /********************************************************************* * 25.03.2014 Walter Pachl * Note: Prime number computations are extended as needed *********************************************************************/ Dcl debug Bit(1) Init('0'b); Dcl run Char(100) Var; Dcl primes(200000) Bin Fixed(31) Init(2,3,5,7,11,13,17,(200000-7)0); Dcl nn Bin Fixed(31) Init(0); Dcl np Bin Fixed(31) Init(7); Dcl hp Bin Fixed(31) Init(17); Dcl ip Bin Fixed(31); Dcl (p,r) Bin Fixed(31); Put Edit('run=',run,'<')(Skip,a,a,a); np=7; call cprimes(20,1,'A');  main_loop: Do ip=1 To 100000;                    /* loop over all primes       */   p=primes(ip);                       /* candidate                  */   If p=0 Then     call cprimes(20,hp+1,'.');   p=primes(ip);                       /* candidate                  */   r=rev(p);                           /* reversed candidate         */   If p=r Then;                        /* skip palindromic prime     */   Else Do;                            /* p is eligible              */     If is_prime(r) Then Do;           /* reversed p is a prime      */       nn=nn+1;                        /* increment number of hits   */       Select;         When(run<='1') Do;           If nn<21 Then Call show_1;  /* call appropriate output    */           If nn=20 Then             Leave main_loop;           End;         When(run='2') Do;           If hp<8000 Then             Call cprimes(1,8000,'B');           If 7700<p & p<8000 Then Call show_2;           If p>8000 Then             Leave main_loop;           End;         When(run='3') Do;           If np<10000 Then             Call cprimes(10000,1,'C');           If nn=10000 Then Do;             Call show_3;             Leave main_loop;             End;           End;         Otherwise Do;           Put skip list('Invoke as pt1 1/2/3');           Return;           End;         End;       End;     End;   End;  show_1: Proc; Dcl first Bit(1) Static Init('1'b); If first Then Do;   Put Edit('the first 20 emirps:')(Skip,a);   first='0'b;   Put Skip;   End; If nn=11 Then   Put Skip; Put Edit(p)(F(4)); End;  show_2: Proc; Dcl first Bit(1) Static Init('1'b); If first Then Do;   Put Edit('emirps between 7700 and 8000:')(Skip,a);   first='0'b;   Put Skip;   End; Put Edit(p)(F(5)); End;  show_3: Proc; Dcl first Bit(1) Static Init('1'b); If first Then Do;   Put Edit('the 10000th emirp:')(Skip,a);   first='0'b;   Put Skip;   End; Put Edit(p)(F(6)); End;  cprimes: Proc(num,mp,s); /********************************************************************* * Fill the array primes with prime numbers * so that it contains at least num primes and all primes<=mp *********************************************************************/ dcl o Char(60) Var; If debug Then   Put String(o) Edit('cprimes: ',s,np,hp)(a,a,2(f(6))); Dcl num Bin Fixed(31);                /* number of primes needed    */ Dcl mp  Bin Fixed(31);                /* max prime must be > mp     */ Dcl p   Bin Fixed(31);                /* candidate for next prime   */ Dcl s   Char(1);                      /* place of invocation        */ loop: Do p=hp+2 By 2 Until(np>=num & hp>mp); /* only odd numbers are elig.*/   If mod(p, 3)=0 Then Iterate;   If mod(p, 5)=0 Then Iterate;   If mod(p, 7)=0 Then Iterate;   If mod(p,11)=0 Then Iterate;   If mod(p,13)=0 Then Iterate;   Do k=7 By 1 While(primes(k)**2<=p);     If mod(p,primes(k))=0 Then       Iterate loop;     End;   np=np+1;   primes(np)=p;   hp=p;   End;  If debug Then    Put Edit(o,' -> ',np,hp)(Skip,a,a,2(f(6)));  End;  rev: Proc(x) Returns(Bin Fixed(31)); /********************************************************************* * reverse the given number *********************************************************************/ Dcl x Bin Fixed(31); Dcl p Pic'ZZZZZZ9'; Dcl qq Char(7) Init(''); Dcl q Pic'ZZZZZZ9' based(addr(qq)); Dcl v Char(8) Var; p=x; v=trim(p); v=reverse(v); substr(qq,8-length(v))=v; Return(q); End;  is_prime: Proc(x) Returns(Bit(1)); /********************************************************************* * check if x is a prime number (binary search in primes) *********************************************************************/ Dcl x  Bin Fixed(31); Dcl lo Bin Fixed(31) Init(1); Dcl hi Bin Fixed(31); Dcl m  Bin Fixed(31); If x>hp Then Do;                 /* x is outside of range in primes */   If debug Then     Put Edit('is_prime x=',x,'hp=',hp)(Skip,2(a,f(8),x(1)));   Call cprimes(1,x,'D');         /* extend range of primes          */   End; hi=np; Do While(lo<=hi);                /* lookup                          */   m=(lo+hi)/2;   Select;     When        (x=primes(m))   Return('1'b); /* x is a prime number*/     When        (x<primes(m))   hi=m-1;     Otherwise /* x>primes(m) */ lo=m+1;     End;   End; Return('0'b);                    /* x is not a prime number         */ End;  End;`
Output:
```run=1 <
the first 20 emirps:
13  17  31  37  71  73  79  97 107 113
149 157 167 179 199 311 337 347 359 389

run=2 <
emirps between 7700 and 8000:
7717 7757 7817 7841 7867 7879 7901 7927 7949 7951 7963

run=3 <
the 10000th emirp:
948349```

## Python

This uses Prime_decomposition#Python:_Using_Croft_Spiral_sieve and so the prime number generator self-extends to generate ever larger primes automatically.

There is no explicit hard-coded ceiling added to the code for the prime generator, which is the reason given for the need to invoke a program three times in the task description.

`from __future__ import print_functionfrom prime_decomposition import primes, is_primefrom heapq import *from itertools import islice def emirp():    largest = set()    emirps = []    heapify(emirps)    for pr in primes():        while emirps and pr > emirps:            yield heappop(emirps)        if pr in largest:            yield pr        else:            rp = int(str(pr)[::-1])            if rp > pr and is_prime(rp):                heappush(emirps, pr)                largest.add(rp) print('First 20:\n  ', list(islice(emirp(), 20)))print('Between 7700 and 8000:\n  [', end='')for pr in emirp():    if pr >= 8000: break    if pr >= 7700: print(pr, end=', ')print(']')print('10000th:\n  ', list(islice(emirp(), 10000-1, 10000)))`
Output:
```First 20:
[13, 17, 31, 37, 71, 73, 79, 97, 107, 113, 149, 157, 167, 179, 199, 311, 337, 347, 359, 389]
Between 7700 and 8000:
[7717, 7757, 7817, 7841, 7867, 7879, 7901, 7927, 7949, 7951, 7963, ]
10000th:
```

## Racket

This implementation seems to have exploded somewhat due to

• the need to "account" for the greatest tested prime
• the need to reset memory between runs
• the need for a main (to support the above)
• and a (possibly misguided) thought that performance might be a consideration
``` (my naive version finds the 10,0000th in ... ms)
```

So there are two versions presented below. The first is minimalist, providing basic functions, unburdened by accounting or (too many) performance considerations (please don't mark this as needing attention... I know it falls short of

`#lang racket(require math/number-theory) (define (stigid n)  (define (inr n a) (if (= 0 n) a (inr (quotient n 10) (+ (* 10 a) (modulo n 10)))))  (inr n 0)) (define (emirp-prime? n)  (define u (stigid n))  (and (not (= u n)) (prime? n) (prime? u))) (printf "\"show the first twenty emirps.\"~%")(for/list ((n (sequence-filter emirp-prime? (in-range 11 +Inf.0 2))) (_ (in-range 20))) n) (printf "\"show all emirps between 7,700 and 8,000\"~%")(for/list ((n (sequence-filter emirp-prime? (in-range 7701 8000 2)))) n) (printf "\"show the 10,000th emirp\"~%")(let loop ((i 10000) (p 9))  (define p+2 (+ p 2))  (cond [(not (emirp-prime? p+2)) (loop i p+2)] [(= i 1) p+2] [else (loop (- i 1) p+2)]))`

The second is somewhat larger and seems to be a playground for all sorts of code.

`#lang racket;; ---------------------------------------------------------------------------------------------------;; There are two distinct requirements here...;; 1. to test for emirp-primality - this can be done as easily as testing for primality.;;    We use math/number-theory's "prime?" for this, which has no bounds;; 2. to find the nth emirp-prime. Even when were doing this with normal primes, we wouldn't test;;    each number; rather sieve them. Prime sieves by their very nature are at least memory bound...;;    so I'm happy in this case that they are kept within the bounds of "fixnum" integers. Once we;;    accept that, we can use the unsafe-ops on fixnums which allow for a performance boost. The;;    fixnum / sieve code is after this simpler stuff.;; ---------------------------------------------------------------------------------------------------(require math/number-theory) ;; this slows things down, having to unbox, test and rebox the m.p.g -- but the task asks for some;; accounting to be performed, so account we do!(define max-prime-tested (box 0)) (define (report-mpg)  (printf "Max prime tested (using math/number-theory): ~a~%" (unbox max-prime-tested))) (define (prime?/remember-max n)  (define rv (prime? n))  (when (and rv (> n (unbox max-prime-tested))) (set-box! max-prime-tested n))  rv) (define (stigid n)  (define (inner-stigid n a) (if (= 0 n) a (inner-stigid (quotient n 10) (+ (* 10 a) (modulo n 10)))))  (inner-stigid n 0)) (define (emirp-prime? n)  (define u (stigid n))  (and (not (= u n)) (prime?/remember-max n) (prime?/remember-max u))) ;; ---------------------------------------------------------------------------------------------------(require  racket/require  (except-in   (filtered-in (lambda (n) (regexp-replace #rx"unsafe-" n "")) racket/unsafe/ops) unbox set-box!)) ;; NB using fixnum below limits stigid to "fixnum" (about 2^60) range of numbers;; but, unleashed, unsafe-fx... are fast(define (fxstigid n)  (define (inner-fxstigid n a)    (if (fx= 0 n) a (inner-fxstigid (fxquotient n 10) (fx+ (fx* 10 a) (fxmodulo n 10)))))  (inner-fxstigid n 0)) ;; Grows the sieve to n (so n is included in the sieve);; Values in the sieve are: = 0 - known non-prime;;                          > 0 - known prime;; The new sieve does not alter non-zero values in the old sieve; to preserve cachceing of e.g. emirps;; Always returns a copy (so it is caller responsibility to determine the necessity of this function)(define (extend-prime-sieve sieve n)  (define sieve-size (bytes-length sieve))  (define sieve-size+ (fx+ 1 n))  (define new-sieve (make-bytes sieve-size+ 1))  (bytes-copy! new-sieve 0 sieve 0 (fxmin sieve-size+ sieve-size))  (for* ((f (in-range 2 (add1 (integer-sqrt sieve-size+))))         #:unless (fx= (bytes-ref new-sieve f) 0) ; the only case of non-prime         (f+ (in-range (fx* f (fxmax 2 (fxquotient sieve-size f))) sieve-size+ f)))    (bytes-set! new-sieve f+ 0))  (values sieve-size+ new-sieve)) ;; task three *needs* a sieve to operate sub-second:;; values in sieve are:;;   0 - known non-prime;;   1 - known prime, unknown emirp-ality (freshly generated from extend-prime-sieve);;   2 - known prime, known non-emirp -- needed for sieve extension;;   3 - known emirp (and .: known prime)(define-values  (emirp-prime?/sieve reset-sieve! report-mpg/sieved extend-sieve!)  (let [(sieve-size 2) (the-sieve (bytes 0 0))]    (define (extend-sieve! n)      (when (fx>= n sieve-size)        (define-values (sieve-size+ new-sieve) (extend-prime-sieve the-sieve n))        (set! the-sieve new-sieve) (set! sieve-size sieve-size+)))    (values     (lambda (n)       (extend-sieve! n)       (case (bytes-ref the-sieve n)         [(0) #f] ; it's not even prime         [(1) ; it's a prime... but is is emirp?          (define u (fxstigid n))          (define new-sieve-n            (cond              [(fx= u n) 2]              [(fx> u n) (if (emirp-prime?/sieve u) 3 2)]              [(fx= (bytes-ref the-sieve u) 1) 3]              [else 2]))          (bytes-set! the-sieve n new-sieve-n)          (fx= new-sieve-n 3)]         [(2) #f] ; we know it's not emirp         [(3) #t])) ; we already knew it's an emirp     (lambda () (set! sieve-size 2) (set! the-sieve (bytes 0 0)))     (lambda () (printf "Sieve size: ~a~%Max prime generated (sieve): ~a~%" sieve-size                        (for/last ((n the-sieve) (p (in-naturals)) #:unless (fx= 0 n)) p)))               extend-sieve!))) ;; ---------------------------------------------------------------------------------------------------;; testing *-primality is a lot cheaper than generating, and we'll use math/number-theory to do;; this... it's fast enough. Because they cannot be palindromic and because 2 is the only even prime;; (and is palindromic), all emirps are odd - hence our sequences starting with an odd (>= 11),;; stepping by 2.(define (task1 (emirp?-test emirp-prime?))  (printf "\"show the first twenty emirps.\" [~s]~%" emirp?-test)  (for/list ((n (sequence-filter emirp?-test (in-range 11 +Inf.0 2))) (_ (in-range 20))) n)) (define (task2 (emirp?-test emirp-prime?))  (printf "\"show all emirps between 7,700 and 8,000\" [~s]~%" emirp?-test)  (for/list ((n (sequence-filter emirp?-test (in-range 7701 8000 2)))) n)) (define (task3 (emirp?-test emirp-prime?) (extend-sieve-fn #f))  (printf "\"show the 10,000th emirp\" [~s]~%" emirp?-test)  (when extend-sieve-fn    (extend-sieve-fn (nth-prime 10000))) ; at a guess, the 10000th emirp will be > the 10000th prime  (let loop ((i 10000) (p 9))    (define p+2 (fx+ p 2))    (cond [(not (emirp?-test p+2)) (loop i p+2)] [(fx= i 1) p+2] [else (loop (fx- i 1) p+2)]))) ;; -| MAIN |------------------------------------------------------------------------------------------(provide main)(define (main task)  ;; to avoid the *necessity* of calling from the command line multiple times, we reset the sieve on  ;; each invocation of main  (reset-sieve!)  (set-box! max-prime-tested 0)  (match task    ["1" (displayln (task1)) (report-mpg)]    ["2" (displayln (task2)) (report-mpg)]    ["3" (displayln (task3 emirp-prime?/sieve extend-sieve!)) (report-mpg/sieved)])) ;; -| TESTS |-----------------------------------------------------------------------------------------(module+ test  (require rackunit)  (check-false (emirp-prime?/sieve 12))  (check-false (emirp-prime?/sieve 23))  (check-true (emirp-prime?/sieve 13))  (check-equal?   (for/list       ((n (sequence-filter emirp-prime?/sieve (in-range 11 100000 2)))        (_ (in-range 3))) n)   '(13 17 31))  (check-equal? (time (task1 emirp-prime?/sieve)) (time (task1)))  (check-equal? (time (task2 emirp-prime?/sieve)) (time (task2)))  (check-equal? (time (task3 emirp-prime?/sieve extend-sieve!)) (time (task3)))) `
Output:
```"show the first twenty emirps."
'(13 17 31 37 71 73 79 97 107 113 149 157 167 179 199 311 337 347 359 389)
"show all emirps between 7,700 and 8,000"
'(7717 7757 7817 7841 7867 7879 7901 7927 7949 7951 7963)
"show the 10,000th emirp"
948349```

Second program, run from Linux bash shell:

```\$ for i in 1 2 3; do racket -t Emirp-primes.rkt -m \$i; echo; done
"show the first twenty emirps." [#<procedure:emirp-prime?>]
(13 17 31 37 71 73 79 97 107 113 149 157 167 179 199 311 337 347 359 389)
Max prime tested (using math/number-theory): 991

"show all emirps between 7,700 and 8,000" [#<procedure:emirp-prime?>]
(7717 7757 7817 7841 7867 7879 7901 7927 7949 7951 7963)
Max prime tested (using math/number-theory): 9787

"show the 10,000th emirp" [#<procedure:...Emirp-primes.rkt:77:5>]
948349
Sieve size: 999998```

## REXX

### version 1

Specifications of arguments note:   The following REXX program accepts:

•   a single number       N,   indicates to display the   Nth   emirp prime
•   two numbers     N     M,   indicates to display the   Nth   ──►   Mth   emirp primes.
•   two numbers     N   -M,   indicates to display the emirp primes between   N   and  M   (inclusive).

Programming note:   the trial division method of generating (regular) primes is a bit on the slow side, so some
memoization was added (assisting with the  j ),   and some of the trial divisions were hard-coded to minimize
the CPU time a bit.

`/*REXX program finds  emirp  primes (base 10):  when a prime reversed, is another prime.*/parse arg x y .                                  /*obtain optional arguments from the CL*/if x=='' | x==","  then do;  x=1;  y=20;  end    /*Not specified?  Then use the default.*/if y==''  then y=x                               /* "      "         "   "   "     "    */r=y<0;    y=abs(y)                               /*display a  range  of  emirp primes ? */rly=length(y) + \r                               /*adjusted length of the  Y  value.    */!.=0;  c=0;   _=2 3 5 7 11 13 17;   \$=           /*isP; emirp count; low primes; emirps.*/    do #=1  for words(_);   p=word(_,#);   @.#=p;    !.p=1;    end  /*#*/#=#-1;   ip=#;  s.#[email protected].#**2                       /*adjust # (for the DO loop);  last P².*/                            /*▒▒▒▒▒▒▒▒▒▒▒▒▒▒ [↓]   generate more primes within range.   */    do [email protected].#+2  by 2                             /*only find  odd  primes from here on. */    if length(#)>rly  then leave                 /*have we enough primes for emirps?    */    if j//3      ==0  then iterate               /*is  J  divisible by three?           */    if right(j,1)==5  then iterate               /*is the right-most digit a "5" ?      */    if j//7      ==0  then iterate               /*is  J  divisible by seven?           */    if j//11     ==0  then iterate               /*is  J  divisible by eleven?          */    if j//13     ==0  then iterate               /*is  J  divisible by thirteen?        */                                                 /*[↑]  the above five lines saves time.*/          do k=ip  while  s.k<=j                 /*divide by the known  odd  primes.    */          if j//@.k==0  then iterate j           /*J divisible by X?  Then ¬prime.   ___*/          end   /*k*/                            /* [↑]  divide by odd primes up to √ j */    #=#+1                                        /*bump the number of primes found.     */    @.#=j;      s.#=j*j;     !.j=1               /*assign to sparse array; prime²; prime*/    end         /*j*/                            /* [↑]  keep generating until enough.  */                            /*▒▒▒▒▒▒▒▒▒▒▒▒▒▒ [↓]    filter  emirps  for the display.    */    do j=6  to @.#;   [email protected].j                      /*traipse through the regular primes.  */    if (r&_>y) | (\r&c==y)  then leave           /*is the prime not within the range?   */    __=reverse(_)                                /*reverse (digits) of the regular prime*/    if \!.__   | _==__    then iterate           /*is the  reverse  a different prime ? */    c=c+1                                        /*bump the emirp prime counter.        */    if (r&_<x) | (\r&c<x) then iterate           /*is  emirp  not within allowed range? */    \$=\$ _                                        /*append prime to the emirpPrime list. */    end   /*j*/                                  /* [↑]  list:  by value  or  by range. */                                                 /* [↓]  display the emirp list.        */say strip(\$);   say;   n=words(\$);   ?=(n\==1)   /*display the  emirp primes  wanted.   */if ?  then say  n   'emirp primes shown.'        /*stick a fork in it,  we're all done. */`

output   when using the following for input:   1   20

```13 17 31 37 71 73 79 97 107 113 149 157 167 179 199 311 337 347 359 389

20 emirp primes shown.
```

output   when using the following for input:   7700   -8000

```7717 7757 7817 7841 7867 7879 7901 7927 7949 7951 7963

11 emirp primes shown.
```

output   when using the following for input:   10000

```948349
```

### version 2

` /********************************************************************* * 27.03.2014 Walter Pachl *********************************************************************/ Parse Arg run first.=1 nn=0 ol='' lb='00'x If run='' Then run=1 call cprimes 20,20,'A' main_loop: Do ip=1 To 1000000                    /* loop over all primes       */   p=primes.ip                         /* candidate                  */   If p=0 Then     call cprimes 20,hp+1,'B'   p=primes.ip                        /* candidate                  */   r=reverse(p)                       /* reversed candidate         */   If p<>r Then Do                    /* not a palindromic prime    */     If is_prime(r) Then Do           /* reversed p is a prime      */       nn=nn+1                        /* increment number of hits   */       Select         When run<='1' Then Do           If nn<21 Then Call show 1,'the first 20 emirps:',4           If nn=20 Then             Leave           End         When(run='2') Then Do           If hp<8000 Then             Call cprimes 1,8000,'C'           If 7700<p & p<8000 Then Call show 2,'emirps between 7700 and 8000:',5           If p>8000 Then             Leave           End         When(run='3') Then Do           If nn=10000 Then Do             Call show 3,'the 10.000th emirp:',6             Leave             End           End         When(run='4') Then Do           Call cprimes 1,999999    /* dirty trick to speed thins up */           If nn=10000 Then Do             Call show 4,'the 10.000th emirp (alternate version):',6             Leave             End           End         Otherwise Do           Say 'Invoke as ptx 1/2/3'           Exit           End         End       End     End  End Call oo Say 'largest prime:' hp Exit  show: Parse Arg task,header,nl If first.task Then Do   Call o header||lb   first.task=0   End Call o right(p,nl) If nn=10 Then   Call o lb Return cprimes: Procedure Expose primes. psquare. is_prime. nprimes hp/********************************************************************** adapted for my needs from REXX's Extensible prime generation* Fill the array primes with prime numbers* so that it contains at least num primes and all primes<=mp*********************************************************************/  Parse Arg num,mp  If symbol('primes.0')=='LIT' Then Do  /* 1st time here? Initialize */    primes.=0                           /* prime numbers             */    is_prime.=0                         /* is_prime.x -> x is prime  */    psquare.=0                          /* psquare.x = square of     */    plist='2 3 5 7 11 13 17 19 23'      /* knows low primes.         */    Do i=1 For words(plist)      p=word(plist,i)      primes.i=p      is_prime.p=1      End    nprimes=i-1    primes.0=nprimes+1    psquare.nprimes=primes.nprimes**2   /* square of this prime      */    End                             /* [?]  done with building low Ps */  Do j=primes.nprimes+2 By 2 While nprimes<num | primes.nprimes<mp    If j//3==0 Then       Iterate    If right(j,1)==5 Then Iterate    If j//7==0 Then       Iterate    If j//11==0 Then      Iterate    If j//13==0 Then      Iterate    If j//17==0 Then      Iterate    If j//19==0 Then      Iterate    If j//23==0 Then      Iterate    Do k=primes.0-1 While psquare.k<=j  /* check for other known primes */      If j//primes.k==0 Then      /* J is divisible by k-th prime    */        Iterate j                 /* j is not prime                  */      End    nprimes=nprimes+1             /* bump number of primes found.    */    primes.nprimes=j    psquare.nprimes=j*j    is_prime.j=1    hp=j    End  Return  is_prime: Procedure Expose primes. psquare. is_prime. nprimes hp /********************************************************************* * check if x is a prime number *********************************************************************/ Parse Arg x If x>hp Then   Call cprimes 1,x Return is_prime.x o: ol=ol||arg(1)   Returnoo: Do While ol<>''      Parse Var ol l (lb) ol      Say l      End    Return`
output
```rexx ptz 1
the first 20 emirps:
13  17  31  37  71  73  79  97 107 113
149 157 167 179 199 311 337 347 359 389
largest prime: 991

rexx ptz 2
emirps between 7700 and 8000:
7717 7757 7817 7841 7867 7879 7901 7927 7949 7951 7963
largest prime: 10007

rexx ptz 3
the 10.000th emirp:
948349
largest prime: 1000003

rexx ptz 4 (slightly faster that rexx ptz 3)
the 10.000th emirp (alternate version):
948349
largest prime: 1000003```

## Ring

` nr = 1m = 2see "first 20 :" + nlwhile nr < 21      emirp = isEmirp(m)      if emirp = 1 see m see " "         nr++ ok      m++end  see nl + nl nr = 1m = 7701see "between 7700 8000 :" + nlwhile m > 7700 and m < 8000      emirp = isEmirp(m)      if emirp = 1 see m see " " nr++ ok      m++endsee nl + nl nr = 1m = 2see "Nth 10000 :" + nlwhile nr > 0 and nr < 101      emirp = isEmirp(m)      if emirp = 1 nr++ ok      m++endsee m + nl func isEmirp n     if not isPrime(n) return false ok     cStr = string(n)       cstr2 = ""     for x = len(cStr) to 1 step -1 cStr2 += cStr[x] next     rev = number(cstr2)     if rev = n return false ok     return isPrime(rev) func isPrime n     if n < 2 return false ok     if n < 4 return true ok     if n % 2 = 0 return false ok     for d = 3 to sqrt(n) step 2          if n % d = 0 return false ok     next     return true `

## Ruby

`require 'prime' emirp = Enumerator.new do |y|  Prime.each do |prime|    rev = prime.to_s.reverse.to_i    y << prime  if rev.prime? and rev != prime  endend puts "First 20 emirps:", emirp.first(20).join(" ")puts "Emirps between 7,700 and 8,000:"emirp.with_index(1) do |prime,i|  print "#{prime} "  if (7700..8000).cover?(prime)  if i==10000    puts "", "10,000th emirp:", prime    break  endend`
Output:
```First 20 emirps:
13 17 31 37 71 73 79 97 107 113 149 157 167 179 199 311 337 347 359 389
Emirps between 7,700 and 8,000:
7717 7757 7817 7841 7867 7879 7901 7927 7949 7951 7963
10,000th emirp:
948349
```

## Rust

`#![feature(iterator_step_by)] extern crate primal; fn is_prime(n: u64) -> bool {    if n == 2 || n == 3 || n == 5 || n == 7 || n == 11 || n == 13 { return true; }    if n % 2 == 0 || n % 3 == 0 || n % 5 == 0 || n % 7 == 0 || n % 11 == 0 || n % 13 == 0 { return false; }    let root = (n as f64).sqrt() as u64 + 1;    (17..root).step_by(2).all(|i| n % i != 0)} fn is_emirp(n: u64) -> bool {    let mut aux = n;    let mut rev_prime = 0;    while aux > 0 {        rev_prime = rev_prime * 10 + aux  % 10;        aux /= 10;    }    if n == rev_prime { return false; }    is_prime(rev_prime)} fn calculate() -> (Vec<usize>, Vec<usize>, usize) {    let mut count = 1;    let mut vec1 = Vec::new();    let mut vec2 = Vec::new();    let mut emirp_10_000 = 0;     for i in primal::Primes::all() {        if is_emirp(i as u64) {            if count < 21 { vec1.push(i) }            if i > 7_700 && i < 8_000 { vec2.push(i) }            if count == 10_000 {                emirp_10_000 = i;                break;            }            count += 1;        }    }     (vec1, vec2, emirp_10_000)} fn main() {    let (vec1, vec2, emirp_10_000) = calculate();     println!("First 20 emirp-s : {:?}", vec1);    println!("Emirps-s between 7700 and 8000 : {:?}", vec2);    println!("10.000-th emirp : {}", emirp_10_000);}`
Output:
```First 20 primes : [13, 17, 31, 37, 71, 73, 79, 97, 107, 113, 149, 157, 167, 179, 199, 311, 337, 347, 359, 389]
Emirps-s between 7700 and 8000 : [7717, 7757, 7817, 7841, 7867, 7879, 7901, 7927, 7949, 7951, 7963]
10.000-th emirp : 948349

real	0m0.040s
user	0m0.036s
sys	0m0.003s
```

## Scala

### Using BigInt's isProbablePrime()

The isProbablePrime() method performs a Miller-Rabin primality test to within a given certainty.

`def isEmirp( v:Long ) : Boolean = { val b = BigInt(v.toLong) val r = BigInt(v.toString.reverse.toLong) b != r && b.isProbablePrime(16) && r.isProbablePrime(16)} // Generate the output{  val (a,b1,b2,c) = (20,7700,8000,10000)  println( "%32s".format(          "First %d emirps: ".format( a )) + Stream.from(2).filter( isEmirp(_) ).take(a).toList.mkString(",") )  println( "%32s".format( "Emirps between %d and %d: ".format( b1, b2 )) + {for( i <- b1 to b2 if( isEmirp(i) ) ) yield i}.mkString(",") )  println( "%32s".format(                "%,d emirp: ".format( c )) + Iterator.from(2).filter( isEmirp(_) ).drop(c-1).next )}`
Output:
```               First 20 emirps: 13,17,31,37,71,73,79,97,107,113,149,157,167,179,199,311,337,347,359,389
Emirps between 7700 and 8000: 7717,7757,7817,7841,7867,7879,7901,7927,7949,7951,7963
10,000 emirp: 948349```

## Sidef

Translation of: Perl
`func forprimes(a, b, callback) {    for (var p = a.dec.next_prime; p <= b; p.next_prime!) {        callback(p)    }} func is_emirp(p) {    var str = Str(p)    var rev = str.reverse    (str != rev) && is_prime(Num(rev))} func emirp_list(count) {    var i = 13    var inc = (100 + 10*count)    var n = []    while (n.len < count) {        forprimes(i, i+inc - 1, {|p|            is_emirp(p) && (n << p)        })        (i, inc) = (i+inc, int(inc * 1.03) + 1000)    }    n.splice(count)    return n} say ("First 20: ", emirp_list(20).join(' '))say ("Between 7700 and 8000: ", gather {        forprimes(7700, 8000, {|p| is_emirp(p) && take(p) })    }.join(' '))say ("The 10,000'th emirp: ", emirp_list(10000)[-1])`
Output:
```First 20: 13 17 31 37 71 73 79 97 107 113 149 157 167 179 199 311 337 347 359 389
Between 7700 and 8000: 7717 7757 7817 7841 7867 7879 7901 7927 7949 7951 7963
The 10,000'th emirp: 948349
```

## Smalltalk

Works with Smalltalk/X

This uses a builtin class called LazyCons, which is useful to implement infinite lists. The code is functional, looking somewhat Scheme'isch (that's what blocks are for).

First an emirp checker:

`isEmirp :=     [:p | |e|         (e := p asString reversed asNumber) isPrime        and:[ e ~= p ]    ].`

an infinite list of primes:

`primeGen :=     [:n |         LazyCons car:n cdr:[primeGen value:(n nextPrime)]    ].`

an infinite list of emirps, taking an infinite list of primes as arg:

`emirpGen :=     [:l | |rest el|        rest := l.        [ el := rest car. rest := rest cdr. isEmirp value:el ] whileFalse.        LazyCons car:el cdr:[emirpGen value:rest]    ].`

two infinite lists:

`listOfPrimes := primeGen value:2.listOfEmirps := emirpGen value:listOfPrimes.`

generating output:

`Transcript     show:'first 20 emirps: ';     showCR:(listOfEmirps take:20) asArray. Transcript     show:'emirps between 7700 and 8000 are: ';     showCR:((7700 to:8000) select:[:n | n isPrime and:[isEmirp value:n]]). Transcript     show:'10000''th emirp: ';     showCR:(listOfEmirps nth:10000).`
Generates:
```first 20 emirps: #(13 17 31 37 71 73 79 97 107 113 149 157 167 179 199 311 337 347 359 389)
emirps between 7700 and 8000 are: OrderedCollection(7717 7757 7817 7841 7867 7879 7901 7927 7949 7951 7963)
10000'th emirp: 948349```

LazyCons is easily defined as:

`Object subclass: #Cons     instancevariableNames:'car cdr'. car:newCar cdr:newCdr    car := newCar. cdr := newCdr car    ^car cdr    ^cdr Cons subclass:#LazyCons cdr    cdr := cdr value.    self changeClassTo:Cons.    ^cdr`

## Stata

`emirp 1000list in 1/20, noobs noh   +-----+  |  13 |  |  17 |  |  31 |  |  37 |  |  71 |  |-----|  |  73 |  |  79 |  |  97 |  | 107 |  | 113 |  |-----|  | 149 |  | 157 |  | 167 |  | 179 |  | 199 |  |-----|  | 311 |  | 337 |  | 347 |  | 359 |  | 389 |  +-----+ emirp 10000list if 7700<p & p<8000, noobs noh   +------+  | 7717 |  | 7757 |  | 7817 |  | 7841 |  | 7867 |  |------|  | 7879 |  | 7901 |  | 7927 |  | 7949 |  | 7951 |  |------|  | 7963 |  +------+ emirp 1000000list if _n==10000, noobs noh   +--------+  | 948349 |  +--------+`

Now the definition of emirp.ado:

`program emirp	args n	qui clear	qui mata: build(`n')	qui save temp, replace	qui replace p=real(strreverse(strofreal(p)))	qui merge 1:1 p using temp, keep(3) nogen	qui drop if real(strreverse(strofreal(p)))==pend matareal colvector sieve(real scalar n) {	real colvector a	real scalar i,j	if (n<2) return(J(0,1,.))	a=J(n,1,1)	a=0	for (i=1; i<=n; i++) {		if (a[i]) {			j=i*i			if (j>n) return(select(1::n,a))			for (; j<=n; j=j+i) a[j]=0		}	}} function build(n) {	a=sieve(n)	st_addobs(rows(a))	st_addvar("long","p")	st_store(.,1,a)}end`

## Tcl

Library: Tcllib (Package: math::numtheory)
`package require math::numtheory # Import only to keep line lengths downnamespace import math::numtheory::isprimeproc emirp? {n} {    set r [string reverse \$n]    expr {\$n != \$r && [isprime \$n] && [isprime \$r]}} # Generate the various emirpsfor {set n 2;set emirps {}} {[llength \$emirps] < 20} {incr n} {    if {[emirp? \$n]} {lappend emirps \$n}}puts "first20: \$emirps" for {set n 7700;set emirps {}} {\$n <= 8000} {incr n} {    if {[emirp? \$n]} {lappend emirps \$n}}puts "7700-8000: \$emirps" for {set n 2;set ne 0} true {incr n} {    if {[emirp? \$n] && [incr ne] == 10000} break}puts "10,000: \$n"`
Output:
```first20: 13 17 31 37 71 73 79 97 107 113 149 157 167 179 199 311 337 347 359 389
7700-8000: 7717 7757 7817 7841 7867 7879 7901 7927 7949 7951 7963
10,000: 948349
```

## VBA

`Option Explicit Private Const MAX As Long = 5000000Private Emirps As New CollectionPrivate CollTemp As New Collection Sub Main()Dim tt = Timer    FillCollectionOfEmirps    Debug.Print "At this point : Execution time = " & Timer - t & " seconds."    Debug.Print "We have a Collection of the " & Emirps.Count & " first Emirps."    Debug.Print "---------------------------"    'show the first   twenty   emirps    Debug.Print "the first 20 emirps: "; ExtractEmirps(1, 20)    'show all emirps between 7,700 and 8,000    Debug.Print "all emirps between 7,700 and 8,000: "; ExtractEmirps(7700, 8000, True)    'show the   10,000th   emirp    Debug.Print "the 10,000th emirp: "; ExtractEmirps(10000, 10000)  End Sub Private Function ExtractEmirps(First As Long, Last As Long, Optional Value = False) As StringDim temp\$, i As Long, e    If First = Last Then        ExtractEmirps = Emirps(First)    Else        If Not Value Then            For i = First To Last                temp = temp & ", " & Emirps(i)            Next        Else            For Each e In Emirps                If e > First And e < Last Then                    temp = temp & ", " & e                End If                If e = Last Then Exit For            Next e        End If        ExtractEmirps = Mid(temp, 3)    End IfEnd Function Private Sub FillCollectionOfEmirps()Dim Primes() As Long, e, i As Long    Primes = Atkin    For i = LBound(Primes) To UBound(Primes)        CollTemp.Add Primes(i), CStr(Primes(i))    Next i    For Each e In CollTemp        If IsEmirp(e) Then Emirps.Add e    NextEnd Sub Private Function Atkin() As Long()Dim MyBool() As BooleanDim SQRT_MAX As Long, i&, j&, N&, cpt&, MAX_TEMP As Long, temp() As Long    ReDim MyBool(MAX)    SQRT_MAX = Sqr(MAX) + 1    MAX_TEMP = Sqr(MAX / 4) + 1    For i = 1 To MAX_TEMP        For j = 1 To SQRT_MAX            N = 4 * i * i + j * j            If N <= MAX And (N Mod 12 = 1 Or N Mod 12 = 5) Then                MyBool(N) = True            End If        Next j    Next i    MAX_TEMP = Sqr(MAX / 3) + 1    For i = 1 To MAX_TEMP        For j = 1 To SQRT_MAX            N = 3 * i * i + j * j            If N <= MAX And N Mod 12 = 7 Then                MyBool(N) = True            End If        Next j    Next i    For i = 1 To SQRT_MAX        For j = 1 To SQRT_MAX            N = 3 * i * i - j * j            If i > j And N <= MAX And N Mod 12 = 11 Then                MyBool(N) = True            End If        Next j    Next i    For i = 5 To SQRT_MAX Step 2        If MyBool(i) Then            For j = i * i To MAX Step i                MyBool(j) = False            Next        End If    Next    ReDim temp(MAX / 2)    temp(0) = 2: temp(1) = 3: cpt = 2    For i = 5 To MAX Step 2        If MyBool(i) Then temp(cpt) = i: cpt = cpt + 1    Next    ReDim Preserve temp(cpt - 1)    Atkin = tempEnd Function Private Function IsEmirp(N) As BooleanDim a As String, b As String    a = StrReverse(CStr(N)): b = CStr(N)    If a <> b Then        On Error Resume Next        CollTemp.Add a, a        If Err.Number > 0 Then            IsEmirp = True        Else            CollTemp.Remove a        End If        On Error GoTo 0    End IfEnd Function`
Output:
```At this point : Execution time = 13,23047 seconds.
We have a Collection of the 29952 first Emirps.
---------------------------
the first 20 emirps: 13, 17, 31, 37, 71, 73, 79, 97, 107, 113, 149, 157, 167, 179, 199, 311, 337, 347, 359, 389
all emirps between 7,700 and 8,000: 7717, 7757, 7817, 7841, 7867, 7879, 7901, 7927, 7949, 7951, 7963
the 10,000th emirp: 948349```

## Visual Basic .NET

Translation of: C#
`Imports System.Runtime.CompilerServices Module Module1    <Extension()>    Function ToHashSet(Of T)(source As IEnumerable(Of T)) As HashSet(Of T)        Return New HashSet(Of T)(source)    End Function     <Extension()>    Function Reverse(number As Integer) As Integer        If number < 0 Then            Return -Reverse(-number)        End If        If number < 10 Then            Return number        End If         Dim rev = 0        While number > 0            rev = rev * 10 + number Mod 10            number = number \ 10        End While         Return rev    End Function     <Extension()>    Function Delimit(Of T)(source As IEnumerable(Of T), Optional seperator As String = " ") As String        Return String.Join(If(seperator, " "), source)    End Function     Iterator Function Primes(bound As Integer) As IEnumerable(Of Integer)        If bound < 2 Then            Return        End If        Yield 2         Dim composite As New BitArray((bound - 1) / 2)        Dim limit As Integer = Int((Int(Math.Sqrt(bound)) - 1) / 2)        For i = 0 To limit - 1            If composite(i) Then                Continue For            End If            Dim prime = 2 * i + 3            Yield prime             For j As Integer = Int((prime * prime - 2) / 2) To composite.Count - 1 Step prime                composite(j) = True            Next        Next        For i = limit To composite.Count - 1            If Not composite(i) Then                Yield 2 * i + 3            End If        Next    End Function     Iterator Function FindEmirpPrimes(limit As Integer) As IEnumerable(Of Integer)        Dim ps = Primes(limit).ToHashSet()         For Each p In ps            Dim rev = p.Reverse()            If rev <> p AndAlso ps.Contains(rev) Then                Yield p            End If        Next    End Function     Sub Main()        Dim limit = 1_000_000        Console.WriteLine("First 20:")        Console.WriteLine(FindEmirpPrimes(limit).Take(20).Delimit())        Console.WriteLine()         Console.WriteLine("Between 7700 and 8000:")        Console.WriteLine(FindEmirpPrimes(limit).SkipWhile(Function(p) p < 7700).TakeWhile(Function(p) p < 8000).Delimit())        Console.WriteLine()         Console.WriteLine("10000th:")        Console.WriteLine(FindEmirpPrimes(limit).ElementAt(9999))    End Sub End Module`
Output:
```First 20:
13 17 31 37 71 73 79 97 107 113 149 157 167 179 199 311 337 347 359 389

Between 7700 and 8000:
7717 7757 7817 7841 7867 7879 7901 7927 7949 7951 7963

10000th:
948349```

## zkl

Uses the solution from task Extensible prime generator#zkl. Saves the primes to a list, which gets pretty big.

`var PS=Import("Src/ZenKinetic/sieve").postponed_sieve;var ps=Utils.Generator(PS), plist=ps.walk(10).copy();   fcn isEmirp(p){ rp:=p.toString().reverse().toInt();   if(p==rp) return(False);   if(plist.holds(rp)) return(True);   tp:=p; mp:=p.max(rp); while(tp<mp) { plist.append(tp=ps.next()) }   return(tp==rp);} Utils.Generator(PS).filter(20,isEmirp); Utils.Generator(PS).filter(fcn(p){if(p>8000)return(Void.Stop); p>7700 and isEmirp(p)}); Utils.Generator(PS).reduce(fcn(N,p){N+=isEmirp(p); (N==10000) and T(Void.Stop,p) or N },0);`
Output:
```L(13,17,31,37,71,73,79,97,107,113,149,157,167,179,199,311,337,347,359,389)
L(7817,7841,7867,7879,7901,7927,7949,7951,7963)
948349
```