# One-two primes

One-two primes is a draft programming task. It is not yet considered ready to be promoted as a complete task, for reasons that should be found in its talk page.

Generate the sequence an where each term in a is the smallest n digit prime number (in base 10) composed entirely of the digits 1 and 2.

It is conjectured, though not proven, that a conforming prime exists for all n. No counter examples have been found up to several thousand digits.

• Find and show here, the first 20 elements in the sequence (from 1 digit through 20 digits), or as many as reasonably supported by your language if it is fewer.

Stretch
• Find and and show the abbreviated values for the elements with n equal to 100 through 2000 in increments of 100.
Shorten the displayed number by replacing any leading 1s in the number with the count of 1s, and show the remainder of the number. (See Raku example for reference)

## ALGOL 68

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

As more than 64 bit numbers are required, this uses Algol 68G's LONG LONG INT which has programmer specified precision. The default precision is sufficient for this task.

```BEGIN # find the lowest 1-2 primes with n digits, 1-2 primes contain the     #
#                                               digits 1 and 2 only    #

PR read "primes.incl.a68" PR                   # include prime utilities #
[ 1 : 20 ]CHAR v12;              # will hold the digits of the 1-2 prime #
FOR i TO UPB v12 DO v12[ i ] := "0" OD;        # in little endian format #
BOOL v12 overflowed := FALSE;

# initialises v12 to digit count 1s                                      #
PROC init v12 = ( INT digit count )VOID:
BEGIN
v12 overflowed := FALSE;
FOR digit TO digit count DO
v12[ digit ] := "1"
OD
END # init v12 # ;
# sets v12 to value - the digits corresponding to the set bits of value  #
#                                sre set to 2, the others 1              #
PROC set v12 = ( INT digit count, INT value )VOID:
BEGIN
INT v := value;
FOR digit TO digit count WHILE v > 0 DO
v12[ digit ] := IF ODD v THEN "2" ELSE "1" FI;
v OVERAB 2
OD;
v12 overflowed := v /= 0
END # set v12 # ;

# converts v12 to a numeric valiue                                       #
PROC v12 value = ( INT digit count )LONG LONG INT:
BEGIN
LONG LONG INT n12 := 0;
FOR digit FROM digit count BY -1 TO LWB v12 DO
n12 *:= 10 +:= IF v12[ digit ] = "2" THEN 2 ELSE 1 FI
OD;
n12
END # v12 value # ;

# returns TRUE if the value in v12 is prime, FALSE otherwise             #
PROC v12 is prime = ( INT digit count )BOOL: is probably prime( v12 value( digit count ) );

# show the first 20 1-2 primes                                           #
FOR digits TO 20 DO
init v12( digits );
# 2 can only be the final digit of the 1 digit 1-2 prime             #
# for all other 1-2 primes, the final digit can't be 2               #
FOR v FROM IF digits = 1 THEN 1 ELSE 2 FI BY 2 WHILE NOT v12 is prime( digits )
AND NOT v12 overflowed
DO
set v12( digits, v )
OD;
print( ( whole( digits, -5 ), ": " ) );
IF v12 overflowed THEN
# couldn't find a prime                                          #
print( ( "-1" ) )
ELSE
# found a prime                                                  #
FOR digit FROM digits BY -1 TO LWB v12 DO
print( ( v12[ digit ] ) )
OD
FI;
print( ( newline ) )
OD

END```
Output:
```    1: 2
2: 11
3: 211
4: 2111
5: 12211
6: 111121
7: 1111211
8: 11221211
9: 111112121
10: 1111111121
11: 11111121121
12: 111111211111
13: 1111111121221
14: 11111111112221
15: 111111112111121
16: 1111111112122111
17: 11111111111112121
18: 111111111111112111
19: 1111111111111111111
20: 11111111111111212121
```

## AppleScript

Like the other solutions here, this assumes that "composed entirely of the digits 1 and 2" actually means "composed entirely of the digits 1 and/or 2".

```on oneTwoPrimes(n)
-- Take the first single-digit prime (the only even one) as read.
set output to {"First " & n & " results:", " 1: 2"}
repeat with n from 2 to n
-- Generate odd one-two numbers by adding 1 to each of the n binary digits
-- of each even number < 2 ^ n, treating the results as decimal digits.
set none to true
repeat with even from 0 to (2 ^ n - 2) by 2
set p10 to 1
set oneTwo to p10 -- even's bit 0 + 1.
repeat (n - 1) times
set even to even div 2
set p10 to p10 * 10
set oneTwo to oneTwo + (even mod 2 + 1) * p10
end repeat
-- Finish for this n if a one-two number proves to be a prime.
if (isPrime(oneTwo)) then
set end of output to text -2 thru -1 of (space & n) & ": " & intToText(oneTwo, "")
set none to false
exit repeat
end if
end repeat
if (none) then set end of output to text -2 thru -1 of (space & n) & ": No prime identified"
end repeat

return join(output, linefeed)
end oneTwoPrimes

on isPrime(n)
if (n < 4) then return (n > 1)
if ((n mod 2 is 0) or (n mod 3 is 0)) then return false
repeat with i from 5 to (n ^ 0.5) div 1 by 6
if ((n mod i is 0) or (n mod (i + 2) is 0)) then return false
end repeat
return true
end isPrime

on intToText(int, separator)
set groups to {}
repeat while (int > 999)
set groups's beginning to ((1000 + (int mod 1000 as integer)) as text)'s text 2 thru 4
set int to int div 1000
end repeat
set groups's beginning to int as integer
return join(groups, separator)
end intToText

on join(lst, delim)
set astid to AppleScript's text item delimiters
set AppleScript's text item delimiters to delim
set txt to lst as text
set AppleScript's text item delimiters to astid
return txt
end join

return oneTwoPrimes(20)
```
Output:

The last four results are due to AppleScript's limited number precision.

```"First 20 results:
1: 2
2: 11
3: 211
4: 2111
5: 12211
6: 111121
7: 1111211
8: 11221211
9: 111112121
10: 1111111121
11: 11111121121
12: 111111211111
13: 1111111121221
14: 11111111112221
15: 111111112111121
16: 1111111112122111
17: No prime identified
18: No prime identified
19: No prime identified
20: No prime identified"
```

## C

Translation of: Wren
Library: GMP
```#include <stdio.h>
#include <stdbool.h>
#include <string.h>
#include <gmp.h>

void firstOneTwo(mpz_t res, int n) {
char *s;
bool found = false;
mpz_t k, r, m, t, one, nine;
mpz_inits(k, r, m, t, one, nine, NULL);
mpz_set_ui(one, 1);
mpz_set_ui(nine, 9);
mpz_set_ui(k, 10);
mpz_pow_ui(k, k, n);
mpz_sub_ui(k, k, 1);
mpz_tdiv_q(k, k, nine);
mpz_mul_2exp(r, one, n);
mpz_sub_ui(r, r, 1);
while (mpz_cmp(m, r) <= 0) {
s = mpz_get_str(NULL, 2, m);
mpz_set_str(t, s, 10);
if (mpz_probab_prime_p(t, 15) > 0) {
found = true;
break;
}
}
if (!found) mpz_set_si(t, -1);
mpz_set(res, t);
mpz_clears(k, r, m, t, one, nine, NULL);
}

int main() {
int n, ix;
char *s;
mpz_t res;
mpz_init(res);
for (n = 1; n < 21; ++n) {
firstOneTwo(res, n);
gmp_printf("%4d: %Zd\n", n, res);
}
for (n = 100; n <= 2000; n += 100) {
firstOneTwo(res, n);
if (!mpz_cmp_si(res, -1)) {
printf("No %d-digit prime found with only digits 1 or 2.", n);
} else {
s = mpz_get_str(NULL, 10, res);
ix = strchr(s, '2') - s;
printf("%4d: (1 x %4d) %s\n", n, ix, s + ix);
}
}
mpz_clear(res);
return 0;
}
```
Output:
```Same as Wren example.
```

## J

Implementation:

```pr12=: {{ for_j. 10x#.1 2{~1|."1#:i.2^y do. if. 1 p:j do. j return. end. end. }}"0
```

```   ,.pr12 1+i.20
2
11
211
2111
12211
111121
1111211
11221211
111112121
1111111121
11111121121
111111211111
1111111121221
11111111112221
111111112111121
1111111112122111
11111111111112121
111111111111112111
1111111111111111111
11111111111111212121
```

## Julia

```""" rosettacode.org/wiki/One-two_primes """

using IntegerMathUtils # for the call to libgmp's gmpz_probab_prime_p

""" From Chai Wah Wu's Python code at oeis.org/A036229 """
function show_oeis36229(wanted = 2000)
for ndig in vcat(1:20, 100:100:wanted)
k, r, m = (big"10"^ndig - 1) ÷ 9, big"2"^ndig - 1, big"0"
while m <= r
t = k + parse(BigInt, string(m, base = 2))
if is_probably_prime(t)
pstr = string(t)
if ndig < 21
else
k = something(findfirst(!=('1'), pstr), ndig) - 1
println(lpad(ndig, 4), ": (1 x \$k) ", pstr[k:end])
end
break
end
m += 1
end
end
end

show_oeis36229()
```
Output:
```   1: 2
2: 11
3: 211
4: 2111
5: 12211
6: 111121
7: 1111211
8: 11221211
9: 111112121
10: 1111111121
11: 11111121121
12: 111111211111
13: 1111111121221
14: 11111111112221
15: 111111112111121
16: 1111111112122111
17: 11111111111112121
18: 111111111111112111
19: 1111111111111111111
20: 11111111111111212121
100: (1 x 92) 121112211
200: (1 x 192) 121112211
300: (1 x 288) 1211121112221
400: (1 x 390) 12111122121
500: (1 x 488) 1221222111111
600: (1 x 590) 12112222221
700: (1 x 689) 121111111111
800: (1 x 787) 12122222221111
900: (1 x 891) 1222221221
1000: (1 x 988) 1222122111121
1100: (1 x 1087) 12112111121111
1200: (1 x 1191) 1211222211
1300: (1 x 1289) 122121221121
1400: (1 x 1388) 1222211222121
1500: (1 x 1489) 121112121121
1600: (1 x 1587) 12121222122111
1700: (1 x 1688) 1212121211121
1800: (1 x 1791) 1221211121
1900: (1 x 1889) 122212212211
2000: (1 x 1989) 122121121211
```

## Nim

Library: Nim-Integers

Based on the Python code in the OEIS A036229 entry.

```import std/[strformat, strutils]
import integers

let
One = newInteger(1)
Ten = newInteger(10)

proc a036229(n: Positive): Integer =
var k = Ten^n div 9
var r = One shl n - 1
var m = newInteger(0)
while m <= r:
let t = k + newInteger(`\$`(m, 2))
if t.isprime: return t
inc m
quit &"No {n}-digit prime found with only digits 1 or 2.", QuitFailure

func compressed(n: Integer): string =
let s = \$n
let idx = s.find('2')
result = &"(1 × {idx}) " & s[idx..^1]

for n in 1..20:
echo &"{n:4}: {a036229(n)}"
echo()
for n in countup(100, 2000, 100):
echo &"{n:4}: {compressed(a036229(n))}"
```
Output:
```   1: 2
2: 11
3: 211
4: 2111
5: 12211
6: 111121
7: 1111211
8: 11221211
9: 111112121
10: 1111111121
11: 11111121121
12: 111111211111
13: 1111111121221
14: 11111111112221
15: 111111112111121
16: 1111111112122111
17: 11111111111112121
18: 111111111111112111
19: 1111111111111111111
20: 11111111111111212121

100: (1 × 92) 21112211
200: (1 × 192) 21112211
300: (1 × 288) 211121112221
400: (1 × 390) 2111122121
500: (1 × 488) 221222111111
600: (1 × 590) 2112222221
700: (1 × 689) 21111111111
800: (1 × 787) 2122222221111
900: (1 × 891) 222221221
1000: (1 × 988) 222122111121
1100: (1 × 1087) 2112111121111
1200: (1 × 1191) 211222211
1300: (1 × 1289) 22121221121
1400: (1 × 1388) 222211222121
1500: (1 × 1489) 21112121121
1600: (1 × 1587) 2121222122111
1700: (1 × 1688) 212121211121
1800: (1 × 1791) 221211121
1900: (1 × 1889) 22212212211
2000: (1 × 1989) 22121121211
```

## Perl

Mostly generalized, but doesn't handle first term of the 0/1 case.

Library: ntheory
```use v5.36;
no warnings 'recursion';
use ntheory 'is_prime';

sub condense(\$n) { \$n =~ /^((.)\2+)/; my \$i = length \$1; \$i>9 ? "(\$2 x \$i) " . substr(\$n,\$i) : \$n }

sub combine (\$d, \$a, \$b, \$s='') {                      # NB: \$a < \$b
if (\$d==1 && is_prime \$s.\$a) { return \$s.\$a }
elsif (\$d==1 && is_prime \$s.\$b) { return \$s.\$b }
elsif (\$d==1                  ) { return 0     }
else                            { return combine(\$d-1,\$a,\$b,\$s.\$a) || combine(\$d-1,\$a,\$b,\$s.\$b) }
}

my(\$a,\$b) = (1,2);
say "Smallest n digit prime using only \$a and \$b (or '0' if none exists):";
printf "%4d: %s\n", \$_,          combine(\$_,\$a,\$b) for             1..20;
printf "%4d: %s\n", \$_, condense combine(\$_,\$a,\$b) for map 100*\$_, 1..20;

(\$a,\$b)   = (7,9);
say "\nSmallest n digit prime using only \$a and \$b (or '0' if none exists):";
printf "%4d: %s\n", \$_, condense combine(\$_,\$a,\$b) for 1..20, 100, 200;

# 1st term missing
#(\$a,\$b) = (0,1);
#printf "%4d: %s\n", \$_+1, combine(\$_,\$a,\$b,1) for 1..19;
```
Output:
```Smallest n digit prime using only 1 and 2 (or '0' if none exists):
1: 2
2: 11
3: 211
4: 2111
5: 12211
6: 111121
7: 1111211
8: 11221211
9: 111112121
10: 1111111121
11: 11111121121
12: 111111211111
13: 1111111121221
14: 11111111112221
15: 111111112111121
16: 1111111112122111
17: 11111111111112121
18: 111111111111112111
19: 1111111111111111111
20: 11111111111111212121
100: (1 x 92) 21112211
200: (1 x 192) 21112211
300: (1 x 288) 211121112221
400: (1 x 390) 2111122121
500: (1 x 488) 221222111111
600: (1 x 590) 2112222221
700: (1 x 689) 21111111111
800: (1 x 787) 2122222221111
900: (1 x 891) 222221221
1000: (1 x 988) 222122111121
1100: (1 x 1087) 2112111121111
1200: (1 x 1191) 211222211
1300: (1 x 1289) 22121221121
1400: (1 x 1388) 222211222121
1500: (1 x 1489) 21112121121
1600: (1 x 1587) 2121222122111
1700: (1 x 1688) 212121211121
1800: (1 x 1791) 221211121
1900: (1 x 1889) 22212212211
2000: (1 x 1989) 22121121211

Smallest n digit prime using only 7 and 9 (or '0' if none exists):
1: 7
2: 79
3: 797
4: 0
5: 77797
6: 777977
7: 7777997
8: 77779799
9: 777777799
10: 7777779799
11: 77777779979
12: 777777779777
13: 7777777779977
14: (7 x 11) 977
15: (7 x 11) 9797
16: (7 x 11) 97799
17: (7 x 15) 97
18: (7 x 13) 97977
19: (7 x 16) 997
20: (7 x 16) 9997
100: (7 x 93) 9979979
200: (7 x 192) 99777779
```

## Phix

```with javascript_semantics
include mpfr.e
function p12(integer n)
string res = repeat('1',n)
mpz p = mpz_init(res)
while not mpz_prime(p) do
for k=max(n-1,1) to 1 by -1 do
if res[k]='1' then
res[k] = '2'
exit
end if
res[k] = '1'
end for
mpz_set_str(p,res)
end while
if n>20 then
integer k = find('2',res)
res[2..k-2] = sprintf("..(%d 1s)..",k-1)
end if
return res
end function
integer hn = iff(platform()=JS?2:20)
for n in tagset(20)&tagstart(100,hn,100) do
printf(1,"%4d: %s\n",{n,p12(n)})
end for
```
Output:
```   1: 2
2: 11
3: 211
4: 2111
5: 12211
6: 111121
7: 1111211
8: 11221211
9: 111112121
10: 1111111121
11: 11111121121
12: 111111211111
13: 1111111121221
14: 11111111112221
15: 111111112111121
16: 1111111112122111
17: 11111111111112121
18: 111111111111112111
19: 1111111111111111111
20: 11111111111111212121
100: 1..(92 1s)..121112211
200: 1..(192 1s)..121112211
300: 1..(288 1s)..1211121112221
400: 1..(390 1s)..12111122121
500: 1..(488 1s)..1221222111111
600: 1..(590 1s)..12112222221
700: 1..(689 1s)..121111111111
800: 1..(787 1s)..12122222221111
900: 1..(891 1s)..1222221221
1000: 1..(988 1s)..1222122111121
1100: 1..(1087 1s)..12112111121111
1200: 1..(1191 1s)..1211222211
1300: 1..(1289 1s)..122121221121
1400: 1..(1388 1s)..1222211222121
1500: 1..(1489 1s)..121112121121
1600: 1..(1587 1s)..12121222122111
1700: 1..(1688 1s)..1212121211121
1800: 1..(1791 1s)..1221211121
1900: 1..(1889 1s)..122212212211
2000: 1..(1989 1s)..122121121211
```

Takes a while past 700 digits, capping it at 200 keeps it below 1s under pwa/p2js

## Python

```""" rosettacode.org/wiki/One-two_primes """
from itertools import permutations
from gmpy2 import is_prime

def oeis36229(wanted=20):
''' get first [wanted] entries in OEIS A036229 '''
for ndig in range(1, wanted + 1):
if ndig < 21 or ndig % 100 == 0:
dig = ['1' for _ in range(ndig)] + ['2' for _ in range(ndig)]
for arr in permutations(dig, ndig):
candidate = int(''.join(arr))
if is_prime(candidate):
print(f'{ndig:4}: ', candidate)
break

oeis36229()
```
Output:
```   1:  2
2:  11
3:  211
4:  2111
5:  12211
6:  111121
7:  1111211
8:  11221211
9:  111112121
10:  1111111121
11:  11111121121
12:  111111211111
13:  1111111121221
14:  11111111112221
15:  111111112111121
16:  1111111112122111
17:  11111111111112121
18:  111111111111112111
19:  1111111111111111111
20:  11111111111111212121
```

Translation of: Julia
```from sympy import isprime

def show_oeis36229(wanted=2000):
''' From Chai Wah Wu's Python code at oeis.org/A036229 '''
for ndig in list(range(1, 21)) + list(range(100, wanted + 1, 100)):
k, i, j = (10**ndig - 1) // 9, 2**ndig - 1, 0
while j <= i:
candidate = k + int(bin(j)[2:])
if isprime(candidate):
pstr = str(candidate)
if ndig < 21:
print(f'{ndig:4}: {pstr}')
else:
k = pstr.index('2')
print(f'{ndig:4}: (1 x {k}) {pstr[k-1:]}')

break

j += 1

show_oeis36229()
```
Output:

same as Wren, etc.

## Quackery

`1-2-prime` returns the first qualifying prime of the specified number of digits, or `-1` if no qualifying prime found.

`prime` is defined at Miller–Rabin primality test#Quackery.

```  [ -1 swap
dup temp put
bit times
[ [] i^
temp share times
[ dup 1 &
rot join swap
1 >> ]
swap witheach
[ 1+ swap 10 * + ]
dup prime iff
[ nip conclude ]
done
drop ]
temp release ]         is 1-2-prime ( n --> n )

[] 20 times [ i^ 1+ 1-2-prime join ]
witheach [ echo cr ]```
Output:
```2
11
211
2111
12211
111121
1111211
11221211
111112121
1111111121
11111121121
111111211111
1111111121221
11111111112221
111111112111121
1111111112122111
11111111111112121
111111111111112111
1111111111111111111
11111111111111212121
```

## Raku

```sub condense (\$n) { my \$i = \$n.index(2); \$i ?? "(1 x \$i) {\$n.substr(\$i)}" !! \$n }

sub onetwo (\$d, \$s='') { take \$s and return unless \$d; onetwo(\$d-1,\$s~\$_) for 1,2 }

sub get-onetwo (\$d) { (gather onetwo \$d).hyper.grep(&is-prime)[0] }

printf "%4d: %s\n", \$_, get-onetwo(\$_) for 1..20;
printf "%4d: %s\n", \$_, condense get-onetwo(\$_) for (1..20) »×» 100;
```
Output:
```   1: 2
2: 11
3: 211
4: 2111
5: 12211
6: 111121
7: 1111211
8: 11221211
9: 111112121
10: 1111111121
11: 11111121121
12: 111111211111
13: 1111111121221
14: 11111111112221
15: 111111112111121
16: 1111111112122111
17: 11111111111112121
18: 111111111111112111
19: 1111111111111111111
20: 11111111111111212121
100: (1 x 92) 21112211
200: (1 x 192) 21112211
300: (1 x 288) 211121112221
400: (1 x 390) 2111122121
500: (1 x 488) 221222111111
600: (1 x 590) 2112222221
700: (1 x 689) 21111111111
800: (1 x 787) 2122222221111
900: (1 x 891) 222221221
1000: (1 x 988) 222122111121
1100: (1 x 1087) 2112111121111
1200: (1 x 1191) 211222211
1300: (1 x 1289) 22121221121
1400: (1 x 1388) 222211222121
1500: (1 x 1489) 21112121121
1600: (1 x 1587) 2121222122111
1700: (1 x 1688) 212121211121
1800: (1 x 1791) 221211121
1900: (1 x 1889) 22212212211
2000: (1 x 1989) 22121121211```

### Generalized

This version will do the task requirements, but will also find (without modification):

Really, the only one that is a little tricky is the first one (0,1). That one required some specialized logic. All of the rest would work with the task specific version with different hard coded digits.

Limited the stretch to keep the run time reasonable. Finishes all in around 12 seconds on my system.

```for 929,(0,1),229,(1,2),930,(1,3),931,(1,4),932,(1,5),933,(1,6),934,(1,7),935,(1,8),
936,(1,9),937,(2,3),938,(2,7),939,(2,9),940,(3,4),941,(3,5),942,(3,7),943,(3,8),
944,(4,7),945,(4,9),946,(5,7),947,(5,9),948,(6,7),949,(7,8),950,(7,9),951,(8,9)
-> \$oeis, \$pair {

say "\nOEIS:A036{\$oeis} - Smallest n digit prime using only {\$pair[0]} and {\$pair[1]} (or '0' if none exists):";

sub condense (\$n) { \$n.subst(/(.) {} :my \$repeat=\$0; (\$repeat**{9..*})/, -> \$/ {"(\$0 x {1+\$1.chars}) "}) }

sub build (\$digit, \$sofar='') { take \$sofar and return unless \$digit; build(\$digit-1,\$sofar~\$_) for |\$pair }

sub get-prime (\$digits) {
(\$pair[0] ?? (gather build \$digits).first: &is-prime
!! (gather build \$digits-1, \$pair[1]).first: &is-prime
) // 0
}

printf "%4d: %s\n", \$_, condense .&get-prime for flat 1..20, 100, 200;
}
```
Output:
```OEIS:A036929 - Smallest n digit prime using only 0 and 1 (or '0' if none exists):
1: 0
2: 11
3: 101
4: 0
5: 10111
6: 101111
7: 1011001
8: 10010101
9: 100100111
10: 1000001011
11: 10000001101
12: 100000001111
13: 1000000111001
14: 10000000001011
15: 100000000100101
16: 1(0 x 10) 11101
17: 1(0 x 12) 1101
18: 1(0 x 11) 100111
19: 1(0 x 13) 10011
20: 1(0 x 12) 1100101
100: 1(0 x 93) 101101
200: 1(0 x 189) 1110101011

OEIS:A036229 - Smallest n digit prime using only 1 and 2 (or '0' if none exists):
1: 2
2: 11
3: 211
4: 2111
5: 12211
6: 111121
7: 1111211
8: 11221211
9: 111112121
10: 1111111121
11: 11111121121
12: 111111211111
13: 1111111121221
14: (1 x 10) 2221
15: 111111112111121
16: 1111111112122111
17: (1 x 13) 2121
18: (1 x 14) 2111
19: (1 x 19)
20: (1 x 14) 212121
100: (1 x 92) 21112211
200: (1 x 192) 21112211

... many lines manually omitted ...

OEIS:A036950 - Smallest n digit prime using only 7 and 9 (or '0' if none exists):
1: 7
2: 79
3: 797
4: 0
5: 77797
6: 777977
7: 7777997
8: 77779799
9: 777777799
10: 7777779799
11: 77777779979
12: 777777779777
13: 7777777779977
14: (7 x 11) 977
15: (7 x 11) 9797
16: (7 x 11) 97799
17: (7 x 15) 97
18: (7 x 13) 97977
19: (7 x 16) 997
20: (7 x 16) 9997
100: (7 x 93) 9979979
200: (7 x 192) 99777779

OEIS:A036951 - Smallest n digit prime using only 8 and 9 (or '0' if none exists):
1: 0
2: 89
3: 0
4: 8999
5: 89899
6: 888989
7: 8888989
8: 88888999
9: 888898889
10: 8888888989
11: 88888888999
12: 888888898999
13: 8888888999899
14: (8 x 13) 9
15: (8 x 10) 98999
16: (8 x 10) 989999
17: (8 x 16) 9
18: (8 x 13) 98889
19: (8 x 16) 989
20: (8 x 13) 9888989
100: (8 x 91) 998998889
200: (8 x 190) 9888898989```

## RPL

Candidate numbers are generated lexically by the `UP12` recursive function to speed up execution: the first 20 terms are found in 1 minute 39 seconds.

Works with: HP version 50g
```≪ ""
1 ROT START "1" + NEXT
≫ ‘REPUNIT’ STO          @ (n  →  "11..1")

≪
IF DUP THEN
DUP2 DUP SUB NUM 99 SWAP - CHR REPL
LASTARG ROT DROP
IF "1" == THEN 1 - UP12      @ factor the carry
ELSE DROP END
ELSE DROP "1" SWAP + END
≫ ‘UP12’ STO             @ ("121..21"  n  →  "122..21")

≪ { } "2"
WHILE OVER SIZE 20 < REPEAT
DUP STR→
IF DUP ISPRIME? THEN
ROT SWAP + SWAP
SIZE 1 + REPUNIT
ELSE
DROP DUP SIZE UP12
END
END DROP
```
Output:
```1: {2 11 211 2111 12211 111121 1111211 11221211 111112121 1111111121 11111121121 111111211111 1111111121221 11111111112221 111111112111121 1111111112122111 11111111111112121 111111111111112111 1111111111111111111 11111111111111212121}
```

## Wren

Library: Wren-gmp
Library: Wren-fmt
Library: Wren-iterate

This is based on the Python code in the OEIS entry. Run time about 52 seconds.

```import "./gmp" for Mpz
import "./fmt" for Fmt
import "./iterate" for Stepped

var firstOneTwo = Fn.new { |n|
var k = Mpz.ten.pow(n).sub(Mpz.one).div(Mpz.nine)
var r = Mpz.one.lsh(n).sub(Mpz.one)
var m = Mpz.zero
while (m <= r) {
var t = k + Mpz.fromStr(m.toString(2))
if (t.probPrime(15) > 0) return t
m.inc
}
return Mpz.minusOne
}

for (n in 1..20) Fmt.print("\$4d: \$i", n, firstOneTwo.call(n))
for (n in Stepped.new(100..2000, 100)) {
var t = firstOneTwo.call(n)
if (t == Mpz.minusOne) {
System.print("No %(n)-digit prime found with only digits 1 or 2.")
} else {
var ts = t.toString
var ix = ts.indexOf("2")
Fmt.print("\$4d: (1 x \$4d) \$s", n, ix, ts[ix..-1])
}
}
```
Output:
```   1: 2
2: 11
3: 211
4: 2111
5: 12211
6: 111121
7: 1111211
8: 11221211
9: 111112121
10: 1111111121
11: 11111121121
12: 111111211111
13: 1111111121221
14: 11111111112221
15: 111111112111121
16: 1111111112122111
17: 11111111111112121
18: 111111111111112111
19: 1111111111111111111
20: 11111111111111212121
100: (1 x   92) 21112211
200: (1 x  192) 21112211
300: (1 x  288) 211121112221
400: (1 x  390) 2111122121
500: (1 x  488) 221222111111
600: (1 x  590) 2112222221
700: (1 x  689) 21111111111
800: (1 x  787) 2122222221111
900: (1 x  891) 222221221
1000: (1 x  988) 222122111121
1100: (1 x 1087) 2112111121111
1200: (1 x 1191) 211222211
1300: (1 x 1289) 22121221121
1400: (1 x 1388) 222211222121
1500: (1 x 1489) 21112121121
1600: (1 x 1587) 2121222122111
1700: (1 x 1688) 212121211121
1800: (1 x 1791) 221211121
1900: (1 x 1889) 22212212211
2000: (1 x 1989) 22121121211
```
Library: Wren-long

Alternatively, we can use Wren-cli to complete the basic task in about 2.7 seconds.

64-bit integers are needed to find the first 20 terms so we need to use the above module here.

```import "./long" for ULong
import "./fmt" for Conv, Fmt

var firstOneTwo = Fn.new { |n|
var k = ULong.new("1" * n)
var r = (ULong.one << n) - 1
var m = 0
while (r >= m) {
var t = k + ULong.new(Conv.bin(m))
if (t.isPrime) return t
m = m + 1
}
return ULong.zero
}

for (n in 1..20) Fmt.print("\$2d: \$i", n, firstOneTwo.call(n))
```
Output:
```As GMP version for first 20.
```

### Generalized

Slower than Raku at about 15.2 seconds though acceptable given that Wren is having to do a lot of string manipulation here.

```import "./gmp" for Mpz
import "./fmt" for Fmt
import "./iterate" for Stepped

var firstPrime = Fn.new { |n, d|
var k = Mpz.ten.pow(n).sub(Mpz.one).div(Mpz.nine)
var r = Mpz.one.lsh(n).sub(Mpz.one)
var m = Mpz.zero
while (m <= r) {
var t = k + Mpz.fromStr(m.toString(2))
var s = t.toString
if (d[0] != 2) {
if (d[0] != 1) s = s.replace("1", d[0].toString)
if (d[1] != 2) s = s.replace("2", d[1].toString)
} else {
s = s.replace("1", "x").replace("2", d[1].toString).replace("x", "2")
}
if (s[0] == "0") s = "1" + s[1..-1]
t.setStr(s)
if (t.probPrime(15) > 0) return t
m.inc
}
return Mpz.zero
}

var digits = [
[0, 1], [1, 2], [1, 3], [1, 4], [1, 5], [1, 6], [1, 7], [1, 8],
[1, 9], [2, 3], [2, 7], [2, 9], [3, 4], [3, 5], [3, 7], [3, 8],
[4, 7], [4, 9], [5, 7], [5, 9], [6, 7], [7, 8], [7, 9], [8, 9]
]
for (d in digits) {
Fmt.print("Smallest n digit prime using only \$d and \$d (or '0' if none exists)", d[0], d[1])
for (n in 1..20) Fmt.print("\$3d: \$i", n, firstPrime.call(n, d))
for (n in Stepped.new(100..200, 100)) {
var t = firstPrime.call(n, d)
var ts = t.toString
if (d[0] != 0) {
var ix = ts.indexOf(d[1].toString)
Fmt.print("\$3d: (\$d x \$3d) \$s", n, d[0], ix, ts[ix..-1])
} else {
var ix = ts[1..-1].indexOf(d[1].toString)
Fmt.print("\$3d: \$d (0 x \$3d) \$s", n, d[1], ix, ts[ix..-1])
}
}
System.print()
}
```
Output:
```Smallest n digit prime using only 0 and 1 (or '0' if none exists)
1: 0
2: 11
3: 101
4: 0
5: 10111
6: 101111
7: 1011001
8: 10010101
9: 100100111
10: 1000001011
11: 10000001101
12: 100000001111
13: 1000000111001
14: 10000000001011
15: 100000000100101
16: 1000000000011101
17: 10000000000001101
18: 100000000000100111
19: 1000000000000010011
20: 10000000000001100101
100: 1 (0 x  93) 0101101
200: 1 (0 x 189) 01110101011

Smallest n digit prime using only 1 and 2 (or '0' if none exists)
1: 2
2: 11
3: 211
4: 2111
5: 12211
6: 111121
7: 1111211
8: 11221211
9: 111112121
10: 1111111121
11: 11111121121
12: 111111211111
13: 1111111121221
14: 11111111112221
15: 111111112111121
16: 1111111112122111
17: 11111111111112121
18: 111111111111112111
19: 1111111111111111111
20: 11111111111111212121
100: (1 x  92) 21112211
200: (1 x 192) 21112211

Smallest n digit prime using only 1 and 3 (or '0' if none exists)
1: 3
2: 11
3: 113
4: 3313
5: 11113
6: 113111
7: 1111333
8: 11111131
9: 111111113
10: 1111113313
11: 11111111113
12: 111111133333
13: 1111111111333
14: 11111111113133
15: 111111111113113
16: 1111111111313131
17: 11111111111131333
18: 111111111111111131
19: 1111111111111111111
20: 11111111111111111131
100: (1 x  94) 331131
200: (1 x 190) 3311311111

....

Smallest n digit prime using only 7 and 9 (or '0' if none exists)
1: 7
2: 79
3: 797
4: 0
5: 77797
6: 777977
7: 7777997
8: 77779799
9: 777777799
10: 7777779799
11: 77777779979
12: 777777779777
13: 7777777779977
14: 77777777777977
15: 777777777779797
16: 7777777777797799
17: 77777777777777797
18: 777777777777797977
19: 7777777777777777997
20: 77777777777777779997
100: (7 x  93) 9979979
200: (7 x 192) 99777779

Smallest n digit prime using only 8 and 9 (or '0' if none exists)
1: 0
2: 89
3: 0
4: 8999
5: 89899
6: 888989
7: 8888989
8: 88888999
9: 888898889
10: 8888888989
11: 88888888999
12: 888888898999
13: 8888888999899
14: 88888888888889
15: 888888888898999
16: 8888888888989999
17: 88888888888888889
18: 888888888888898889
19: 8888888888888888989
20: 88888888888889888989
100: (8 x  91) 998998889
200: (8 x 190) 9888898989
```