Find largest left truncatable prime in a given base: Difference between revisions

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{{task|Prime Numbers}}
:ref 1: https://oeis.org/A103443
:related task 1: http://rosettacode.org/wiki/Truncatable_primes
 
A [[Truncatable primes|truncatable prime]] is one where all non-empty substrings that finish at the end of the number (right-substrings) are also primes ''when understood as numbers in a particular base''. The largest such prime in a given (integer) base is therefore computable, provided the base is larger than 2.
Obviously the right most digit must be prime, so in base 10 candidate are 2,3,5,7. Putting a digit in the range 1 to base-1 in front of each candidate must result in a prime. So 2 and 5 like the whale and the carnations in Hitchickers come into existance only to be extinguished before they have time to realize it. 13,17, ... 83,97 are candidates. Again Putting a digit in the range 1 to base-1 in front of each candidate must be a prime. Repeating until there are no larger candidates finds the largest left truncatable prime.
 
Let's consider what happens in base 10. Obviously the right most digit must be prime, so in base 10 candidates are 2,3,5,7. Putting a digit in the range 1 to base-1 in front of each candidate must result in a prime. So 2 and 5, like the whale and the petunias in ''The Hitchhiker's Guide to the Galaxy'', come into existence only to be extinguished before they have time to realize it, because 2 and 5 preceded by any digit in the range 1 to base-1 is not prime. Some numbers formed by preceding 3 or 7 by a digit in the range 1 to base-1 are prime. So 13,17,23,37,43,47,53,67,73,83,97 are candidates. Again, putting a digit in the range 1 to base-1 in front of each candidate must be a prime. Repeating until there are no larger candidates finds the largest left truncatable prime.
 
Let's work base 3 by hand:
 
0 and 1 are not prime so the last digit must be 2. 12<sub>3</sub> = 5<sub>10</sub> which is prime, 22<sub>3</sub> = 8<sub>10</sub> which is not so 12<sub>3</sub> is the only candidate. 112<sub>3</sub> = 14<sub>10</sub> which is not prime, 212<sub>3</sub> = 23<sub>10</sub> which is, so 212<sub>3</sub> is the only candidate. 1212<sub>3</sub> = 50<sub>10</sub> which is not prime, 2212<sub>3</sub> = 77<sub>10</sub> which also is not prime. So there are no more candidates, therefore 23 is the largest left truncatable prime in base 3.
 
The task is to reconstruct as much, and possibly more, of the table in ref[https://oeis.org/A103443 1the OEIS] as you are able.
 
Related Tasks:
* [[Miller-Rabin primality test]]
<br><br>
 
=={{header|BBC BASIC}}==
{{works with|BBC BASIC for Windows}}
Uses the '''H'''uge '''I'''nteger '''M'''ath & '''E'''ncryption library from http://devotechs.com/
<syntaxhighlight lang="bbcbasic"> HIMEM = PAGE + 3000000
INSTALL @lib$+"HIMELIB"
PROC_himeinit("HIMEkey")
DIM old$(20000), new$(20000)
h1% = 1 : h2% = 2 : h3% = 3 : h4% = 4
FOR base% = 3 TO 17
PRINT "Base "; base% " : " FN_largest_left_truncated_prime(base%)
NEXT
END
DEF FN_largest_left_truncated_prime(base%)
LOCAL digit%, i%, new%, old%, prime%, fast%, slow%
fast% = 1 : slow% = 50
old$() = ""
PROC_hiputdec(1, STR$(base%))
PROC_hiputdec(2, "1")
REPEAT
new% = 0 : new$() = ""
PROC_hiputdec(3, "0")
FOR digit% = 1 TO base%-1
SYS `hi_Add`, ^h2%, ^h3%, ^h3%
FOR i% = 0 TO old%-1
PROC_hiputdec(4, old$(i%))
SYS `hi_Add`, ^h3%, ^h4%, ^h4%
IF old% OR digit% > 1 THEN
IF old% > 100 THEN
SYS `hi_IsPrime_RB`, ^fast%, ^h4% TO prime%
ELSE
SYS `hi_IsPrime_RB`, ^slow%, ^h4% TO prime%
ENDIF
IF prime% THEN new$(new%) = FN_higetdec(4) : new% += 1
ENDIF
NEXT
NEXT
SYS `hi_Mul`, ^h1%, ^h2%, ^h2%
SWAP old$(), new$()
SWAP old%, new%
UNTIL old% = 0
= new$(new%-1)</syntaxhighlight>
'''Output:'''
<pre>
Base 3 : 23
Base 4 : 4091
Base 5 : 7817
Base 6 : 4836525320399
Base 7 : 817337
Base 8 : 14005650767869
Base 9 : 1676456897
Base 10 : 357686312646216567629137
Base 11 : 2276005673
Base 12 : 13092430647736190817303130065827539
Base 13 : 812751503
Base 14 : 615419590422100474355767356763
Base 15 : 34068645705927662447286191
Base 16 : 1088303707153521644968345559987
Base 17 : 13563641583101
</pre>
 
=={{header|C}}==
{{libheader|GMP}}
<syntaxhighlight lang="c">#include <stdio.h>
#include <gmp.h>
 
typedef unsigned long ulong;
 
ulong small_primes[] = {2,3,5,7,11,13,17,19,23,29,31,37,41,
43,47,53,59,61,67,71,73,79,83,89,97};
 
#define MAX_STACK 128
mpz_t tens[MAX_STACK], value[MAX_STACK], answer;
 
ulong base, seen_depth;
 
void add_digit(ulong i)
{
ulong d;
for (d = 1; d < base; d++) {
mpz_set(value[i], value[i-1]);
mpz_addmul_ui(value[i], tens[i], d);
if (!mpz_probab_prime_p(value[i], 1)) continue;
 
if (i > seen_depth ||
(i == seen_depth && mpz_cmp(value[i], answer) == 1))
{
if (!mpz_probab_prime_p(value[i], 50)) continue;
 
mpz_set(answer, value[i]);
seen_depth = i;
gmp_fprintf(stderr, "\tb=%lu d=%2lu | %Zd\n", base, i, answer);
}
 
add_digit(i+1);
}
}
 
void do_base()
{
ulong i;
mpz_set_ui(answer, 0);
mpz_set_ui(tens[0], 1);
for (i = 1; i < MAX_STACK; i++)
mpz_mul_ui(tens[i], tens[i-1], base);
 
for (seen_depth = i = 0; small_primes[i] < base; i++) {
fprintf(stderr, "\tb=%lu digit %lu\n", base, small_primes[i]);
mpz_set_ui(value[0], small_primes[i]);
add_digit(1);
}
gmp_printf("%d: %Zd\n", base, answer);
}
 
int main(void)
{
ulong i;
for (i = 0; i < MAX_STACK; i++) {
mpz_init_set_ui(tens[i], 0);
mpz_init_set_ui(value[i], 0);
}
mpz_init_set_ui(answer, 0);
 
for (base = 22; base < 30; base++) do_base();
 
return 0;
}</syntaxhighlight>
{{out}}
<pre>
$ ./a.out 2&>/dev/null
22: 33389741556593821170176571348673618833349516314271
23: 116516557991412919458949
24: 10594160686143126162708955915379656211582267119948391137176997290182218433
25: 8211352191239976819943978913
26: 12399758424125504545829668298375903748028704243943878467
27: 10681632250257028944950166363832301357693
28: 720639908748666454129676051084863753107043032053999738835994276213
29: 4300289072819254369986567661
...
</pre>
 
=={{header|C sharp|C#}}==
<syntaxhighlight lang="csharp">using Mpir.NET; // 0.4.0
using System; // 4790@3.6
using System.Collections.Generic;
class MaxLftTrP_B
{
static void Main()
{
mpz_t p; var sw = System.Diagnostics.Stopwatch.StartNew(); L(3);
for (uint b = 3; b < 13; b++)
{
sw.Restart(); p = L(b);
Console.WriteLine("{0} {1,2} {2}", sw.Elapsed, b, p);
}
Console.Read();
}
 
static mpz_t L(uint b)
{
var p = new List<mpz_t>(); mpz_t np = 0;
while ((np = nxtP(np)) < b) p.Add(np);
int i0 = 0, i = 0, i1 = p.Count - 1; mpz_t n0 = b, n, n1 = b * (b - 1);
for (; i < p.Count; n0 *= b, n1 *= b, i0 = i1 + 1, i1 = p.Count - 1)
for (n = n0; n <= n1; n += n0)
for (i = i0; i <= i1; i++)
if (mpir.mpz_probab_prime_p(np = n + p[i], 15) > 0) p.Add(np);
return p[p.Count - 1];
}
 
static mpz_t nxtP(mpz_t n) { mpz_t p = 0; mpir.mpz_nextprime(p, n); return p; }
}</syntaxhighlight>
{{out}}
<pre>
00:00:00.0000082 3 23
00:00:00.0000267 4 4091
00:00:00.0000299 5 7817
00:00:00.0027235 6 4836525320399
00:00:00.0000533 7 817337
00:00:00.0026306 8 14005650767869
00:00:00.0004923 9 1676456897
00:00:00.0514316 10 357686312646216567629137
00:00:00.0003609 11 2276005673
00:00:03.3792076 12 13092430647736190817303130065827539
</pre>
 
=={{header|C++}}==
{{trans|C}}
<syntaxhighlight lang="cpp">#include <gmpxx.h>
 
#include <algorithm>
#include <cassert>
#include <functional>
#include <iostream>
#include <vector>
 
using big_int = mpz_class;
 
const unsigned int small_primes[] = {2, 3, 5, 7, 11, 13, 17, 19, 23,
29, 31, 37, 41, 43, 47, 53, 59, 61,
67, 71, 73, 79, 83, 89, 97};
 
bool is_probably_prime(const big_int& n, int reps) {
return mpz_probab_prime_p(n.get_mpz_t(), reps) != 0;
}
 
big_int largest_left_truncatable_prime(unsigned int base) {
std::vector<big_int> powers = {1};
std::vector<big_int> value = {0};
big_int result = 0;
 
std::function<void(unsigned int)> add_digit = [&](unsigned int i) {
if (i == value.size()) {
value.resize(i + 1);
powers.push_back(base * powers.back());
}
for (unsigned int d = 1; d < base; ++d) {
value[i] = value[i - 1] + powers[i] * d;
if (!is_probably_prime(value[i], 1))
continue;
if (value[i] > result) {
if (!is_probably_prime(value[i], 50))
continue;
result = value[i];
}
add_digit(i + 1);
}
};
 
for (unsigned int i = 0; small_primes[i] < base; ++i) {
value[0] = small_primes[i];
add_digit(1);
}
return result;
}
 
int main() {
for (unsigned int base = 3; base < 18; ++base) {
std::cout << base << ": " << largest_left_truncatable_prime(base)
<< '\n';
}
for (unsigned int base = 19; base < 32; base += 2) {
std::cout << base << ": " << largest_left_truncatable_prime(base)
<< '\n';
}
}</syntaxhighlight>
 
{{out}}
<pre>
3: 23
4: 4091
5: 7817
6: 4836525320399
7: 817337
8: 14005650767869
9: 1676456897
10: 357686312646216567629137
11: 2276005673
12: 13092430647736190817303130065827539
13: 812751503
14: 615419590422100474355767356763
15: 34068645705927662447286191
16: 1088303707153521644968345559987
17: 13563641583101
19: 546207129080421139
21: 391461911766647707547123429659688417
23: 116516557991412919458949
25: 8211352191239976819943978913
27: 10681632250257028944950166363832301357693
29: 4300289072819254369986567661
31: 645157007060845985903112107793191
</pre>
 
=={{header|Eiffel}}==
As there is currently no implementation for arbitrary precision integers this example only works for base 3 to base 9. Respectively for bases where the Result fits into a INTEGER_64.
<syntaxhighlight lang="eiffel">
class
LARGEST_LEFT_TRUNCABLE_PRIME
 
create
make
 
feature
 
make
-- Tests find_prime for different bases.
local
i: INTEGER
decimal: INTEGER_64
do
from
i := 3
until
i = 10
loop
largest := 0
find_prime ("", i)
decimal := convert_to_decimal (largest, i)
io.put_string (i.out + ":%T" + decimal.out)
io.new_line
i := i + 1
end
end
 
find_prime (right_part: STRING; base: INTEGER)
-- Largest left truncable prime for a given 'base'.
local
i, larger, larger_dec: INTEGER_64
right: STRING
prime: BOOLEAN
do
from
i := 1
until
i = base
loop
create right.make_empty
right.deep_copy (right_part)
right.prepend (i.out)
larger := right.to_integer_64
if base /= 10 then
larger_dec := convert_to_decimal (larger, base)
if larger_dec < 0 then
io.put_string ("overflow")
prime := False
else
prime := is_prime (larger_dec)
end
else
prime := is_prime (larger)
end
if prime = TRUE then
find_prime (larger.out, base)
else
if right_part.count > 0 and right_part.to_integer_64 > largest then
largest := right_part.to_integer_64
end
end
i := i + 1
end
end
 
largest: INTEGER_64
 
convert_to_decimal (given, base: INTEGER_64): INTEGER_64
-- 'given' converted to base ten.
require
local
n, i: INTEGER
st_digits: STRING
dec: REAL_64
do
n := given.out.count
dec := 0
st_digits := given.out
from
i := 1
until
n < 0 or i > given.out.count
loop
n := n - 1
dec := dec + st_digits.at (i).out.to_integer * base ^ n
i := i + 1
end
Result := dec.truncated_to_integer_64
end
 
is_prime (n: INTEGER_64): BOOLEAN
--Is 'n' a prime number?
require
positiv_input: n > 0
local
i: INTEGER
max: REAL_64
math: DOUBLE_MATH
do
create math
if n = 2 then
Result := True
elseif n <= 1 or n \\ 2 = 0 then
Result := False
else
Result := True
max := math.sqrt (n)
from
i := 3
until
i > max
loop
if n \\ i = 0 then
Result := False
end
i := i + 2
end
end
end
 
end
</syntaxhighlight>
{{out}}
<pre>
3: 23
4: 4091
5: 7817
6: 4836525320399
7: 817337
8: 14005650767869
9: 1676456897
</pre>
 
=={{header|F_Sharp|F#}}==
<syntaxhighlight lang="fsharp">
(* Find some probable candidates for The Largest Left Trucatable Prime in a given base
Nigel Galloway: April 25th., 2017 *)
let snF Fbase pZ =
let rec fn i g (e:bigint) l =
match e with
| _ when e.IsZero -> i=1I
| _ when e.IsEven -> fn i ((g*g)%l) (e/2I) l
| _ -> fn ((i*g)%l) ((g*g)%l) (e/2I) l
let rec fi n i =
let g = n|>Array.Parallel.collect(fun n->[|for g in 1I..(Fbase-1I) do yield g*i+n|])|>Array.filter(fun n->fn 1I 2I (n-1I) n)
if (Array.isEmpty g) then n else (fi g (i*Fbase))
pZ |> Array.Parallel.map (fun n -> fi [|n|] Fbase)|>Seq.concat|>Seq.max
</syntaxhighlight>
{{out}}
<pre>
> printfn "%A" (snF 3I [2I]);; -> 23
> printfn "%A" (snF 4I [2I;3I]);; -> 4091
> printfn "%A" (snF 5I [2I;3I]);; -> 7817
> printfn "%A" (snF 6I [2I;3I;5I]);; -> 4836525320399
> printfn "%A" (snF 7I [2I;3I;5I]);; -> 817337
> printfn "%A" (snF 8I [2I;3I;5I;7I]);; -> 14005650767869
> printfn "%A" (snF 9I [2I;3I;5I;7I]);; -> 1676456897
> printfn "%A" (snF 10I [2I;3I;5I;7I]);; -> 357686312646216567629137
> printfn "%A" (snF 11I [2I;3I;5I;7I]);; -> 2276005673
> printfn "%A" (snF 12I [2I;3I;5I;7I;11I]);; -> 13092430647736190817303130065827539
Real: 00:00:43.776, CPU: 00:03:43.106, GC gen0: 4381, gen1: 3
> printfn "%A" (snF 13I [2I;3I;5I;7I;11I]);; -> 812751503
> printfn "%A" (snF 14I [2I;3I;5I;7I;11I;13I]);; -> 615419590422100474355767356763
> printfn "%A" (snF 15I [2I;3I;5I;7I;11I;13I]);; -> 34068645705927662447286191
> printfn "%A" (snF 16I [2I;3I;5I;7I;11I;13I]);; -> 1088303707153521644968345559987
> printfn "%A" (snF 17I [2I;3I;5I;7I;11I;13I]);; -> 13563641583101
> printfn "%A" (snF 18I [|2I;3I;5I;7I;11I;13I;17I|]);; -> 571933398724668544269594979167602382822769202133808087
Real: 04:50:58.748, CPU: 14:55:48.221, GC gen0: 1180413, gen1: 62
> printfn "%A" (snF 19I [|2I;3I;5I;7I;11I;13I;17I|]);; -> 546207129080421139
> printfn "%A" (snF 20I [|2I;3I;5I;7I;11I;13I;17I;19I|]);; -> 1073289911449776273800623217566610940096241078373
Real: 00:38:37.354, CPU: 02:42:24.086, GC gen0: 237504, gen1: 30
> printfn "%A" (snF 21I [|2I;3I;5I;7I;11I;13I;17I;19I|]);; -> 391461911766647707547123429659688417
> printfn "%A" (snF 22I [|2I;3I;5I;7I;11I;13I;17I;19I|]);; -> 33389741556593821170176571348673618833349516314271
Real: 00:22:22.206, CPU: 01:34:02.565, GC gen0: 138489, gen1: 24
> printfn "%A" (snF 23I [|2I;3I;5I;7I;11I;13I;17I;19I|]);; -> 116516557991412919458949
</pre>
 
=={{header|Fortran}}==
The initial idea is to see how far 32-bit integers will suffice, to try out the logic for the search. The basic idea is to maintain a "horde" of digit sequences that represent a prime number, then for each survivor in the horde, try adding a possible digit at the high-order end and checking that the resulting number is a prime. If so, add this sequence to the horde. When all trials have been made, if there was an addition, purge the earlier entries, and have another go, which is the next level up. If no addition had been made then the sequence is ended and the largest value amongst the survivors is printed.
 
Fortran does not offer a "list" data structure, so as ever, fiddling with arrays is needed. The situation at the end of a level is that there are entries 1:LH, the "starters" for that level, and following that are entries LH + 1:NH, the added entries. The "starters" are no longer needed and to save on storage, this hole is to be filled. The entire horde could be shifted down LH slots, but there could be many of them. Instead, the tail end entries are copied from the end into the hole. There are of course many variations possible, such as using linked-lists with an "available entry" list so that only links need be messed with rather than copying content, etc.
 
The source file uses the F90 style, mainly because module PRIMEBAG (from [[Extensible_prime_generator#Fortran]]) is available to supply some prime numbers and check whether a number is prime or not. This works up to the 32-bit integer limit: although INTEGER*8 variables are available, that seemed a reasonable stopping point. Otherwise, the source is older-style, except for a few conveniences: the use of "CYCLE" rather than a "GO TO", some array assignments rather than explicit DO-loops, and the special function MAXLOC to locate the index of the maximum value in an array. Although F90 also allows arrays of compound data, the entries are stored via a two-dimensional array, and to keep related digits adjacent in storage the indexing is (digit,entry) rather than (entry,digit) since fortran uses that ordering.
 
Unfortunately, the modernisers have abandoned a feature of First Fortran (1957): the <code>IF OVERFLOW ... </code> statement, or similar. In its place are ad-hoc tests on whether a number has suddenly become zero or negative: there is roughly a 50:50 chance that an overflow in two's-complement integer arithmetic will produce such a result - if positive, the value will still be wrong after an overflow. Such checks are tedious, but not bothering to check will mean that odd behaviour will ensue, and worse, incorrect results. <syntaxhighlight lang="fortran"> USE PRIMEBAG !Gain access to NEXTPRIME and ISPRIME.
Calculates the largest "left-truncatable" digit sequence that is a prime number, in various bases.
INTEGER LBASE,MANY,ENUFF !Some sizes.
PARAMETER (LBASE = 13, MANY = 66666, ENUFF = 66)
INTEGER NS,START(LBASE) !A list of single-digit prime numbers for starters.
INTEGER NH,LH !Counters for the horde.
INTEGER N,HORDEN(MANY) !Numerical value of a digit sequence.
INTEGER*1 HORDED(ENUFF,MANY) !Single-digit values only.
INTEGER B,D,DB !The base, a digit, some power of the base.
INTEGER L !The length of the digit sequence: DB = B**L.
INTEGER P !Possibly a prime number.
INTEGER I !A stepper.
 
MSG = 6 !I/O unit number for "standard output".
IF (.NOT.GRASPPRIMEBAG(66)) STOP "Gan't grab my file!" !Attempt in hope.
NS = 0 !No starters.
P = 1 !Start looking for some primes.
1 P = NEXTPRIME(P) !Thus skipping non-primes.
IF (P.LE.LBASE) THEN !Beyond the limit?
NS = NS + 1 !No. Count another starter.
START(NS) = P !Save its value.
GO TO 1 !And seek further.
END IF !One could insted prepare some values, the primes being well-known.
WRITE (MSG,2) LBASE,NS,START(1:NS) !But, parameterisation is easy enough.
2 FORMAT ("Working in bases 3 to ",I0," there are ",I0, !Announce the known.
* " single-digit primes: ",666(I0:", ")) !The : sez stop if the list is exhausted.
WRITE (MSG,3) !Produce a heading for the tabular output.
3 FORMAT (/"Base Digits Count Max. Value = (in base)")
 
10 DO B = 3,LBASE !Work through the bases.
NH = 0 !The horde is empty.
DO I = 1,NS !Prepare the starters for base B.
IF (START(I).GE.B) EXIT !Like, they're single-digits in base B.
NH = NH + 1 !So, count another in.
HORDEN(NH) = START(I) !Its numerical value.
HORDED(1,NH) = START(I) !Its digits. Just one.
END DO !On to the next single-digit prime number.
L = 0 !Syncopation. The length of the digit sequences.
DB = 1 !The power for the incoming digit.
 
20 L = L + 1 !We're about to add another digit.
IF (L.GE.ENUFF) STOP "Too many digits!" !Hopefully, there's room.
DB = DB*B !The new power of B.
IF (DB.LE.0) GO TO 29 !Integer overflow?
LH = NH !The live ones, awaiting extension.
DO I = 1,LH !Step through each starter.
N = HORDEN(I) !Grab its numerical value.
DO D = 1,B - 1 !Consider all possible lead digits.
P = D*DB + N !Place it at the start of the number.
IF (P.LE.0) GO TO 29 !Oh for IF OVERFLOW ...
IF (ISPRIME(P)) THEN !And if it is a prime,
IF (NH.GE.MANY) STOP "Too many sequences!" !Add a sequence.
NH = NH + 1 !Count in a survivor.
HORDEN(NH) = P !The numerical value.
HORDED(1:L,NH) = HORDED(1:L,I) !The digits.
HORDED(L + 1,NH) = D !Plus the added high-order digit.
END IF !So much for persistent primality.
END DO !On to the next lead digit.
END DO !On to the next starter.
 
N = NH - LH !The number of entries added to the horde.
IF (N.GT.0) THEN !Were there any?
DO I = 1,MIN(LH,N) !Yes. Overwrite the starters.
HORDEN(I) = HORDEN(NH) !From the tail end of the horde.
HORDED(1:L + 1,I) = HORDED(1:L + 1,NH) !Digit sequences as well.
NH = NH - 1 !One snipped off.
END DO !Thus fill the gap at the start.
NH = N !The new horde count.
LH = NH !All are starters for the next level.
GO TO 20 !See how it goes.
END IF !So much for further progress.
GO TO 30 !But if none, done.
29 WRITE (MSG,28) B,L,NH,DB,P !Curses. Offer some details.
28 FORMAT (I4,I7,I6,28X,"Integer overflow!",2I12)
CYCLE !Or, GO TO the end of the loop.
 
30 I = MAXLOC(HORDEN(1:NH),DIM = 1) !Finger the mostest number.
WRITE (MSG,31) B,L,NH,HORDEN(I),HORDED(L:1:-1,I) !Results!
31 FORMAT (I4,I7,I6,I11," = "666(I0:".")) !See Format 3.
 
END DO !On to the next base.
END !Simple enough.</syntaxhighlight>
 
Results:
<pre>
Working in bases 3 to 13 there are 6 single-digit primes: 2, 3, 5, 7, 11, 13
 
Base Digits Count Max. Value = (in base)
3 3 1 23 = 2.1.2
4 6 3 4091 = 3.3.3.3.2.3
5 6 1 7817 = 2.2.2.2.3.2
6 11 42 Integer overflow! 362797056 -2138904587
7 7 1 817337 = 6.6.4.2.6.2.3
8 10 27 Integer overflow! 1073741824 -1763182509
9 9 5 Integer overflow! 387420489 -1971761312
10 9 546 Integer overflow! 1000000000 -1299575929
11 8 2 Integer overflow! 214358881 -2107742185
12 8 7712 Integer overflow! 429981696 -1718612639
13 8 4 Integer overflow! 815730721 -1993454625
</pre>
 
So, there being no type declarations such as INTEGER*600, multi-precision arithmetic is needed to go further. There is no universally-used library for this, but thanks to previous effort in [[Sequence_of_primorial_primes#Fortran]] a collection is available, another F90 "module". This however works with a fixed value of BIGBASE, which is expected to be a large number and a power of ten. While there would be no great difficulty in converting from the digit sequences in the current base into a BIGNUM in base BIGBASE, it is more interesting to work with the desired base so that the digit sequences are manipulated directly. Accordingly, a variation, with the module starting <syntaxhighlight lang="fortran"> MODULE BIGNUMBERVB !Limited services: integers, no negative numbers, variable base possible.
INTEGER BIGORDER !A limited attempt at generality.
PARAMETER (BIGORDER = 1) !This is the order of the base of the big number arithmetic.
INTEGER BIGBASE,BIGLIMIT !Sized thusly.
c PARAMETER (BIGBASE = 10**BIGORDER, BIGLIMIT = 8888/BIGORDER) !Enough?
PARAMETER (BIGLIMIT = 666)
TYPE BIGNUM !So, a big number is simple.
INTEGER LAST !This many digits (of size BIGBASE) are in use.
INTEGER DIGIT(BIGLIMIT) !The digits, in ascending power order.
END TYPE BIGNUM !So much for that.
</syntaxhighlight>
 
As checked via earlier tests, using a fixed value for BIGLIMIT that is "surely big enough" enables faster execution than variable sizes. Now, BIGBASE is a variable, with a view to <code>DO BIGBASE = 3,17</code> and almost everything else remains the same, though with BIGBASE being a rather small number, there is no need to employ 64-bit variables via INTEGER*8 at certain places. The use of BIGORDER is disrupted and routines employing it should be avoided or adjusted, thus in BIGTASTE, adding <syntaxhighlight lang="fortran"> IF (MOD(BIGBASE,10).NE.0) STOP "BIGTASTE expects powers of 10" !Alas. Otherwise the "E" formalism fails.</syntaxhighlight> for example. The changes produce <syntaxhighlight lang="fortran"> SUBROUTINE BIGWRITE(F,B) !Show B.
INTEGER F !I/O unit number.
TYPE(BIGNUM) B !The number.
WRITE (F,1,ADVANCE="NO") B.DIGIT(B.LAST:1:-1) !Roll the digits in base BIGBASE.
1 FORMAT (666(I0:".")) !Not bothering with using letters for digits above nine.
END SUBROUTINE BIGWRITE !Simple, but messy.
 
SUBROUTINE BIGTEN(B,TEXT) !Produce a base ten digit string.
TYPE(BIGNUM) B !The number.
CHARACTER*(*) TEXT !The digits.
TYPE(BIGNUM) N !A copy I can mess with.
INTEGER L,D !Assistants.
N.LAST = B.LAST !So, make my copy.
N.DIGIT(1:N.LAST) = B.DIGIT(1:B.LAST) !Only the live digits are wanted.
TEXT = "" !Clear for action.
L = LEN(TEXT) !Find the far end.
10 D = BIGDIVRN(N,10) !Digits emerge from the low-order end of the number.
TEXT(L:L) = CHAR(ICHAR("0") + D) !Convert a digit to text, usual assumptions.
IF (N.LAST.EQ.1 .AND. N.DIGIT(1).EQ.0) RETURN !If zero, N is finished.
L = L - 1 !Otherwise, another digits will emerge.
IF (L.GT.0) GO TO 10 !If there is space, go for it.
TEXT(1:1) = "!" !Otherwise, signify overflow.
END SUBROUTINE BIGTEN !No negatives, so no sign is needed.
 
LOGICAL FUNCTION BIGISPRIME(B) !Ad-hoc report.
TYPE(BIGNUM) B !The number.
BIGISPRIME = ABS(BIGFACTOR(B,2800)).EQ.1 !Condensed report.
END FUNCTION BIGISPRIME !Can't be bothered with ISPRIME from PRIMEBAG.
</syntaxhighlight>
Which is to say that BIGWRITE will show the digits of a number as decimal numbers separated by periods rather than involving letters as additional digit symbols, while BIGTEN will prepare a text version in base ten, whatever BIGBASE is. Finally, BIGMRPRIME used to quit if BIGBASE were less than four, because it wants to test numbers not exceeding four by only inspecting a single digit of the big number, so that it can for larger numbers perform a direct test for divisibility by two and three without rejecting those numbers as primes just in case it is invoked for them. So ... <syntaxhighlight lang="fortran">Catch some annoying cases, to protect the direct tests for divisibility by two and three...
IF (N.LAST.LE.2) THEN !A smallish number? I want to compare to four, but BIGBASE might be two.
NR = BIGVALUE(N) !Surely so.
IF (NR.LE.4) THEN !Some special values are known.
BIGMRPRIME = NR.GE.2 .AND. NR.LE.3 !Like, the neighbours.
RETURN !Thus allow 2 to be reported as prime.
END IF !Yet, test for 2 as a possible factor for larger numbers.
END IF !Without struggling over SQRT and suchlike.
BIGMRPRIME = .FALSE. !Most numbers are not primes.
IF (BIGMOD2(N).EQ.0) RETURN !A single expression using .OR. risks always evaluating BOTH parts, damnit,
IF (BIGMODN(N,3).EQ.0) RETURN !Even for even numbers. Possibly doing so "in parallel" is no consolation.
</syntaxhighlight>
With all this in hand, the job can be done by <syntaxhighlight lang="fortran"> USE PRIMEBAG !Gain access to NEXTPRIME and ISPRIME.
USE BIGNUMBERVB !Alas, INTEGER*600 is not available.
Calculates the largest "left-truncatable" digit sequence that is a prime number, in various bases.
INTEGER LBASE,MANY !Some sizes.
PARAMETER (LBASE = 17, MANY = 66666)
INTEGER NS,START(LBASE) !A list of single-digit prime numbers for starters.
TYPE(BIGNUM) HORDE(MANY) !A collection.
INTEGER N,NH,LH !Counters for the horde.
INTEGER L !The length of the digit sequence.
INTEGER I,D !Steppers.
CHARACTER*42 TEXT !A scratchpad, for decimal values.
REAL T0,T1 !In memory of lost time.
 
MSG = 6 !I/O unit number for "standard output".
IF (.NOT.GRASPPRIMEBAG(66)) STOP "Gan't grab my file!" !Attempt in hope.
NS = 0 !No starters.
N = 1 !Start looking for some primes.
1 N = NEXTPRIME(N) !Thus skipping non-primes.
IF (N.LE.LBASE) THEN !Beyond the limit?
NS = NS + 1 !No. Count another starter.
START(NS) = N !Save its value.
GO TO 1 !And seek further.
END IF !One could insted prepare some values, the primes being well-known.
WRITE (MSG,2) LBASE,NS,START(1:NS) !But, parameterisation is easy enough.
2 FORMAT ("Working in bases 3 to ",I0," there are ",I0, !Announce the known.
* " single-digit primes: ",666(I0:", ")) !The : sez stop if the list is exhausted.
WRITE (MSG,3) !Produce a heading for the tabular output.
3 FORMAT (/"Base Digits Count",29X," Maximum Value = (in base)") !See Format 31.
 
Chug through the various bases to be used for the numerology.
CALL CPU_TIME(T0) !Start the timing.
10 DO BIGBASE = 3,LBASE !Not really very BIG bases.
NH = 0 !The horde is empty.
DO I = 1,NS !Prepare the starters for base BIGBASE.
IF (START(I).GE.BIGBASE) EXIT !Like, they're single-digits in base BIGBASE which may exceed ten...
NH = NH + 1 !So, count another in.
HORDE(NH).DIGIT(1) = START(I) !Its numerical value.
HORDE(NH).LAST = 1 !Its digit count. Just one.
END DO !On to the next single-digit prime number in BIGBASE.
L = 1 !The numbers all have one digit.
Consider each starter for extension via another high-order digit, to be placed at DIGIT(L + 1).
20 L = L + 1 !We're about to add another digit, now at DIGIT(L).
IF (L.GT.BIGLIMIT) STOP "Too many digits!" !Hopefully, there's room.
HORDE(1:NH).LAST = L !There is. Advise the BIGNUM horde of this.
LH = NH !The live ones, awaiting extension.
DO I = 1,LH !Step through each starter.
DO D = 1,BIGBASE - 1 !Consider all possible lead digits.
HORDE(I).DIGIT(L) = D !Place it at the start of the number.
IF (BIGISPRIME(HORDE(I))) THEN !And if it is a prime, or seems likely to be ...
IF (NH.GE.MANY) STOP "Too many sequences!" !Add a sequence.
NH = NH + 1 !Count in a survivor.
HORDE(NH).LAST = L !Its digit count.
HORDE(NH).DIGIT(1:L) = HORDE(I).DIGIT(1:L) !Its digits.
END IF !So much for persistent primality.
END DO !On to the next lead digit.
END DO !On to the next starter.
Check for added entries and compact the collection if there are some.
N = NH - LH !The number of entries added to the horde.
IF (N.GT.0) THEN !Were there any?
DO I = 1,MIN(LH,N) !Yes. Overwrite the starters.
HORDE(I).LAST = HORDE(NH).LAST !From the tail end of the horde.
HORDE(I).DIGIT(1:L) = HORDE(NH).DIGIT(1:L) !Copying only the live digits.
NH = NH - 1 !One snipped off.
END DO !Thus fill the gap at the start.
NH = N !The new horde count.
GO TO 20 !See how it goes.
END IF !So much for further progress.
Cast forth the mostest of the starters.
30 HORDE(1:NH).LAST = L - 1 !The testing involved an extra digit, which was not accepted.
L = 1 !Now go looking for the mostest of the survivors.
DO I = 2,NH !By comparing all the rest.
IF (BIGSIGN(HORDE(L),HORDE(I)).LT.0) L = I !Consider A - B.
END DO !On to the next.
CALL BIGTEN(HORDE(L),TEXT) !Get a decimal digit string.
WRITE (MSG,31) BIGBASE,HORDE(L).LAST,NH,TEXT !Some auxiliary details.
31 FORMAT (I4,I7,I6,1X,A," = ",$) !See Format 3.
CALL BIGWRITE(MSG,HORDE(L)) !The number at last!
WRITE (MSG,*) !Finish the line.
END DO !On to the next base.
CALL CPU_TIME(T1) !Completed the run.
 
Closedown.
200 WRITE (MSG,201) !First, some statistics.
201 FORMAT (/,"The MR prime test makes a series of trials, "
1 "stopping early",/'only when a "definitely composite" ',
2 "result is encountered.")
WRITE (MSG,202) "Trial",(I,I = 1,BIGMRTRIALS) !Roll the trial number.
WRITE (MSG,202) "Count",BIGMRCOUNT !Now the counts.
202 FORMAT (A6,": ",666I8) !This should do.
WRITE (MSG,*) "CPU time:",T1 - T0 !The cost.
END !Simple enough.
</syntaxhighlight>
 
And the results, slightly edited to remove six columns of spaces...
<pre>
Working in bases 3 to 17 there are 7 single-digit primes: 2, 3, 5, 7, 11, 13, 17
 
Base Digits Count Maximum Value = (in base)
3 3 1 23 = 2.1.2
4 6 3 4091 = 3.3.3.3.2.3
5 6 1 7817 = 2.2.2.2.3.2
6 17 1 4836525320399 = 1.4.1.4.1.5.1.1.4.1.4.4.5.1.4.3.5
7 7 1 817337 = 6.6.4.2.6.2.3
8 15 1 14005650767869 = 3.1.3.6.3.6.1.6.5.5.3.7.7.7.5
9 10 3 1676456897 = 4.2.8.4.4.8.4.4.6.5
10 24 1 357686312646216567629137 = 3.5.7.6.8.6.3.1.2.6.4.6.2.1.6.5.6.7.6.2.9.1.3.7
11 9 1 2276005673 = 10.6.8.8.2.2.8.2.7
12 32 1 13092430647736190817303130065827539 = 4.7.1.10.3.4.10.1.6.4.2.5.9.11.10.1.6.11.3.2.4.10.11.8.10.3.2.11.7.8.1.7
13 8 4 812751503 = 12.12.4.12.8.12.6.5
14 26 2 615419590422100474355767356763 = 13.9.6.7.12.12.13.6.3.3.8.8.5.2.2.6.1.9.8.8.3.10.7.13.2.3
15 22 1 34068645705927662447286191 = 6.12.6.12.2.12.14.2.12.14.14.14.10.4.8.2.6.14.6.4.2.11
16 25 1 1088303707153521644968345559987 = 13.11.12.7.15.11.10.2.4.15.14.6.10.14.12.4.6.2.10.11.15.6.3.11.3
17 11 1 13563641583101 = 6.12.6.6.12.12.4.12.12.8.3
 
The MR prime test makes a series of trials, stopping early
only when a "definitely composite" result is encountered.
Trial: 1 2 3 4 5 6
Count: 517641 235380 235380 235380 235380 235380
CPU time: 599.2188
</pre>
 
So, once again, it is plain that using a large BIGBASE is beneficial. The plain number version first given works in the computer's own arithmetic base, and preparing such values from the digit strings in the given base is not difficult. Despite the inconvenience of messing with digit sequences not in the same base as used for calculation, a trial run using base 10000 required 260 seconds instead - and gave the same numbers. Bignumber arithmetic via assembler to fully exploit the available hardware would surely do better still.
 
Going further will require MANY to be enlarged. Already, base twelve required just over nineteen thousand entries, and base eighteen overflowed MANY = 66666. This suggests that a lot of data is being shifted about, so some sort of linked-list scheme might reduce that. Incidentally, in <code>B.LAST = A.LAST; B.DIGIT(1:N) = A.DIGIT(1:N)</code> and similar, because the storage for .DIGIT immediately follows that for .LAST, one might hope that an advanced compiler would combine the two statements into one sequential copy... Alas, the Compaq F90/95 compiler produces two wads of code, of 20 operations and then 92. Bounds checking is active, but still...
 
And, as one who can recall when one was admitted to the status of "prime" (like, being divisible only by itself and one), what about allowing numbers to end with the digit one...
<pre>
Working in bases 3 to 17
 
Base Digits Count Maximum Value = (in base)
3 2 1 7 = 2.1
4 7 1 9829 = 2.1.2.1.2.1.1
5 4 1 311 = 2.2.2.1
6 19 1 580639005096133 = 5.4.1.4.5.2.5.5.1.3.1.5.5.1.3.1.4.2.1
7 8 1 3602999 = 4.2.4.2.4.2.4.1
8 9 1 104056657 = 6.1.4.7.4.3.5.2.1
9 5 3 41023 = 6.2.2.4.1
10 20 1 89726156799336363541 = 8.9.7.2.6.1.5.6.7.9.9.3.3.6.3.6.3.5.4.1
11 7 1 11750399 = 6.6.10.6.2.8.1
12 30 1 57434208867139354150297607357437 = 2.10.10.1.1.11.6.4.4.7.11.8.2.2.8.10.8.7.6.5.9.5.2.2.10.2.1.5.9.1
13 10 1 66073331221 = 6.2.12.12.10.8.12.10.12.1
14 25 1 39607537776359469390989456509 = 12.4.7.3.5.12.3.11.6.3.11.5.11.9.13.9.3.7.10.12.6.4.3.8.1
15 20 4 319674182915416424428051 = 14.6.4.8.6.8.12.6.2.4.10.8.6.4.8.14.14.14.10.1
16 23 2 2208955789035921681292672241 = 7.2.3.3.4.9.5.4.5.1.5.7.8.15.9.3.3.4.3.3.12.15.1
17 11 2 16408729108033 = 8.2.6.4.4.6.14.2.12.6.1
 
The MR prime test makes a series of trials, stopping early
only when a "definitely composite" result is encountered.
Trial: 1 2 3 4 5 6
Count: 97667 44905 44904 44904 44904 44904
CPU time: 111.2656
</pre>
 
=={{header|Go}}==
{{trans|C}}
 
 
Note that the use of ProbablyPrime(0) requires Go 1.8 or later.
<syntaxhighlight lang="go">package main
 
import (
"fmt"
"math/big"
)
 
var smallPrimes = [...]int{2, 3, 5, 7, 11, 13, 17, 19, 23, 29}
 
const maxStack = 128
 
var (
tens, values [maxStack]big.Int
bigTemp, answer = new(big.Int), new(big.Int)
base, seenDepth int
)
 
func addDigit(i int) {
for d := 1; d < base; d++ {
values[i].Set(&values[i-1])
bigTemp.SetUint64(uint64(d))
bigTemp.Mul(bigTemp, &tens[i])
values[i].Add(&values[i], bigTemp)
if !values[i].ProbablyPrime(0) {
continue
}
if i > seenDepth || (i == seenDepth && values[i].Cmp(answer) == 1) {
if !values[i].ProbablyPrime(0) {
continue
}
answer.Set(&values[i])
seenDepth = i
}
addDigit(i + 1)
}
}
 
func doBase() {
answer.SetUint64(0)
tens[0].SetUint64(1)
bigTemp.SetUint64(uint64(base))
seenDepth = 0
for i := 1; i < maxStack; i++ {
tens[i].Mul(&tens[i-1], bigTemp)
}
for i := 0; smallPrimes[i] < base; i++ {
values[0].SetUint64(uint64(smallPrimes[i]))
addDigit(1)
}
fmt.Printf("%2d: %s\n", base, answer.String())
}
 
func main() {
for base = 3; base <= 17; base++ {
doBase()
}
}</syntaxhighlight>
 
{{out}}
<pre>
3: 23
4: 4091
5: 7817
6: 4836525320399
7: 817337
8: 14005650767869
9: 1676456897
10: 357686312646216567629137
11: 2276005673
12: 13092430647736190817303130065827539
13: 812751503
14: 615419590422100474355767356763
15: 34068645705927662447286191
16: 1088303707153521644968345559987
17: 13563641583101
</pre>
 
=={{header|Haskell}}==
Miller-Rabin test code from [http://www.haskell.org/haskellwiki/Testing_primality#Miller-Rabin_Primality_Test HaskellWiki], with modifications.
<langsyntaxhighlight lang="haskell">primesTo100 = [2,3,5,7,11,13,17,19,23,29,31,37,41,43,47,53,59,61,67,71,73,79,83,89,97]
 
-- (eq. to) find2km (2^k * n) = (k,n)
Line 21 ⟶ 877:
| r == 1 = (k,m)
| otherwise = f (k+1) q
where (q,r) = quotRem m 2
 
-- n is the number to test; a is the (presumably randomly chosen) witness
Line 78 ⟶ 934:
addDigit a = filter (is_prime [3]) $ map (a*b+) x
 
main = mapM_ print $ map (\x->(x, left_trunc x)) [3..21]</langsyntaxhighlight>
<pre>
(3,23)
Line 101 ⟶ 957:
</pre>
 
=={{header|Perl 6J}}==
Using the Miller-Rabin definition of <tt>is-prime</tt> from [http://rosettacode.org/wiki/Miller-Rabin_primality_test], or the built-in, if available. For speed we do a single try, which allows a smattering of composites in, but this does not appear to damage the algorithm much, and the correct answer appears to be within a few candidates of the end all the time. We keep the last few hundred candidates, sort them into descending order, then pick the largest that passes Miller-Rabin with 100 tries.
 
<syntaxhighlight lang="j">ltp=:3 :0
I believe the composites don't hurt the algorithm because they do not confer any selective advantage on their "children", so their average length is no longer than the correct solution. In addition, the candidates in the finalist set all appear to be largely independent of each other, so any given composite tends to contribute only a single candidate, if any. I have no proof of this; I prefer my math to have more vigor than rigor. <tt>:-)</tt>
probe=. i.1 0
<lang perl6>for 3..* -> $base {
while. #probe do.
say "Starting base $base...";
probe=. (#~ 1 p: y #.]),/(}.i.y),"0 _1/have=. probe
my @stems = grep { is-prime($_, 100)}, ^$base;
end.
my @runoff;
>./y#.have
for 1 .. * -> $digits {
)</syntaxhighlight>
print ' ', @stems.elems;
 
my @new;
Quick example:
my $place = $base ** $digits;
 
my $tries = 1;
<syntaxhighlight lang="j"> (,ltp)"0]3 4 5 6 7 8 9 10 11
for @stems -> $stem {
3 23
for 1 ..^ $base -> $digit {
4 my $new = $digit * $place + $stem; 4091
5 7817
@new.push($new) if is-prime($new, $tries);
6 }4836525320399
7 } 817337
8 14005650767869
last unless @new;
9 1676456897
push @runoff, @stems if @new < @stems and @new < 100;
10 992429121339693967
@stems = @new;
11 2276005673</syntaxhighlight>
 
Representation of a current longer effort:
 
<syntaxhighlight lang="j"> (,ltp)"0]3}.i.20x
3 23
4 4091
5 7817
6 4836525320399
7 817337
8 14005650767869
9 1676456897
10 357686312646216567629137
11 2276005673
12 13092430647736190817303130065827539
13 812751503
14 615419590422100474355767356763
15 34068645705927662447286191
16 1088303707153521644968345559987
17 13563641583101
18 571933398724668544269594979167602382822769202133808087
19 546207129080421139</syntaxhighlight>
 
=={{header|Java}}==
'''Code:'''
<syntaxhighlight lang="java">import java.math.BigInteger;
import java.util.*;
 
class LeftTruncatablePrime
{
private static List<BigInteger> getNextLeftTruncatablePrimes(BigInteger n, int radix, int millerRabinCertainty)
{
List<BigInteger> probablePrimes = new ArrayList<BigInteger>();
String baseString = n.equals(BigInteger.ZERO) ? "" : n.toString(radix);
for (int i = 1; i < radix; i++)
{
BigInteger p = new BigInteger(Integer.toString(i, radix) + baseString, radix);
if (p.isProbablePrime(millerRabinCertainty))
probablePrimes.add(p);
}
pushreturn @runoff, @stemsprobablePrimes;
}
say "\n Finalists: ", +@runoff;
public static BigInteger getLargestLeftTruncatablePrime(int radix, int millerRabinCertainty)
{
List<BigInteger> lastList = null;
List<BigInteger> list = getNextLeftTruncatablePrimes(BigInteger.ZERO, radix, millerRabinCertainty);
while (!list.isEmpty())
{
lastList = list;
list = new ArrayList<BigInteger>();
for (BigInteger n : lastList)
list.addAll(getNextLeftTruncatablePrimes(n, radix, millerRabinCertainty));
}
if (lastList == null)
return null;
Collections.sort(lastList);
return lastList.get(lastList.size() - 1);
}
public static void main(String[] args)
{
if (args.length != 2) {
System.err.println("There must be exactly two command line arguments.");
return;
}
int maxRadix;
try {
maxRadix = Integer.parseInt(args[0]);
if (maxRadix < 3) throw new NumberFormatException();
} catch (NumberFormatException e) {
System.err.println("Radix must be an integer greater than 2.");
return;
}
int millerRabinCertainty;
try {
millerRabinCertainty = Integer.parseInt(args[1]);
} catch (NumberFormatException e) {
System.err.println("Miiller-Rabin Certainty must be an integer.");
return;
}
for (int radix = 3; radix <= maxRadix; radix++)
{
BigInteger largest = getLargestLeftTruncatablePrime(radix, millerRabinCertainty);
System.out.print("n=" + radix + ": ");
if (largest == null)
System.out.println("No left-truncatable prime");
else
System.out.println(largest + " (in base " + radix + "): " + largest.toString(radix));
}
}
}</syntaxhighlight>
 
'''Example:'''
for @runoff.sort(-*) -> $finalist {
my $b = $finalist.base($base);
say " Checking :$base\<", $b, '>';
my $ok = True;
for $base ** 2, $base ** 3, $base ** 4 ... $base ** $b.chars -> $place {
my $f = $finalist % $place;
if not is-prime($f, 100) {
say " Oops, :$base\<", $f.base($base), '> is not a prime!!!';
$ok = False;
last;
}
}
next unless $ok;
 
<pre>java LeftTruncatablePrime 17 100
say " Largest ltp in base $base = $finalist";
n=3: 23 (in base 3): 212
last;
n=4: 4091 (in base 4): 333323
n=5: 7817 (in base 5): 222232
n=6: 4836525320399 (in base 6): 14141511414451435
n=7: 817337 (in base 7): 6642623
n=8: 14005650767869 (in base 8): 313636165537775
n=9: 1676456897 (in base 9): 4284484465
n=10: 357686312646216567629137 (in base 10): 357686312646216567629137
n=11: 2276005673 (in base 11): a68822827
n=12: 13092430647736190817303130065827539 (in base 12): 471a34a164259ba16b324ab8a32b7817
n=13: 812751503 (in base 13): cc4c8c65
n=14: 615419590422100474355767356763 (in base 14): d967ccd63388522619883a7d23
n=15: 34068645705927662447286191 (in base 15): 6c6c2ce2ceeea4826e642b
n=16: 1088303707153521644968345559987 (in base 16): dbc7fba24fe6aec462abf63b3
n=17: 13563641583101 (in base 17): 6c66cc4cc83</pre>
 
=={{header|Julia}}==
This solution keeps candidate values in an array. A new digit is added with each generation, removing the previous generation's primes from the front of the list (<tt>popfirst!</tt>) and adding new candidates to the end of the list (<tt>append!</tt>) with each generation. The maximum value yielded in each generation is tracked as a provisional answer. Once the array is emptied (because no digit could be added to any of the previous generation's primes to yield a prime), the algorithm is finished and the answer found.
 
This solution is limited to a base of 17, to keep the processing time to well under a minute (about 15 seconds on an old but decent quality notebook). (I've let it run to as high as 23, but that took something like 20 minutes as I was otherwise occupied.) I did attempt a few optimizations of this general approach (such as moving the logic of <tt>addmsdigit</tt> into <tt>lefttruncprime</tt> and being clever about identifying the maximum of a given generation) but none of these tweaks resulted in a significant improvement in efficiency.
<syntaxhighlight lang="julia">using Primes, Printf
function addmsdigit(p::Integer, b::Integer, s::Integer)
a = Vector{typeof(p)}()
q = p
for i in 1:(b-1)
q += s
isprime(q) || continue
push!(a, q)
end
return a
end
function lefttruncprime(pbase::Integer)
a = Vector{BigInt}()
append!(a, primes(pbase - 1))
mlt = zero(BigInt)
s = one(BigInt)
while !isempty(a)
mlt = maximum(a)
s *= pbase
for i in 1:length(a)
p = popfirst!(a)
append!(a, addmsdigit(p, pbase, s))
end
end
return mlt
end
lo, hi = 3, 17
println("The largest left truncatable primes for bases", @sprintf(" %d to %d.", lo, hi))
for i in lo:hi
mlt = lefttruncprime(i)
@printf("%10d %-30d (%s)\n", i, mlt, string(mlt, base=i))
end
</syntaxhighlight>{{out}}
<pre>
The largest left truncatable primes for bases 3 to 17.
3 23 (212)
4 4091 (333323)
5 7817 (222232)
6 4836525320399 (14141511414451435)
7 817337 (6642623)
8 14005650767869 (313636165537775)
9 1676456897 (4284484465)
10 357686312646216567629137 (357686312646216567629137)
11 2276005673 (a68822827)
12 13092430647736190817303130065827539 (471a34a164259ba16b324ab8a32b7817)
13 812751503 (cc4c8c65)
14 615419590422100474355767356763 (d967ccd63388522619883a7d23)
15 34068645705927662447286191 (6c6c2ce2ceeea4826e642b)
16 1088303707153521644968345559987 (dbc7fba24fe6aec462abf63b3)
17 13563641583101 (6c66cc4cc83)
</pre>
 
=={{header|Kotlin}}==
{{trans|Java}}
<syntaxhighlight lang="scala">// version 1.1.2
 
import java.math.BigInteger
 
fun nextLeftTruncatablePrimes(n: BigInteger, radix: Int, certainty: Int): List<BigInteger> {
val probablePrimes = mutableListOf<BigInteger>()
val baseString = if (n == BigInteger.ZERO) "" else n.toString(radix)
for (i in 1 until radix) {
val p = BigInteger(i.toString(radix) + baseString, radix)
if (p.isProbablePrime(certainty)) probablePrimes.add(p)
}
return probablePrimes
}</lang>
}
 
fun largestLeftTruncatablePrime(radix: Int, certainty: Int): BigInteger? {
var lastList: List<BigInteger>? = null
var list = nextLeftTruncatablePrimes(BigInteger.ZERO, radix, certainty)
while (!list.isEmpty()) {
lastList = list
list = mutableListOf()
for (n in lastList) list.addAll(nextLeftTruncatablePrimes(n, radix, certainty))
}
if (lastList == null) return null
return lastList.sorted().last()
}
 
fun main(args: Array<String>) {
print("Enter maximum radix : ")
val maxRadix = readLine()!!.toInt()
print("Enter certainty : ")
val certainty = readLine()!!.toInt()
println()
for (radix in 3..maxRadix) {
val largest = largestLeftTruncatablePrime(radix, certainty)
print("Base = ${"%-2d".format(radix)} : ")
if (largest == null)
println("No left truncatable prime")
else
println("${largest.toString().padEnd(35)} -> ${largest.toString(radix)}")
}
}</syntaxhighlight>
 
{{out}}
Sampe input/output - expect run time of about 3.5 minutes on a typical laptop:
<pre>
Enter maximum radix : 17
Enter certainty : 100
 
Base = 3 : 23 -> 212
Base = 4 : 4091 -> 333323
Base = 5 : 7817 -> 222232
Base = 6 : 4836525320399 -> 14141511414451435
Base = 7 : 817337 -> 6642623
Base = 8 : 14005650767869 -> 313636165537775
Base = 9 : 1676456897 -> 4284484465
Base = 10 : 357686312646216567629137 -> 357686312646216567629137
Base = 11 : 2276005673 -> a68822827
Base = 12 : 13092430647736190817303130065827539 -> 471a34a164259ba16b324ab8a32b7817
Base = 13 : 812751503 -> cc4c8c65
Base = 14 : 615419590422100474355767356763 -> d967ccd63388522619883a7d23
Base = 15 : 34068645705927662447286191 -> 6c6c2ce2ceeea4826e642b
Base = 16 : 1088303707153521644968345559987 -> dbc7fba24fe6aec462abf63b3
Base = 17 : 13563641583101 -> 6c66cc4cc83
</pre>
 
=={{header|Maple}}==
<syntaxhighlight lang="maple">MaxLeftTruncatablePrime := proc(b, $)
local i, j, c, p, sdprimes;
local tprimes := table();
sdprimes := select(isprime, [seq(1..b-1)]);
for p in sdprimes do
if assigned(tprimes[p]) then
next;
end if;
i := ilog[b](p)+1;
j := 1;
do
c := j*b^i + p;
if j >= b then
# we have tried all 1 digit extensions of p, add p to tprimes and move back 1 digit
tprimes[p] := p;
if i = 1 then
# if we are at the first digit, go to the next 1 digit prime
break;
end if;
i := i - 1;
j := 1;
p := p - iquo(p, b^i)*b^i;
elif assigned(tprimes[c]) then
j := j + 1;
elif isprime(c) then
p := c;
i := i + 1;
j := 1;
else
j := j+1;
end if;
end do;
end do;
return max(indices(tprimes, 'nolist'));
end proc;</syntaxhighlight>
 
=={{header|Mathematica}} / {{header|Wolfram Language}}==
 
<syntaxhighlight lang="text">LargestLeftTruncatablePrimeInBase[n_] :=
Max[NestWhile[{Select[
Flatten@Outer[Function[{a, b}, #[[2]] a + b],
Range[1, n - 1], #[[1]]], PrimeQ], n #[[2]]} &, {{0},
1}, #[[1]] != {} &, 1, Infinity, -1][[1]]]</syntaxhighlight>
 
Example:
 
<syntaxhighlight lang="text">Do[Print[n, "\t", LargestLeftTruncatablePrimeInBase@n], {n, 3, 17}]</syntaxhighlight>
 
Output:
 
<pre>3 23
 
4 4091
 
5 7817
 
6 4836525320399
 
7 817337
 
8 14005650767869
 
9 1676456897
 
10 357686312646216567629137
 
11 2276005673
 
12 13092430647736190817303130065827539
 
13 812751503
 
14 615419590422100474355767356763
 
15 34068645705927662447286191
 
16 1088303707153521644968345559987
 
17 13563641583101</pre>
 
=={{header|Nim}}==
{{libheader|bignum}}
<syntaxhighlight lang="nim">import bignum, strformat
 
const
 
Primes = [2, 3, 5, 7, 11, 13, 17]
Digits = "0123456789abcdefghijklmnopqrstuvwxyz"
 
#---------------------------------------------------------------------------------------------------
 
func isProbablyPrime(n: Int): bool =
## Return true if "n" is not definitively composite.
probablyPrime(n, 25) != 0
 
#---------------------------------------------------------------------------------------------------
 
func maxLeftTruncablePrime(base: int): Int =
## Return the maximum left truncable prime for given base.
 
let base = base.int32
var primes: seq[Int]
 
# Initialize primes with one digit in given base.
for p in Primes:
if p < base:
primes.add(newInt(p))
else:
break
 
# Build prime list with one more digit per generation.
var next: seq[Int]
while true:
 
# Build the next generation (with one more digit).
for p in primes:
var pstr = ' ' & `$`(p, base) # ' ' as a placeholder for next digit.
for i in 1..<base:
pstr[0] = Digits[i]
let n = newInt(pstr, base)
if n.isProbablyPrime():
next.add(n)
 
if next.len == 0:
# No primes with this number of digits.
# Return the greatest prime in previous generation.
return max(primes)
 
# Prepare to build next generation.
primes = next
next.setLen(0)
 
#———————————————————————————————————————————————————————————————————————————————————————————————————
 
echo "Base Greatest left truncable prime"
echo "====================================="
for base in 3..17:
let m = maxLeftTruncablePrime(base)
echo &"{base:>3} {m}", if base > 10: " (" & `$`(m, base.int32) & ')' else: ""</syntaxhighlight>
 
{{out}}
<pre>Base Greatest left truncable prime
=====================================
3 23
4 4091
5 7817
6 4836525320399
7 817337
8 14005650767869
9 1676456897
10 357686312646216567629137
11 2276005673 (a68822827)
12 13092430647736190817303130065827539 (471a34a164259ba16b324ab8a32b7817)
13 812751503 (cc4c8c65)
14 615419590422100474355767356763 (d967ccd63388522619883a7d23)
15 34068645705927662447286191 (6c6c2ce2ceeea4826e642b)
16 1088303707153521644968345559987 (dbc7fba24fe6aec462abf63b3)
17 13563641583101 (6c66cc4cc83)</pre>
 
=={{header|PARI/GP}}==
Takes half a second to find the terms up to 10, with proofs of primality. The time can be halved without proofs (use <code>ispseudoprime</code> in place of <code>isprime</code>).
<syntaxhighlight lang="parigp">a(n)=my(v=primes(primepi(n-1)),u,t,b=1,best); while(#v, best=vecmax(v); b*=n; u=List(); for(i=1,#v,for(k=1,n-1,if(isprime(t=v[i]+k*b), listput(u,t)))); v=Vec(u)); best</syntaxhighlight>
=={{header|Pascal}}==
==={{header|Free Pascal}}===
Using gmp and depth first search, like by [[Find_largest_left_truncatable_prime_in_a_given_base#Phix|Phix]].<br>Use preset numbers and powers of base to max values expecting that no memory-reallocations during runtime are needed.<br>
HTOP shows no variance in used memory.
<syntaxhighlight lang="pascal">
program TruncatablePrimes;
//https://rosettacode.org/wiki/Find_largest_left_truncatable_prime_in_a_given_base
{$IFDEF FPC}{$MODE DELPHI}{$OPTIMIZATION ON,ALL}{$ENDIF}
{$IFDEF Windows}{$APPTYPE CONSOLE}{$ENDIF}
uses
sysutils,gmp;// http://rosettacode.org/wiki/Extensible_prime_generator#Pascal
const
DgtToChar : array[0..10+26+26-1] of char =
'0123456789ABCDEFGHIJKLMNOPQRSTUVWXYZabcdefghijklmnopqrstuvwxyz';
MaxDgtCnt = 50;
var
pot_Base : array[0..MaxDgtCnt] of mpz_t;
Numbers : array[0..MaxDgtCnt] of mpz_t;
MAX_Found : mpz_t;
Digits,
Digits_Found: array[0..MaxDgtCnt] of byte;
gbl_Count,
Max_Pot : Uint32;
 
procedure InitAll;
var
pot : mpz_t;
MaxBase,
i : integer;
begin
MaxBase := MaxDgtCnt;
mpz_init_set_ui(pot,1);
For i := 0 to High(pot_Base) do
begin
mpz_mul_ui(pot,pot,MaxBase);
mpz_init_set(pot_Base[i],Pot);
mpz_init_set(Numbers[i],Pot);
end;
mpz_init_set(MAX_Found,pot);
mpz_set_ui(MAX_Found,0);
mpz_clear(pot);
end;
 
procedure ClearAll;
var
i : integer;
begin
For i := High(pot_Base) downto 0 do
begin
mpz_clear(pot_Base[i]);
mpz_clear(Numbers[i]);
end;
mpz_clear(MAX_Found);
end;
 
procedure InitPot(Base : byte);
var
pot : mpz_t;
i : integer;
begin
mpz_init_set_ui(pot,1);
For i := 0 to High(pot_Base) do
begin
mpz_set(pot_Base[i],Pot);
mpz_mul_ui(pot,pot,base);
end;
mpz_clear(pot);
mpz_set_ui(MAX_Found,0);
Fillchar(Digits,SizeOf(Digits),#0);
end;
 
procedure Next_Number(Base,pot : byte);
var
i : integer;
begin
inc(gbl_Count);
if pot = 0 then
mpz_set_ui(Numbers[pot],0)
else
mpz_set(Numbers[pot],Numbers[pot-1]);
For i := 1 to Base-1 do
begin
Digits[pot] := i;
mpz_add(Numbers[pot],Numbers[pot],pot_Base[pot]);
if mpz_probab_prime_p(Numbers[pot],5)>0 then
Begin
IF mpz_cmp(MAX_Found,Numbers[pot])<0 then
Begin
mpz_set(Max_Found,Numbers[pot]);
Max_pot := pot;
Digits_Found := Digits;
end;
Next_Number(Base,pot+1);
end;
end;
end;
 
var
base,i : NativeUint;
sol : pChar;
Begin
GetMem(sol,10000);
InitAll;
try
For base := 3 to 31 do
begin
IF (Base>17) AND Not(Odd(Base)) then
continue;
InitPot(base);
gbl_Count := 0;
write('Base ',base:2,' digits ');
Next_Number(base,0);
write(Max_Pot+1:4,' checks ',gbl_Count:8,' ');
if mpz_fits_ulong_p(Max_Found)<> 0 then
write(mpz_get_ui(Max_Found),' ')
else
Begin
mpz_get_str(Sol,10,Max_Found);
write(Sol,' ');
end;
For i := Max_Pot downto 0 do
write(DgtToChar[Digits_Found[i]]);
writeln;
end;
except
ClearAll;
end;
ClearAll;
FreeMem(sol);
end.
</syntaxhighlight>
{{out|@TIO.RUN}}
<pre>
Base 3 digits 3 checks 4 23 212
Base 4 digits 6 checks 17 4091 333323
Base 5 digits 6 checks 16 7817 222232
Base 6 digits 17 checks 455 4836525320399 14141511414451435
Base 7 digits 7 checks 23 817337 6642623
Base 8 digits 15 checks 447 14005650767869 313636165537775
Base 9 digits 10 checks 109 1676456897 4284484465
Base 10 digits 24 checks 4261 357686312646216567629137 357686312646216567629137
Base 11 digits 9 checks 76 2276005673 A68822827
Base 12 digits 32 checks 170054 13092430647736190817303130065827539 471A34A164259BA16B324AB8A32B7817
Base 13 digits 8 checks 101 812751503 CC4C8C65
Base 14 digits 26 checks 34394 615419590422100474355767356763 D967CCD63388522619883A7D23
Base 15 digits 22 checks 9358 34068645705927662447286191 6C6C2CE2CEEEA4826E642B
Base 16 digits 25 checks 27983 1088303707153521644968345559987 DBC7FBA24FE6AEC462ABF63B3
Base 17 digits 11 checks 363 13563641583101 6C66CC4CC83
Base 19 digits 14 checks 686 546207129080421139 CIEG86GCEA2C6H
Base 21 digits 27 checks 59132 391461911766647707547123429659688417 G8AGG2GCA8CAK4K68GEA4G2K22H
Base 23 digits 17 checks 1373 116516557991412919458949 IMMGM6C6IMCI66A4H
Base 25 digits 20 checks 10485 8211352191239976819943978913 ME6OM6OECGCC24C6EG6D
Base 27 digits 28 checks 98337 10681632250257028944950166363832301357693 O2AKK6EKG844KAIA4MACK6C2ECAB
Base 29 digits 19 checks 3927 4300289072819254369986567661 KCG66AGSCKEIASMCKKJ
Base 31 digits 22 checks 11315 645157007060845985903112107793191 UUAUIKUC4UI6OCECI642SD
 
Real time: 4.274 s User time: 4.155 s Sys. time: 0.047 s CPU share: 98.32 %</pre>
 
=={{header|Perl}}==
Similar to the Pari solution. Uses ntheory for primality tests and Math::GMPz for bigints that aren't dog slow. We use <tt>is_prob_prime</tt> in the loop which does the BPSW test, then generate a proof for the selected result.
 
a(18) has a max candidate list of 1,449,405 entries and takes a bit over 20 minutes to solve.
{{libheader|ntheory}}
<syntaxhighlight lang="perl">use ntheory qw/:all/;
use Math::GMPz;
 
sub lltp {
my($n, $b, $best) = (shift, Math::GMPz->new(1));
my @v = map { Math::GMPz->new($_) } @{primes($n-1)};
while (@v) {
$best = vecmax(@v);
$b *= $n;
my @u;
foreach my $vi (@v) {
push @u, grep { is_prob_prime($_) } map { $vi + $_*$b } 1 .. $n-1;
}
@v = @u;
}
die unless is_provable_prime($best);
$best;
}
 
printf "%2d %s\n", $_, lltp($_) for 3 .. 17;</syntaxhighlight>
{{out}}
<pre>
3 23
4 4091
5 7817
6 4836525320399
7 817337
8 14005650767869
9 1676456897
10 357686312646216567629137
11 2276005673
12 13092430647736190817303130065827539
13 812751503
14 615419590422100474355767356763
15 34068645705927662447286191
16 1088303707153521644968345559987
17 13563641583101
</pre>
 
=={{header|Phix}}==
{{libheader|Phix/mpfr}}
{{trans|C}}
Depth-first search uses 1% of the memory of width-first search, and as a result runs about 30% faster (while still doing exactly the same actual work).
<!--<syntaxhighlight lang="phix">(phixonline)-->
<span style="color: #008080;">with</span> <span style="color: #008080;">javascript_semantics</span>
<span style="color: #008080;">include</span> <span style="color: #004080;">mpfr</span><span style="color: #0000FF;">.</span><span style="color: #000000;">e</span>
<span style="color: #004080;">sequence</span> <span style="color: #000000;">tens</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">mpz_inits</span><span style="color: #0000FF;">(</span><span style="color: #000000;">1</span><span style="color: #0000FF;">,</span><span style="color: #000000;">1</span><span style="color: #0000FF;">),</span>
<span style="color: #000000;">vals</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">mpz_inits</span><span style="color: #0000FF;">(</span><span style="color: #000000;">1</span><span style="color: #0000FF;">),</span>
<span style="color: #000000;">digits</span> <span style="color: #0000FF;">=</span> <span style="color: #0000FF;">{</span><span style="color: #000000;">0</span><span style="color: #0000FF;">}</span>
<span style="color: #004080;">mpz</span> <span style="color: #000000;">answer</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">mpz_init</span><span style="color: #0000FF;">(</span><span style="color: #000000;">0</span><span style="color: #0000FF;">)</span>
<span style="color: #004080;">integer</span> <span style="color: #000000;">base</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">seen_depth</span>
<span style="color: #008080;">procedure</span> <span style="color: #000000;">add_digit</span><span style="color: #0000FF;">(</span><span style="color: #004080;">integer</span> <span style="color: #000000;">i</span><span style="color: #0000FF;">)</span>
<span style="color: #004080;">atom</span> <span style="color: #000000;">t1</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">time</span><span style="color: #0000FF;">()+</span><span style="color: #000000;">1</span>
<span style="color: #008080;">if</span> <span style="color: #000000;">i</span><span style="color: #0000FF;">></span><span style="color: #7060A8;">length</span><span style="color: #0000FF;">(</span><span style="color: #000000;">vals</span><span style="color: #0000FF;">)</span> <span style="color: #008080;">then</span>
<span style="color: #000000;">vals</span> <span style="color: #0000FF;">&=</span> <span style="color: #7060A8;">mpz_init_set</span><span style="color: #0000FF;">(</span><span style="color: #000000;">vals</span><span style="color: #0000FF;">[</span><span style="color: #000000;">i</span><span style="color: #0000FF;">-</span><span style="color: #000000;">1</span><span style="color: #0000FF;">])</span>
<span style="color: #000000;">tens</span> <span style="color: #0000FF;">&=</span> <span style="color: #7060A8;">mpz_init_set</span><span style="color: #0000FF;">(</span><span style="color: #000000;">tens</span><span style="color: #0000FF;">[</span><span style="color: #000000;">i</span><span style="color: #0000FF;">-</span><span style="color: #000000;">1</span><span style="color: #0000FF;">])</span>
<span style="color: #7060A8;">mpz_mul_si</span><span style="color: #0000FF;">(</span><span style="color: #000000;">tens</span><span style="color: #0000FF;">[</span><span style="color: #000000;">i</span><span style="color: #0000FF;">],</span><span style="color: #000000;">tens</span><span style="color: #0000FF;">[</span><span style="color: #000000;">i</span><span style="color: #0000FF;">],</span><span style="color: #000000;">base</span><span style="color: #0000FF;">)</span>
<span style="color: #000000;">digits</span> <span style="color: #0000FF;">&=</span> <span style="color: #000000;">0</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
<span style="color: #008080;">for</span> <span style="color: #000000;">d</span><span style="color: #0000FF;">=</span><span style="color: #000000;">1</span> <span style="color: #008080;">to</span> <span style="color: #000000;">base</span><span style="color: #0000FF;">-</span><span style="color: #000000;">1</span> <span style="color: #008080;">do</span>
<span style="color: #000000;">digits</span><span style="color: #0000FF;">[</span><span style="color: #000000;">i</span><span style="color: #0000FF;">]</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">d</span>
<span style="color: #7060A8;">mpz_set</span><span style="color: #0000FF;">(</span><span style="color: #000000;">vals</span><span style="color: #0000FF;">[</span><span style="color: #000000;">i</span><span style="color: #0000FF;">],</span> <span style="color: #000000;">vals</span><span style="color: #0000FF;">[</span><span style="color: #000000;">i</span><span style="color: #0000FF;">-</span><span style="color: #000000;">1</span><span style="color: #0000FF;">])</span>
<span style="color: #7060A8;">mpz_addmul_ui</span><span style="color: #0000FF;">(</span><span style="color: #000000;">vals</span><span style="color: #0000FF;">[</span><span style="color: #000000;">i</span><span style="color: #0000FF;">],</span> <span style="color: #000000;">tens</span><span style="color: #0000FF;">[</span><span style="color: #000000;">i</span><span style="color: #0000FF;">],</span> <span style="color: #000000;">d</span><span style="color: #0000FF;">)</span>
<span style="color: #008080;">if</span> <span style="color: #7060A8;">mpz_prime</span><span style="color: #0000FF;">(</span><span style="color: #000000;">vals</span><span style="color: #0000FF;">[</span><span style="color: #000000;">i</span><span style="color: #0000FF;">])</span> <span style="color: #008080;">then</span>
<span style="color: #008080;">if</span> <span style="color: #000000;">i</span><span style="color: #0000FF;">></span><span style="color: #000000;">seen_depth</span>
<span style="color: #008080;">or</span> <span style="color: #0000FF;">(</span><span style="color: #000000;">i</span><span style="color: #0000FF;">==</span><span style="color: #000000;">seen_depth</span> <span style="color: #008080;">and</span> <span style="color: #7060A8;">mpz_cmp</span><span style="color: #0000FF;">(</span><span style="color: #000000;">vals</span><span style="color: #0000FF;">[</span><span style="color: #000000;">i</span><span style="color: #0000FF;">],</span> <span style="color: #000000;">answer</span><span style="color: #0000FF;">)></span><span style="color: #000000;">0</span><span style="color: #0000FF;">)</span> <span style="color: #008080;">then</span>
<span style="color: #7060A8;">mpz_set</span><span style="color: #0000FF;">(</span><span style="color: #000000;">answer</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">vals</span><span style="color: #0000FF;">[</span><span style="color: #000000;">i</span><span style="color: #0000FF;">])</span>
<span style="color: #000000;">seen_depth</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">i</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
<span style="color: #000000;">add_digit</span><span style="color: #0000FF;">(</span><span style="color: #000000;">i</span><span style="color: #0000FF;">+</span><span style="color: #000000;">1</span><span style="color: #0000FF;">)</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
<span style="color: #008080;">if</span> <span style="color: #7060A8;">time</span><span style="color: #0000FF;">()></span><span style="color: #000000;">t1</span> <span style="color: #008080;">and</span> <span style="color: #7060A8;">platform</span><span style="color: #0000FF;">()!=</span><span style="color: #004600;">JS</span> <span style="color: #008080;">then</span>
<span style="color: #7060A8;">printf</span><span style="color: #0000FF;">(</span><span style="color: #000000;">1</span><span style="color: #0000FF;">,</span><span style="color: #008000;">" base %d: (%d) %v \r"</span><span style="color: #0000FF;">,</span> <span style="color: #0000FF;">{</span><span style="color: #000000;">base</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">seen_depth</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">digits</span><span style="color: #0000FF;">[</span><span style="color: #000000;">1</span><span style="color: #0000FF;">..</span><span style="color: #000000;">i</span><span style="color: #0000FF;">]})</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">for</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">procedure</span>
<span style="color: #008080;">procedure</span> <span style="color: #000000;">do_base</span><span style="color: #0000FF;">()</span>
<span style="color: #004080;">atom</span> <span style="color: #000000;">t0</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">time</span><span style="color: #0000FF;">()</span>
<span style="color: #000000;">seen_depth</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">0</span>
<span style="color: #7060A8;">mpz_set_si</span><span style="color: #0000FF;">(</span><span style="color: #000000;">answer</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">0</span><span style="color: #0000FF;">)</span>
<span style="color: #7060A8;">mpz_set_si</span><span style="color: #0000FF;">(</span><span style="color: #000000;">tens</span><span style="color: #0000FF;">[</span><span style="color: #000000;">1</span><span style="color: #0000FF;">],</span> <span style="color: #000000;">1</span><span style="color: #0000FF;">)</span>
<span style="color: #008080;">for</span> <span style="color: #000000;">i</span><span style="color: #0000FF;">=</span><span style="color: #000000;">2</span> <span style="color: #008080;">to</span> <span style="color: #7060A8;">length</span><span style="color: #0000FF;">(</span><span style="color: #000000;">tens</span><span style="color: #0000FF;">)</span> <span style="color: #008080;">do</span>
<span style="color: #7060A8;">mpz_mul_si</span><span style="color: #0000FF;">(</span><span style="color: #000000;">tens</span><span style="color: #0000FF;">[</span><span style="color: #000000;">i</span><span style="color: #0000FF;">],</span> <span style="color: #000000;">tens</span><span style="color: #0000FF;">[</span><span style="color: #000000;">i</span><span style="color: #0000FF;">-</span><span style="color: #000000;">1</span><span style="color: #0000FF;">],</span> <span style="color: #000000;">base</span><span style="color: #0000FF;">)</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">for</span>
<span style="color: #008080;">for</span> <span style="color: #000000;">i</span><span style="color: #0000FF;">=</span><span style="color: #000000;">1</span> <span style="color: #008080;">to</span> <span style="color: #000000;">base</span> <span style="color: #008080;">do</span>
<span style="color: #004080;">integer</span> <span style="color: #000000;">pi</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">get_prime</span><span style="color: #0000FF;">(</span><span style="color: #000000;">i</span><span style="color: #0000FF;">)</span>
<span style="color: #008080;">if</span> <span style="color: #000000;">pi</span><span style="color: #0000FF;">>=</span><span style="color: #000000;">base</span> <span style="color: #008080;">then</span> <span style="color: #008080;">exit</span> <span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
<span style="color: #7060A8;">mpz_set_si</span><span style="color: #0000FF;">(</span><span style="color: #000000;">vals</span><span style="color: #0000FF;">[</span><span style="color: #000000;">1</span><span style="color: #0000FF;">],</span> <span style="color: #000000;">pi</span><span style="color: #0000FF;">)</span>
<span style="color: #000000;">digits</span><span style="color: #0000FF;">[</span><span style="color: #000000;">1</span><span style="color: #0000FF;">]</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">pi</span>
<span style="color: #000000;">add_digit</span><span style="color: #0000FF;">(</span><span style="color: #000000;">2</span><span style="color: #0000FF;">)</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">for</span>
<span style="color: #004080;">string</span> <span style="color: #000000;">rd</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">mpz_get_str</span><span style="color: #0000FF;">(</span><span style="color: #000000;">answer</span><span style="color: #0000FF;">),</span>
<span style="color: #000000;">rb</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">mpz_get_str</span><span style="color: #0000FF;">(</span><span style="color: #000000;">answer</span><span style="color: #0000FF;">,</span><span style="color: #000000;">base</span><span style="color: #0000FF;">)</span>
<span style="color: #000000;">t0</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">time</span><span style="color: #0000FF;">()-</span><span style="color: #000000;">t0</span>
<span style="color: #004080;">string</span> <span style="color: #000000;">t</span> <span style="color: #0000FF;">=</span> <span style="color: #008080;">iff</span><span style="color: #0000FF;">(</span><span style="color: #000000;">t0</span><span style="color: #0000FF;">></span><span style="color: #000000;">0.1</span><span style="color: #0000FF;">?</span><span style="color: #008000;">" ["</span><span style="color: #0000FF;">&</span><span style="color: #7060A8;">elapsed</span><span style="color: #0000FF;">(</span><span style="color: #000000;">t0</span><span style="color: #0000FF;">)&</span><span style="color: #008000;">"]"</span><span style="color: #0000FF;">:</span><span style="color: #008000;">""</span><span style="color: #0000FF;">)</span>
<span style="color: #7060A8;">printf</span><span style="color: #0000FF;">(</span><span style="color: #000000;">1</span><span style="color: #0000FF;">,</span><span style="color: #008000;">"%3d %-41s (%s, %d digits)%s\n"</span><span style="color: #0000FF;">,</span> <span style="color: #0000FF;">{</span><span style="color: #000000;">base</span><span style="color: #0000FF;">,</span><span style="color: #000000;">rd</span><span style="color: #0000FF;">,</span><span style="color: #000000;">rb</span><span style="color: #0000FF;">,</span><span style="color: #7060A8;">length</span><span style="color: #0000FF;">(</span><span style="color: #000000;">rb</span><span style="color: #0000FF;">),</span><span style="color: #000000;">t</span><span style="color: #0000FF;">})</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">procedure</span>
<span style="color: #004080;">atom</span> <span style="color: #000000;">t0</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">time</span><span style="color: #0000FF;">()</span>
<span style="color: #008080;">for</span> <span style="color: #000000;">b</span><span style="color: #0000FF;">=</span><span style="color: #000000;">3</span> <span style="color: #008080;">to</span> <span style="color: #000000;">31</span> <span style="color: #008080;">do</span>
<span style="color: #008080;">if</span> <span style="color: #0000FF;">(</span><span style="color: #7060A8;">platform</span><span style="color: #0000FF;">()!=</span><span style="color: #004600;">JS</span> <span style="color: #008080;">or</span> <span style="color: #008080;">not</span> <span style="color: #7060A8;">find</span><span style="color: #0000FF;">(</span><span style="color: #000000;">b</span><span style="color: #0000FF;">,{</span><span style="color: #000000;">12</span><span style="color: #0000FF;">,</span><span style="color: #000000;">14</span><span style="color: #0000FF;">,</span><span style="color: #000000;">16</span><span style="color: #0000FF;">,</span><span style="color: #000000;">21</span><span style="color: #0000FF;">,</span><span style="color: #000000;">25</span><span style="color: #0000FF;">,</span><span style="color: #000000;">27</span><span style="color: #0000FF;">,</span><span style="color: #000000;">31</span><span style="color: #0000FF;">}))</span>
<span style="color: #008080;">and</span> <span style="color: #0000FF;">(</span><span style="color: #000000;">b</span><span style="color: #0000FF;"><=</span><span style="color: #000000;">17</span> <span style="color: #008080;">or</span> <span style="color: #7060A8;">remainder</span><span style="color: #0000FF;">(</span><span style="color: #000000;">b</span><span style="color: #0000FF;">,</span><span style="color: #000000;">2</span><span style="color: #0000FF;">)=</span><span style="color: #000000;">1</span><span style="color: #0000FF;">)</span> <span style="color: #008080;">then</span>
<span style="color: #000000;">base</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">b</span>
<span style="color: #000000;">do_base</span><span style="color: #0000FF;">()</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">for</span>
<span style="color: #0000FF;">?</span><span style="color: #7060A8;">elapsed</span><span style="color: #0000FF;">(</span><span style="color: #7060A8;">time</span><span style="color: #0000FF;">()-</span><span style="color: #000000;">t0</span><span style="color: #0000FF;">)</span>
<!--</syntaxhighlight>-->
{{out}}
Even (as in multiples of 2) bases above 18 omitted, so that it completes in a reasonable timeframe, although it does show progress.<br>
Likewise several further numbers are omitted under pwa/p2js so you can stare at a blank screen for about 6s instead of 4&frac12; mins.<br>
I once managed to get 18, but it took over 40 minutes, likewise 22 took 3 mins 46s, 33 over 8 mins, and 35 nearly 2 mins.
<pre>
3 23 (212, 3 digits)
4 4091 (333323, 6 digits)
5 7817 (222232, 6 digits)
6 4836525320399 (14141511414451435, 17 digits)
7 817337 (6642623, 7 digits)
8 14005650767869 (313636165537775, 15 digits)
9 1676456897 (4284484465, 10 digits)
10 357686312646216567629137 (357686312646216567629137, 24 digits) [0.2s]
11 2276005673 (a68822827, 9 digits)
12 13092430647736190817303130065827539 (471a34a164259ba16b324ab8a32b7817, 32 digits) [12.0s]
13 812751503 (cc4c8c65, 8 digits)
14 615419590422100474355767356763 (d967ccd63388522619883a7d23, 26 digits) [2.3s]
15 34068645705927662447286191 (6c6c2ce2ceeea4826e642b, 22 digits) [0.6s]
16 1088303707153521644968345559987 (dbc7fba24fe6aec462abf63b3, 25 digits) [2.0s]
17 13563641583101 (6c66cc4cc83, 11 digits)
19 546207129080421139 (cieg86gcea2c6h, 14 digits)
21 391461911766647707547123429659688417 (g8agg2gca8cak4k68gea4g2k22h, 27 digits) [5.8s]
23 116516557991412919458949 (immgm6c6imci66a4h, 17 digits)
25 8211352191239976819943978913 (me6om6oecgcc24c6eg6d, 20 digits) [1s]
27 10681632250257028944950166363832301357693 (o2akk6ekg844kaia4mack6c2ecab, 28 digits) [12.0s]
29 4300289072819254369986567661 (kcg66agsckeiasmckkj, 19 digits) [0.4s]
31 645157007060845985903112107793191 (uuauikuc4ui6oceci642sd, 22 digits) [1.3s]
"38s"
</pre>
 
=={{header|Python}}==
<syntaxhighlight lang="python">import random
 
def is_probable_prime(n,k):
#this uses the miller-rabin primality test found from rosetta code
if n==0 or n==1:
return False
if n==2:
return True
if n % 2 == 0:
return False
s = 0
d = n-1
 
while True:
quotient, remainder = divmod(d, 2)
if remainder == 1:
break
s += 1
d = quotient
 
def try_composite(a):
if pow(a, d, n) == 1:
return False
for i in range(s):
if pow(a, 2**i * d, n) == n-1:
return False
return True # n is definitely composite
for i in range(k):
a = random.randrange(2, n)
if try_composite(a):
return False
return True # no base tested showed n as composite
def largest_left_truncatable_prime(base):
radix = 0
candidates = [0]
while True:
new_candidates=[]
multiplier = base**radix
for i in range(1,base):
new_candidates += [x+i*multiplier for x in candidates if is_probable_prime(x+i*multiplier,30)]
if len(new_candidates)==0:
return max(candidates)
candidates = new_candidates
radix += 1
 
for b in range(3,24):
print("%d:%d\n" % (b,largest_left_truncatable_prime(b)))
</syntaxhighlight>
 
Output:
<pre>3:23
 
4:4091
 
5:7817
 
6:4836525320399
 
7:817337
 
8:14005650767869
 
9:1676456897
 
10:357686312646216567629137
 
11:2276005673
 
12:13092430647736190817303130065827539
 
13:812751503
</pre>
 
=={{header|Racket}}==
<syntaxhighlight lang="racket">#lang racket
(require math/number-theory)
 
(define (prepend-digit b d i n)
(+ (* d (expt b i)) n))
 
(define (extend b i ts)
(define ts*
(for/list ([t (in-set ts)])
(for/set ([d (in-range 1 b)]
#:when (prime? (prepend-digit b d i t)))
(prepend-digit b d i t))))
(apply set-union ts*))
 
(define (truncables b n)
; return set of truncables of length n in base b
(if (= n 1)
(for/set ([d (in-range 1 b)] #:when (prime? d)) d)
(extend b (- n 1) (truncables b (- n 1)))))
 
(define (largest b)
(let loop ([ts (truncables b 1)]
[n 1])
(define ts* (extend b n ts))
(if (set-empty? ts*)
(apply max (set->list ts))
(loop ts* (+ n 1)))))
 
 
(for/list ([b (in-range 3 18)])
(define l (largest b))
; (displayln (list b l))
(list b l))
 
; Output:
'((3 23)
(4 4091)
(5 7817)
(6 4836525320399)
(7 817337)
(8 14005650767869)
(9 1676456897)
(10 357686312646216567629137)
(11 2276005673)
(12 13092430647736190817303130065827539)
(13 812751503)
(14 615419590422100474355767356763)
(15 34068645705927662447286191)
(16 1088303707153521644968345559987)
(17 13563641583101))</syntaxhighlight>
 
=={{header|Raku}}==
(formerly Perl 6)
{{works with|Rakudo|2018.12}}
Pretty fast for bases 3 .. 11. 12 is slow. 18 is glacial.
<syntaxhighlight lang="raku" line>use ntheory:from<Perl5> <is_prime>;
 
for 3 .. 11 -> $base {
say "Starting base $base...";
my @stems = grep { .is-prime }, ^$base;
for 1 .. * -> $digits {
print ' ', @stems.elems;
my @new;
my $place = $base ** $digits;
for 1 ..^ $base -> $digit {
my $left = $digit * $place;
@new.append: (@stems »+» $left).grep: { is_prime("$_") }
}
last unless +@new;
@stems = @new;
}
say "\nLargest ltp in base $base = {@stems.max} or :$base\<@stems.max.base($base)}>\n";
}</syntaxhighlight>
{{out}}
<pre>Starting base 3...
1 1 1
Largest ltp in base 3 = 23 or :3<212>
Finalists: 1
 
Checking :3<212>
Largest ltp in base 3 = 23
Starting base 4...
2 2 3 53 53 6 4 23
Largest ltp in base 4 = 4091 or :4<333323>
Finalists: 12
 
Checking :4<3323233>
Oops, :4<33> is not a prime!!!
Checking :4<1323233>
Oops, :4<33> is not a prime!!!
Checking :4<333323>
Largest ltp in base 4 = 4091
Starting base 5...
2 4 4 3 1 1
Largest ltp in base 5 = 7817 or :5<222232>
Finalists: 8
 
Checking :5<222232>
Largest ltp in base 5 = 7817
Starting base 6...
3 4 12 2625 4544 5654 6560 6762 6659 5751 4035 2520 1412 97 43 2 1
Largest ltp in base 6 = 4836525320399 or :6<14141511414451435>
Finalists: 285
 
Checking :6<14141511414451435>
Largest ltp in base 6 = 4836525320399
Starting base 7...
3 6 6 4 1 1 1
Largest ltp in base 7 = 817337 or :7<6642623>
Finalists: 11
 
Checking :7<6642623>
Largest ltp in base 7 = 817337
Starting base 8...
4 12 29 50 6766 7977 6561 5251 3938 2827 17 8 3 2 1
Largest ltp in base 8 = 14005650767869 or :8<313636165537775>
Finalists: 294
 
Checking :8<313636165537775>
Largest ltp in base 8 = 14005650767869
Starting base 9...
4 9 15 17 24 16 9 6 5 3
Largest ltp in base 9 = 1676456897 or :9<4284484465>
Finalists: 63
 
Checking :9<4284484465>
Largest ltp in base 9 = 1676456897
Starting base 10...
4 11 4039 10199 197192 335326 439429 536521 556545 528517 456448 358354 278276 212 117 72 42 24 13 6 5 4 3 1
Largest ltp in base 10 = 357686312646216567629137 or :10<357686312646216567629137>
Finalists: 287
 
Checking :10<357686312646216567629137>
Largest ltp in base 10 = 357686312646216567629137
Starting base 11...
4 8 15 18 15 8 4 2 1
Largest ltp in base 11 = 2276005673 or :11<A68822827>
Finalists: 48
 
Checking :11<A68822827>
Largest ltp in base 11 = 2276005673
Starting base 12...
5 2423 124119 431409 11791124 26162496 49484733 80547711 1173211231 1546014826 1810017341 1954618787 1977719001 1828017567 1577115169 1257412085 96369272 68666606 46254451 29902882 18671796 11521108 627601 357346 189181 107103 5049 2019 98 32 1 1
Largest ltp in base 12 = 13092430647736190817303130065827539 or :12<471A34A164259BA16B324AB8A32B7817>
Finalists: 190
 
Checking :12<471A34A164259BA16B324AB8A32B7817>
Largest ltp in base 12 = 13092430647736190817303130065827539
Starting base 13...
5 13 2120 23 17 11 7 4
Largest ltp in base 13 = 812751503 or :13<CC4C8C65>
Finalists: 62
 
Checking :13<CC4C8C65>
Largest ltp in base 13 = 812751503
Starting base 14...
6 2826 103101 308300 694678 13291299 21482093 30893017 38443751 42904196 43114197 39233823 32903206 26092549 19481908 12981269 809783 529507 332322 167163 97 55 27 13 5 2
Largest ltp in base 14 = 615419590422100474355767356763 or :14<D967CCD63388522619883A7D23>
Finalists: 366
 
Checking :14<D967CCD63388522619883A7D23>
Largest ltp in base 14 = 615419590422100474355767356763
Starting base 15...
6 22 8079 213207 401391 658644 955934 12081177 13071275 12091167 10751039 841816 626608 434424 268261 144142 7574 4645 2725 1413 7 1
Largest ltp in base 15 = 34068645705927662447286191 or :15<6C6C2CE2CEEEA4826E642B>
Finalists: 314
 
Checking :15<6C6C2CE2CEEEA4826E642B>
Largest ltp in base 15 = 34068645705927662447286191
Starting base 16...
6 3231 131124 355337 792749 13691292 20931973 28622695 34053210 36933490 35193335 31402980 26602525 19301840 13421278 910878 571556 335326 180174 9593 50 25 9 5 1
Largest ltp in base 16 = 1088303707153521644968345559987 or :16<DBC7FBA24FE6AEC462ABF63B3>
Finalists: 365
 
Checking :16<DBC7FBA24FE6AEC462ABF63B3>
Largest ltp in base 16 = 1088303707153521644968345559987
Starting base 17...
6 2322 4743 6455 8074 6258 4341 3231 23 8 1
Largest ltp in base 17 = 13563641583101 or :17<6C66CC4CC83>
Finalists: 249
...</pre>
Checking :17<6C66CC4CC83>
Largest ltp in base 17 = 13563641583101
Starting base 18...
7 49 311 1396 5117 15243 38818 85684 167133 293518 468456 680171 911723 1133959 1313343 1423597 1449405 1395514 1270222 1097353 902477 707896 529887 381239 263275 174684 111046 67969 40704 23201 12793 6722 3444 1714 859 422 205 98 39 14 7 1 1
Finalists: 364
Checking :18<AF93E41A586HE75A7HHAAB7HE12FG79992GA7741B3D>
Largest ltp in base 18 = 571933398724668544269594979167602382822769202133808087
Starting base 19...
7 29 61 106 122 117 93 66 36 18 10 10 6 4
Finalists: 350
Checking :19<CIEG86GCEA2C6H>
Largest ltp in base 19 = 546207129080421139
Starting base 20...
8 56 321 1311 4156 10963 24589 47737 83011 129098 181707 234685 278792 306852 315113 302684 273080 233070 188331 145016 105557 73276 48819 31244 19237 11209 6209 3383 1689 917 430 196 80 44 20 7 2
Finalists: 349
Checking :20<FC777G3CG1FIDI9I31IE5FFB379C7A3F6EFID>
Largest ltp in base 20 = 1073289911449776273800623217566610940096241078373
Starting base 21...
8 41 165 457 1079 2072 3316 4727 6003 6801 7051 6732 5862 4721 3505 2474 1662 1039 628 369 219 112 52 17 13 4 2
Finalists: 200
Checking :21<G8AGG2GCA8CAK4K68GEA4G2K22H>
Largest ltp in base 21 = 391461911766647707547123429659688417
Starting base 22...
8 48 261 1035 3259 8247 17727 33303 55355 82380 111919 137859 158048 167447 165789 153003 132516 108086 83974 61950 43453 29212 18493 11352 6693 3738 2053 1125 594 293 145 70 31 13 6 2 1
Finalists: 268
Checking :22<FFHALC8JFB9JKA2AH9FAB4I9L5I9L3GF8D5L5>
Largest ltp in base 22 = 33389741556593821170176571348673618833349516314271
Starting base 23...
8 30 78 137 181 200 186 171 121 100 67 41 24 16 9 2 1
Finalists: 260
Checking :23<IMMGM6C6IMCI66A4H>
Largest ltp in base 23 = 116516557991412919458949</pre>
 
=={{header|Ruby}}==
===Ruby Ruby===
<langsyntaxhighlight lang="ruby">
# Compute the largest left truncatable prime
#
Line 281 ⟶ 1,922:
}
puts "The largest left truncatable prime #{"less than #{BASE ** MAX} " if MAX < 500}in base #{BASE} is #{stems.max}"
</syntaxhighlight>
</lang>
By changing BASE from 3 to 14 this produces the solutions in 'Number of left truncatable primes in a given base' on the Discussion Page for bases except 10, 12 and 14.
 
Line 291 ⟶ 1,932:
===JRuby===
I require a fast probably prime test. Java has one, is it any good? Let's find out. Ruby can borrow from Java using JRuby. Modifying the Ruby solution:
<syntaxhighlight lang="ruby">
<lang Ruby>
# Compute the largest left truncatable prime
#
Line 315 ⟶ 1,956:
}
puts "\nThe largest left truncatable prime #{"less than #{BASE ** MAX} " if MAX < 500}in base #{BASE} is #{stems.max}"
</syntaxhighlight>
</lang>
Produces all the reults in 'Number of left truncatable primes in a given base' on the discussion page. For bases 18, 20, and 22 I changed the confidence level from 100 to 5 and checked the final answer. Even so base 18 takes a while. For base 24:
<pre>
Line 324 ⟶ 1,965:
</pre>
That is going to be big!
 
=={{header|Scala}}==
<syntaxhighlight lang="scala">import scala.collection.parallel.immutable.ParSeq
 
object LeftTruncatablePrime extends App {
private def leftTruncatablePrime(maxRadix: Int, millerRabinCertainty: Int) {
def getLargestLeftTruncatablePrime(radix: Int, millerRabinCertainty: Int): BigInt = {
def getNextLeftTruncatablePrimes(n: BigInt, radix: Int, millerRabinCertainty: Int) = {
def baseString = if (n == 0) "" else n.toString(radix)
 
for {i <- (1 until radix).par
p = BigInt(Integer.toString(i, radix) + baseString, radix)
if p.isProbablePrime(millerRabinCertainty)
} yield p
}
 
def iter(list: ParSeq[BigInt], lastList: ParSeq[BigInt]): ParSeq[BigInt] = {
if (list.isEmpty) lastList
else
iter((for (n <- list.par) yield getNextLeftTruncatablePrimes(n, radix, millerRabinCertainty)).flatten, list)
}
 
iter(getNextLeftTruncatablePrimes(0, radix, millerRabinCertainty), ParSeq.empty).max
}
 
for (radix <- (3 to maxRadix).par) {
val largest = getLargestLeftTruncatablePrime(radix, millerRabinCertainty)
println(f"n=$radix%3d: " +
(if (largest == null) "No left-truncatable prime"
else f"$largest%35d (in base $radix%3d) ${largest.toString(radix)}"))
 
}
}
 
val argu: Array[String] = if (args.length >=2 ) args.slice(0, 2) else Array("17", "100")
val maxRadix = argu(0).toInt.ensuring(_ > 2, "Radix must be an integer greater than 2.")
 
try {
val millerRabinCertainty = argu(1).toInt
 
println(s"Run with maxRadix = $maxRadix and millerRabinCertainty = $millerRabinCertainty")
 
leftTruncatablePrime(maxRadix, millerRabinCertainty)
println(s"Successfully completed without errors. [total ${scala.compat.Platform.currentTime - executionStart} ms]")
}
catch {
case _: NumberFormatException => Console.err.println("Miller-Rabin Certainty must be an integer.")
}
 
}</syntaxhighlight>
 
=={{header|Sidef}}==
{{trans|Perl}}
<syntaxhighlight lang="ruby">func lltp(n) {
var b = 1
var best = nil
var v = (n-1 -> primes)
 
while (v) {
best = v.max
b *= n
v.map! { |vi|
{|i| i*b + vi }.map(1..^n).grep{.is_prime}...
}
}
 
return best
}
 
for i in (3..17) {
printf("%2d %s\n", i, lltp(i))
}</syntaxhighlight>
{{out}}
<pre>
3 23
4 4091
5 7817
6 4836525320399
7 817337
8 14005650767869
9 1676456897
10 357686312646216567629137
11 2276005673
12 13092430647736190817303130065827539
13 812751503
14 615419590422100474355767356763
15 34068645705927662447286191
16 1088303707153521644968345559987
17 13563641583101
</pre>
 
Alternative solution:
<syntaxhighlight lang="ruby">func digits2num(digits, base) {
digits.map_kv {|k,v| base**k * v }.sum
}
 
func generate_from_suffix(p, base) {
 
var seq = [p]
 
for n in (1 ..^ base) {
var t = [p..., n]
if (is_prime(digits2num(t, base))) {
seq << __FUNC__(t, base)...
}
}
 
return seq
}
 
func left_truncatable_primes(base) {
 
var prime_digits = (base-1 -> primes)
 
prime_digits.map {|p| generate_from_suffix([p], base)... }\
.map {|t| digits2num(t, base) }\
.sort
}
 
for n in (3..11) {
var ltp = left_truncatable_primes(n)
say ("There are #{'%4d' % ltp.len} left-truncatable primes in base #{'%2d' % n}, where largest is #{ltp.max}")
}</syntaxhighlight>
{{out}}
<pre>
There are 3 left-truncatable primes in base 3, where largest is 23
There are 16 left-truncatable primes in base 4, where largest is 4091
There are 15 left-truncatable primes in base 5, where largest is 7817
There are 454 left-truncatable primes in base 6, where largest is 4836525320399
There are 22 left-truncatable primes in base 7, where largest is 817337
There are 446 left-truncatable primes in base 8, where largest is 14005650767869
There are 108 left-truncatable primes in base 9, where largest is 1676456897
There are 4260 left-truncatable primes in base 10, where largest is 357686312646216567629137
There are 75 left-truncatable primes in base 11, where largest is 2276005673
</pre>
 
=={{header|Swift}}==
{{trans|Python}}
{{libheader|Attaswift BigInt}}
<syntaxhighlight lang="text">import BigInt
 
func largestLeftTruncatablePrime(_ base: Int) -> BigInt {
var radix = 0
var candidates = [BigInt(0)]
 
while true {
let multiplier = BigInt(base).power(radix)
var newCandidates = [BigInt]()
 
for i in 1..<BigInt(base) {
newCandidates += candidates.map({ ($0+i*multiplier, ($0+i*multiplier).isPrime(rounds: 30)) })
.filter({ $0.1 })
.map({ $0.0 })
}
 
if newCandidates.count == 0 {
return candidates.max()!
}
 
candidates = newCandidates
radix += 1
}
}
 
for i in 3..<18 {
print("\(i): \(largestLeftTruncatablePrime(i))")
}</syntaxhighlight>
 
{{out}}
<pre>3: 23
4: 4091
5: 7817
6: 4836525320399
7: 817337
8: 14005650767869
9: 1676456897
10: 357686312646216567629137
11: 2276005673
12: 13092430647736190817303130065827539
13: 812751503
14: 615419590422100474355767356763
15: 34068645705927662447286191
16: 1088303707153521644968345559987
17: 13563641583101
 
real 1m17.433s
user 1m16.915s
sys 0m0.252s</pre>
 
=={{header|Tcl}}==
<syntaxhighlight lang="tcl">package require Tcl 8.5
 
proc tcl::mathfunc::modexp {a b n} {
for {set c 1} {$b} {set a [expr {$a*$a%$n}]} {
if {$b & 1} {
set c [expr {$c*$a%$n}]
}
set b [expr {$b >> 1}]
}
return $c
}
# Based on Miller-Rabin primality testing, but with small prime check first
proc is_prime {n {count 10}} {
# fast check against small primes
foreach p {
2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 73 79 83 89 97
} {
if {$n == $p} {return true}
if {$n % $p == 0} {return false}
}
 
# write n-1 as 2^s·d with d odd by factoring powers of 2 from n-1
set d [expr {$n - 1}]
for {set s 0} {$d & 1 == 0} {incr s} {
set d [expr {$d >> 1}]
}
for {} {$count > 0} {incr count -1} {
set a [expr {2 + int(rand()*($n - 4))}]
set x [expr {modexp($a, $d, $n)}]
if {$x == 1 || $x == $n - 1} continue
for {set r 1} {$r < $s} {incr r} {
set x [expr {modexp($x, 2, $n)}]
if {$x == 1} {return false}
if {$x == $n - 1} break
}
if {$x != $n-1} {return false}
}
return true
}
 
proc max_left_truncatable_prime {base} {
set stems {}
for {set i 2} {$i < $base} {incr i} {
if {[is_prime $i]} {
lappend stems $i
}
}
set primes $stems
set size 0
for {set b $base} {[llength $stems]} {set b [expr {$b * $base}]} {
# Progress monitoring is nice once we get to 10 and beyond...
if {$base > 9} {
puts "\t[llength $stems] candidates at length [incr size]"
}
set primes $stems
set certainty [expr {[llength $primes] > 100 ? 1 : 5}]
set stems {}
foreach s $primes {
for {set i 1} {$i < $base} {incr i} {
set n [expr {$b*$i + $s}]
if {[is_prime $n $certainty]} {
lappend stems $n
}
}
}
}
# Could be several at same length; choose largest
return [tcl::mathfunc::max {*}$primes]
}
 
for {set i 3} {$i <= 20} {incr i} {
puts "$i: [max_left_truncatable_prime $i]"
}</syntaxhighlight>
{{out|Output up to base 12 (tab-indented parts are progress messages)}}
<pre>
3: 23
4: 4091
5: 7817
6: 4836525320399
7: 817337
8: 14005650767869
9: 1676456897
4 candidates at length 1
11 candidates at length 2
39 candidates at length 3
99 candidates at length 4
192 candidates at length 5
326 candidates at length 6
429 candidates at length 7
521 candidates at length 8
545 candidates at length 9
517 candidates at length 10
448 candidates at length 11
354 candidates at length 12
276 candidates at length 13
212 candidates at length 14
117 candidates at length 15
72 candidates at length 16
42 candidates at length 17
24 candidates at length 18
13 candidates at length 19
6 candidates at length 20
5 candidates at length 21
4 candidates at length 22
3 candidates at length 23
1 candidates at length 24
10: 357686312646216567629137
4 candidates at length 1
8 candidates at length 2
15 candidates at length 3
18 candidates at length 4
15 candidates at length 5
8 candidates at length 6
4 candidates at length 7
2 candidates at length 8
1 candidates at length 9
11: 2276005673
5 candidates at length 1
23 candidates at length 2
119 candidates at length 3
409 candidates at length 4
1126 candidates at length 5
2504 candidates at length 6
4746 candidates at length 7
7727 candidates at length 8
11257 candidates at length 9
14860 candidates at length 10
17375 candidates at length 11
18817 candidates at length 12
19027 candidates at length 13
17594 candidates at length 14
15192 candidates at length 15
12106 candidates at length 16
9292 candidates at length 17
6621 candidates at length 18
4466 candidates at length 19
2889 candidates at length 20
1799 candidates at length 21
1109 candidates at length 22
601 candidates at length 23
346 candidates at length 24
181 candidates at length 25
103 candidates at length 26
49 candidates at length 27
19 candidates at length 28
8 candidates at length 29
2 candidates at length 30
1 candidates at length 31
1 candidates at length 32
12: 13092430647736190817303130065827539
5 candidates at length 1
13 candidates at length 2
20 candidates at length 3
23 candidates at length 4
17 candidates at length 5
11 candidates at length 6
7 candidates at length 7
4 candidates at length 8
13: 812751503
</pre>
I think I'll need to find a faster computer to calculate much more of the sequence, but memory consumption is currently negligible so there's no reason to expect there to be any major problems.
 
=={{header|Wren}}==
{{trans|Kotlin}}
{{libheader|Wren-big}}
{{libheader|Wren-fmt}}
{{libheader|Wren-sort}}
{{libheader|Wren-ioutil}}
A tough task for the Wren interpreter which doesn't have the benefit of GMP.
 
A tad below 39 minutes to process up to base 17 at the lowest ''certainty'' level.
<syntaxhighlight lang="wren">import "./big" for BigInt
import "./fmt" for Conv, Fmt
import "./sort" for Sort
import "./ioutil" for Input
 
var nextLeftTruncatablePrimes = Fn.new { |n, radix, certainty|
var probablePrimes = []
var baseString = (n == BigInt.zero) ? "" : n.toBaseString(radix)
for (i in 1...radix) {
var p = BigInt.fromBaseString(Conv.itoa(i, radix) + baseString, radix)
if (p.isProbablePrime(certainty)) probablePrimes.add(p)
}
return probablePrimes
}
 
var largestLeftTruncatablePrime = Fn.new { |radix, certainty|
var lastList = null
var list = nextLeftTruncatablePrimes.call(BigInt.zero, radix, certainty)
while (!list.isEmpty) {
lastList = list
list = []
for (n in lastList) list.addAll(nextLeftTruncatablePrimes.call(n, radix, certainty))
}
if (!lastList) return null
Sort.quick(lastList)
return lastList[-1]
}
 
var maxRadix = Input.integer("Enter maximum radix : ", 3, 36)
var certainty = Input.integer("Enter certainty : ", 1, 100)
System.print()
for (radix in 3..maxRadix) {
var largest = largestLeftTruncatablePrime.call(radix, certainty)
Fmt.write("Base = $-2d : ", radix)
if (!largest) {
System.print("No left truncatable prime")
} else {
Fmt.print("$-35i -> $s", largest, largest.toBaseString(radix))
}
}</syntaxhighlight>
 
{{out}}
<pre>
Enter maximum radix : 17
Enter certainty : 1
 
Base = 3 : 23 -> 212
Base = 4 : 4091 -> 333323
Base = 5 : 7817 -> 222232
Base = 6 : 4836525320399 -> 14141511414451435
Base = 7 : 817337 -> 6642623
Base = 8 : 14005650767869 -> 313636165537775
Base = 9 : 1676456897 -> 4284484465
Base = 10 : 833757579699383379513367 -> 833757579699383379513367
Base = 11 : 2276005673 -> a68822827
Base = 12 : 13092430647736190817303130065827539 -> 471a34a164259ba16b324ab8a32b7817
Base = 13 : 812751503 -> cc4c8c65
Base = 14 : 615419590422100474355767356763 -> d967ccd63388522619883a7d23
Base = 15 : 34068645705927662447286191 -> 6c6c2ce2ceeea4826e642b
Base = 16 : 1088303707153521644968345559987 -> dbc7fba24fe6aec462abf63b3
Base = 17 : 13563641583101 -> 6c66cc4cc83
</pre>
 
=={{header|zkl}}==
<syntaxhighlight lang="zkl">var [const] BN=Import("zklBigNum"); // libGMP
fcn largest_lefty_prime(base){
primes,p:=List(),BN(1); while(p.nextPrime()<base){ primes.append(p.copy()) }
b,biggest := BN(1),0;
while(primes){
b*=base; // base,base^2,base^3... gets big
ps:=List();
foreach p,n in (primes,[1..base-1]){
if((z:=(p + b*n)).probablyPrime()){
ps.append(z);
if(z>biggest) biggest=z;
}
}
primes=ps; // the number of lists is small
}
biggest
}
 
foreach n in ([3..17]){ println("%2d %s".fmt(n,largest_lefty_prime(n))) }</syntaxhighlight>
I've included 18,19 & 20 here but 18 & 20 are very very slow to compute, it is seconds to compute all the others.
{{out}}
<pre>
3 23
4 4091
5 7817
6 4836525320399
7 817337
8 14005650767869
9 1676456897
10 357686312646216567629137
11 2276005673
12 13092430647736190817303130065827539
13 812751503
14 615419590422100474355767356763
15 34068645705927662447286191
16 1088303707153521644968345559987
17 13563641583101
18 571933398724668544269594979167602382822769202133808087
19 546207129080421139
20 1073289911449776273800623217566610940096241078373
</pre>
9,488

edits