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Line 1: {{~~draft~~ 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 ((gg)%l) (e/2I) l \| _ -> fn ((ig)%l) ((gg)%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 gi+n\|])\|>Array.filter(fun n->fn 1I 2I (n-1I) n) if (Array.isEmpty g) then n else (fi g (iFbase)) 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 INTEGER8 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. INTEGER1 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 = DBB !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 = DDB + 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 INTEGER600, 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 = 10BIGORDER, 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 INTEGER8 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, INTEGER600 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. CHARACTER42 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. ~~Primes generation stolen from [[Sieve of Eratosthenes]].~~ <syntaxhighlight 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) ~~<lang haskell>primes = 2 : 3 : gaps 5 (unionAll [[pp, pp+2p..] \| p <- tail primes])~~ find2km :: Integral a => a -> (Int,a) find2km n = f 0 n where f k m \| 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 millerRabinPrimality :: Integer -> Integer -> Bool millerRabinPrimality n a \| a >= n_ = True \| b0 == 1 \|\| b0 == n_ = True \| otherwise = iter (tail b) where n_ = n-1 (k,m) = find2km n_ b0 = powMod n a m b = take k $ iterate (squareMod n) b0 iter [] = False iter (x:xs) \| x == 1 = False \| x == n_ = True \| otherwise = iter xs -- (eq. to) pow_ () (^2) n k = n^k pow_ :: (Num a, Integral b) => (a->a->a) -> (a->a) -> a -> b -> a pow_ _ _ _ 0 = 1 pow_ mul sq x_ n_ = f x_ n_ 1 where f x n y \| n == 1 = x `mul` y \| r == 0 = f x2 q y \| otherwise = f x2 q (x `mul` y) where (q,r) = quotRem n 2 x2 = sq x mulMod :: Integral a => a -> a -> a -> a mulMod a b c = (b * c) `mod` a squareMod :: Integral a => a -> a -> a squareMod a b = (b * b) `rem` a -- (eq. to) powMod m n k = n^k `mod` m powMod :: Integral a => a -> a -> a -> a powMod m = pow_ (mulMod m) (squareMod m) -- Caller supplies a witness list w, which may be used for MR test. -- Use faster trial division against a small primes list first, to -- weed out more obvious composites. is_prime w n \| n < 100 = n `elem` primesTo100 \| any ((==0).(n`mod`)) primesTo100 = False \| otherwise = all (millerRabinPrimality n) w -- final result gets a more thorough Miller-Rabin left_trunc base = head $ filter (is_prime primesTo100) (reverse hopeful) where hopeful = extend base $ takeWhile (<base) primesTo100 where extend b x = if null d then x else extend (bbase) d where d = concatMap addDigit [1..base-1] -- we do one* prime test, which seems good enough in practice addDigit a = filter (is_prime [3]) $ map (ab+) x main = mapM_ print $ map (\x->(x, left_trunc x)) [3..21]</syntaxhighlight> <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) (21,391461911766647707547123429659688417) </pre> =={{header\|J}}== <syntaxhighlight lang="j">ltp=:3 :0 probe=. i.1 0 while. #probe do. probe=. (#~ 1 p: y #.]),/(}.i.y),"0 _1/have=. probe end. >./y#.have )</syntaxhighlight> Quick example: <syntaxhighlight lang="j"> (,ltp)"0]3 4 5 6 7 8 9 10 11 3 23 4 4091 5 7817 6 4836525320399 7 817337 8 14005650767869 9 1676456897 10 992429121339693967 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); } return probablePrimes; } 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:''' <pre>java LeftTruncatablePrime 17 100 n=3: 23 (in base 3): 212 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) ~~gaps k s@(x:xs) \| k < x = k:gaps (k+2) s~~ a = Vector{typeof(p)}() ~~\| otherwise = gaps (k+2) xs~~ q = p for i in 1:(b-1) q += s isprime(q) \|\| continue push!(a, q) end return a end function lefttruncprime(pbase::Integer) ~~unionAll ((x:xs):t) = x : union xs (unionAll (pairs t)) where~~ a = Vector{BigInt}() ~~pairs ((x:xs):ys:t) = (x : union xs ys) : pairs t~~ 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 ~~union a@(x:xs) b@(y:ys) = case compare x y of~~ println("The largest left truncatable primes for bases", @sprintf(" %d to %d.", lo, hi)) ~~LT -> x : union xs b~~ for i in lo:hi ~~EQ -> x : union xs ys~~ mlt = lefttruncprime(i) ~~GT -> y : union a ys~~ @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}}== ~~isPrime n = all ((/=0).(n`mod`)) $ takeWhile ((<=n).(^2)) primes~~ {{trans\|Java}} <syntaxhighlight lang="scala">// version 1.1.2 import java.math.BigInteger ~~left_trunc base = maximum $ expand base $ takeWhile (<base) primes where~~ ~~expand b x = if null d then x else expand (bbase) d where~~ ~~d = filter isPrime $ concatMap (addDigit b) x~~ ~~addDigit b x = map ((+x).(b)) [1..base-1]~~ fun nextLeftTruncatablePrimes(n: BigInteger, radix: Int, certainty: Int): List<BigInteger> { ~~main = mapM_ print $ map (\x->(x, left_trunc x)) ([3..9]++[11,13])</lang>~~ val probablePrimes = mutableListOf<BigInteger>() ~~=={{header\|Perl 6}}==~~ val baseString = if (n == BigInteger.ZERO) "" else n.toString(radix) ~~Using the Miller-Rabin definition of <tt>is_prime</tt> from [http://rosettacode.org/wiki/Miller-Rabin_primality_test]:~~ ~~<lang~~ ~~perl6>~~ for 3..(i in 1 ->until ~~$base~~radix) { val p = BigInteger(i.toString(radix) + baseString, radix) ~~say "Starting base $base...";~~ if (p.isProbablePrime(certainty)) probablePrimes.add(p) ~~my @stems = grep { is_prime($_, 100)}, ^$base;~~ ~~my @runoff;~~ ~~for 1 .. * -> $digits {~~ ~~print ' ', @stems.elems;~~ ~~my @new;~~ ~~my $place = $base ** $digits;~~ ~~my $tries = 1; # one try seems to be sufficient~~ ~~for @stems -> $stem {~~ ~~for 1 ..^ $base -> $digit {~~ ~~my $new = $digit * $place + $stem;~~ ~~@new.push($new) if is_prime($new, $tries);~~ } } ~~push @runoff if @new < @stems and @new < 10;~~ ~~last unless @new;~~ ~~@stems = @new;~~ } return probablePrimes ~~push @runoff, @stems;~~ } fun largestLeftTruncatablePrime(radix: Int, certainty: Int): BigInteger? { ~~for @runoff.sort(-) -> $finalist {~~ var lastList: List<BigInteger>? = null ~~my $b = $finalist.base($base);~~ var list = nextLeftTruncatablePrimes(BigInteger.ZERO, radix, certainty) ~~say "\n Checking :$base\<", $b, '>';~~ while (!list.isEmpty()) { ~~my $ok = True;~~ lastList = list ~~for $base * 2, $base 3, $base 4 ... $base ** $b.chars -> $place {~~ list = mutableListOf() ~~my $f = $finalist % $place;~~ for (n in lastList) list.addAll(nextLeftTruncatablePrimes(n, radix, certainty)) ~~if not is_prime($f, 100) {~~ } ~~say " Oops, :$base\<", $f.base($base), '> is not a prime!!!';~~ if (lastList == null) return null ~~$ok = False;~~ return lastList.sorted().last() ~~last;~~ } } } ~~next unless $ok;~~ fun main(args: Array<String>) { ~~say " Largest ltp in base $base = $finalist";~~ print("Enter maximum radix : ") ~~last;~~ 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)}") } }</~~lang~~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 := jb^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: " (" & `$`(m, base.int32) & ')' else: ""</syntaxhighlight> {{out}} <pre>Base Greatest left truncable prime ===================================== 3 23 4 4091 5 7817 6 4836525320399 7 817337 8 14005650767869 9 1676456897 10 357686312646216567629137 11 2276005673 (a68822827) 12 13092430647736190817303130065827539 (471a34a164259ba16b324ab8a32b7817) 13 812751503 (cc4c8c65) 14 615419590422100474355767356763 (d967ccd63388522619883a7d23) 15 34068645705927662447286191 (6c6c2ce2ceeea4826e642b) 16 1088303707153521644968345559987 (dbc7fba24fe6aec462abf63b3) 17 13563641583101 (6c66cc4cc83)</pre> =={{header\|PARI/GP}}== Takes half a second to find the terms up to 10, with proofs of primality. The time can be halved without proofs (use <code>ispseudoprime</code> in place of <code>isprime</code>). <syntaxhighlight lang="parigp">a(n)=my(v=primes(primepi(n-1)),u,t,b=1,best); while(#v, best=vecmax(v); b=n; u=List(); for(i=1,#v,for(k=1,n-1,if(isprime(t=v[i]+kb), listput(u,t)))); v=Vec(u)); best</syntaxhighlight> =={{header\|Pascal}}== ==={{header\|Free Pascal}}=== Using gmp and depth first search, like by [[Find_largest_left_truncatable_prime_in_a_given_base#Phix\|Phix]].<br>Use preset numbers and powers of base to max values expecting that no memory-reallocations during runtime are needed.<br> HTOP shows no variance in used memory. <syntaxhighlight lang="pascal"> program TruncatablePrimes; //https://rosettacode.org/wiki/Find_largest_left_truncatable_prime_in_a_given_base {$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 4½ 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+imultiplier for x in candidates if is_probable_prime(x+imultiplier,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 ~~Checking~~in base 3 = 23 or :3<212> ~~Largest ltp in base 3 = 23~~ Starting base 4... 2 2 3 3 43 43 Largest ltp ~~Checking~~in base 4 = 4091 or :4<333323> ~~Largest ltp in base 4 = 4091~~ Starting base 5... 2 4 54 53 1 1 Largest ltp ~~Checking~~in base 5 = 7817 or :5<222232> ~~Largest ltp in base 5 = 7817~~ Starting base 6... 3 4 12 25 4544 5654 6460 6762 6359 5451 3835 2120 1312 97 3 2 1 Largest ltp ~~Checking~~in base 6 = 4836525320399 or :6<14141511414451435> ~~Largest ltp in base 6 = 4836525320399~~ Starting base 7... 3 6 76 54 21 1 1 Largest ltp ~~Checking~~in base 7 = 817337 or :7<6642623> ~~Largest ltp in base 7 = 817337~~ Starting base 8... 4 12 29 50 66 77 61 51 38 27 17 8 3 2 1 Largest ltp ~~Checking~~in base 8 = 14005650767869 or :8<313636165537775> ~~Largest ltp in base 8 = 14005650767869~~ Starting base 9... 4 9 1615 1917 2724 1716 109 6 5 3 Largest ltp ~~Checking~~in base 9 = 1676456897 or :9<4284484465> ~~Largest ltp in base 9 = 1676456897~~ Starting base 10... 4 1211 39 99 192 ~~327~~326 ~~430~~429 521 545 517 448 354 276 212 117 72 42 24 13 6 5 4 3 1 Largest ltp ~~Checking~~in base 10 = 357686312646216567629137 or :10<357686312646216567629137> ~~Largest ltp in base 10 = 357686312646216567629137~~ Starting base 11... 4 98 1615 18 15 8 4 2 1 Largest ltp ~~Checking~~in base 11 = 2276005673 or :11<A68822827> ~~Largest ltp in base 11 = 2276005673~~ Starting base 12... 5 23 ~~120~~119 ~~418~~409 ~~1146~~1124 ~~2547~~2496 ~~4831~~4733 ~~7862~~7711 ~~11449~~11231 ~~15119~~14826 ~~17678~~17341 ~~19161~~18787 ~~19359~~19001 ~~17910~~17567 ~~15460~~15169 ~~12327~~12085 ~~9472~~9272 ~~6765~~6606 ~~4547~~4451 ~~2935~~2882 ~~1827~~1796 ~~1131~~1108 ~~616~~601 ~~353~~346 ~~185~~181 ~~104~~103 49 19 8 2 1 1 Largest ltp ~~Checking~~in base 12 = 13092430647736190817303130065827539 or :12<471A34A164259BA16B324AB8A32B7817> ~~Largest ltp in base 12 = 13092430647736190817303130065827539~~ Starting base 13... 5 13 2120 2423 17 11 7 4 Largest ltp ~~Checking~~in base 13 = 812751503 or :13<CC4C8C65> ~~Largest ltp in base 13 = 812751503~~ Starting base 14... 6 2826 ~~108~~101 ~~321~~300 ~~724~~678 ~~1392~~1299 ~~2238~~2093 ~~3215~~3017 ~~3987~~3751 ~~4454~~4196 ~~4450~~4197 ~~4056~~3823 ~~3407~~3206 ~~2708~~2549 ~~2015~~1908 ~~1348~~1269 ~~827~~783 ~~535~~507 ~~338~~322 ~~172~~163 ~~105~~97 6055 3027 1513 75 32 Largest ltp ~~Checking~~in base 14 = 615419590422100474355767356763 or :14<D967CCD63388522619883A7D23> ~~Largest ltp in base 14 = 615419590422100474355767356763~~ Starting base 15... 6 22 8079 ~~212~~207 ~~399~~391 ~~655~~644 ~~950~~934 ~~1192~~1177 ~~1293~~1275 ~~1179~~1167 ~~1049~~1039 ~~828~~816 ~~617~~608 ~~427~~424 261 142 74 45 25 13 7 1 Largest ltp ~~Checking~~in base 15 = 34068645705927662447286191 or :15<6C6C2CE2CEEEA4826E642B> ~~Largest ltp in base 15 = 34068645705927662447286191~~ Starting base 16... 6 3231 ~~132~~124 ~~362~~337 ~~811~~749 ~~1395~~1292 ~~2148~~1973 ~~2934~~2695 ~~3506~~3210 ~~3820~~3490 ~~3649~~3335 ~~3253~~2980 ~~2752~~2525 ~~1996~~1840 ~~1395~~1278 ~~955~~878 ~~605~~556 ~~353~~326 ~~193~~174 ~~102~~93 5450 2725 119 5 1 Largest ltp ~~Checking~~in base 16 = 1088303707153521644968345559987 or :16<DBC7FBA24FE6AEC462ABF63B3> ~~Largest ltp in base 16 = 1088303707153521644968345559987~~ Starting base 17... 6 22 43 55 74 58 41 31 23 8 1 Largest ltp ~~Checking~~in base 17 = 13563641583101 or :17<6C66CC4CC83> ...</pre> ~~Largest ltp in base 17 = 13563641583101</pre>~~ =={{header\|Ruby}}== ===Ruby Ruby=== <~~lang~~syntaxhighlight lang="ruby"> # Compute the largest left truncatable prime # Line 163 ⟶ 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. The maximum left truncatable prime in bases 10 , 12, and 14 are very large. By changing MAX ~~from~~to 6 toand 11BASE ~~(note~~to ~~that 6~~10 solves related task 1~~) the following are produced quickly with BASE = 10~~: <pre> The largest left truncatable prime less than 1000000 in base 10 is 998443 ~~The largest left truncatable prime less than 10000000 in base 10 is 9986113~~ ~~The largest left truncatable prime less than 100000000 in base 10 is 99979337~~ ~~The largest left truncatable prime less than 1000000000 in base 10 is 999962683~~ ~~The largest left truncatable prime less than 10000000000 in base 10 is 9987983617~~ ~~The largest left truncatable prime less than 100000000000 in base 10 is 999783972~~ 83 </pre> ===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 208 ⟶ 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 217 ⟶ 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\| ib + 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+imultiplier, ($0+imultiplier).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 + bn)).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>