Data commit

Task/Repunit-primes/00-META.yaml

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+---
+from: http://rosettacode.org/wiki/Repunit_primes

Task/Repunit-primes/00-TASK.txt

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+[[wp:Repunit|Repunit]] is a [[wp:Portmanteau|portmanteau]] of the words "repetition" and "unit", with unit being "unit value"... or in laymans terms, '''1'''.  So 1, 11, 111, 1111 & 11111 are all repunits.
+
+Every standard integer base has repunits since every base has the digit 1. This task involves finding the repunits in different bases that are prime.
+
+In base two, the repunits 11, 111, 11111, 1111111, etc. are prime. (These correspond to the [[wp:Mersenne_prime|Mersenne primes]].) 
+
+In base three: 111, 1111111, 1111111111111, etc.
+
+''Repunit primes, by definition, are also [[circular primes]].''
+
+Any repunit in any base having a composite number of digits is necessarily composite. Only repunits (in any base) having a prime number of digits ''might'' be prime.
+
+
+Rather than expanding the repunit out as a giant list of '''1'''s or converting to base 10, it is common to just list the ''number'' of '''1'''s in the repunit; effectively the digit count. The base two repunit primes listed above would be represented as: 2, 3, 5, 7, etc.
+
+Many of these sequences exist on [[oeis:|OEIS]], though they aren't specifically listed as "repunit prime digits" sequences. 
+
+Some bases have very few repunit primes. Bases 4, 8, and likely 16 have only one. Base 9 has none at all. Bases above 16 may have repunit primes as well... but this task is getting large enough already.
+
+
+;Task
+
+* For bases 2 through 16, Find and show, here on this page, the repunit primes as digit counts, up to a limit of 1000.
+
+
+;Stretch
+
+* Increase the limit to 2700 (or as high as you have patience for.)
+
+
+;See also
+
+;* [[wp:Repunit#Repunit_primes|Wikipedia: Repunit primes]]
+;* [[oeis:A000043|OEIS:A000043 - Mersenne exponents: primes p such that 2^p - 1 is prime. Then 2^p - 1 is called a Mersenne prime]] (base 2)
+;* [[oeis:A028491|OEIS:A028491 - Numbers k such that (3^k - 1)/2 is prime]] (base 3)
+;* [[oeis:A004061|OEIS:A004061 - Numbers n such that (5^n - 1)/4 is prime]] (base 5)
+;* [[oeis:A004062|OEIS:A004062 - Numbers n such that (6^n - 1)/5 is prime]] (base 6)
+;* [[oeis:A004063|OEIS:A004063 - Numbers k such that (7^k - 1)/6 is prime]] (base 7)
+;* [[oeis:A004023|OEIS:A004023 - Indices of prime repunits: numbers n such that 11...111 (with n 1's) = (10^n - 1)/9 is prime]] (base 10)
+;* [[oeis:A005808|OEIS:A005808 - Numbers k such that (11^k - 1)/10 is prime]] (base 11)
+;* [[oeis:A004064|OEIS:A004064 - Numbers n such that (12^n - 1)/11 is prime]] (base 12)
+;* [[oeis:A016054|OEIS:A016054 - Numbers n such that (13^n - 1)/12 is prime]] (base 13)
+;* [[oeis:A006032|OEIS:A006032 - Numbers k such that (14^k - 1)/13 is prime]] (base 14)
+;* [[oeis:A006033|OEIS:A006033 - Numbers n such that (15^n - 1)/14 is prime]] (base 15)
+;* [[Circular primes|Related task: Circular primes]]
+
+
+

Task/Repunit-primes/ALGOL-68/repunit-primes.alg

@@ -0,0 +1,22 @@
+BEGIN # find repunit (all digits are 1 ) primes in various bases #
+    INT max base           = 16;
+    INT max repunit digits = 1000;
+    PR precision 3000 PR        # set precision of LONG LONG INT #
+    # 16^1000 has ~1200 digits but the primality test needs more #
+    PR read "primes.incl.a68" PR       # include prime utilities #
+    []BOOL prime = PRIMESIEVE max repunit digits;
+    FOR base FROM 2 TO max base DO
+        LONG LONG INT repunit := 1;
+        print( ( whole( base, -2 ), ":" ) );
+        FOR digits TO max repunit digits DO
+            IF prime[ digits ] THEN
+                IF is probably prime( repunit ) THEN
+                    # found a prime repunit in the current base #
+                    print( ( " ", whole( digits, 0 ) ) )
+                FI
+            FI;
+            repunit *:= base +:= 1
+        OD;
+        print( ( newline ) )
+    OD
+END

Task/Repunit-primes/Arturo/repunit-primes.arturo

@@ -0,0 +1,15 @@
+getRepunit: function [n,b][
+    result: 1
+    loop 1..dec n 'z ->
+        result: result + b^z
+    return result
+]
+
+loop 2..16 'base [
+    print [
+        pad (to :string base) ++ ":" 4
+        join.with:", " to [:string] select 2..1001 'x ->
+            and? -> prime? x
+                 -> prime? getRepunit x base
+    ]
+]

Task/Repunit-primes/C++/repunit-primes.cpp

@@ -0,0 +1,35 @@
+#include <future>
+#include <iomanip>
+#include <iostream>
+#include <vector>
+
+#include <gmpxx.h>
+#include <primesieve.hpp>
+
+std::vector<uint64_t> repunit_primes(uint32_t base,
+                                     const std::vector<uint64_t>& primes) {
+    std::vector<uint64_t> result;
+    for (uint64_t prime : primes) {
+        mpz_class repunit(std::string(prime, '1'), base);
+        if (mpz_probab_prime_p(repunit.get_mpz_t(), 25) != 0)
+            result.push_back(prime);
+    }
+    return result;
+}
+
+int main() {
+    std::vector<uint64_t> primes;
+    const uint64_t limit = 2700;
+    primesieve::generate_primes(limit, &primes);
+    std::vector<std::future<std::vector<uint64_t>>> futures;
+    for (uint32_t base = 2; base <= 36; ++base) {
+        futures.push_back(std::async(repunit_primes, base, primes));
+    }
+    std::cout << "Repunit prime digits (up to " << limit << ") in:\n";
+    for (uint32_t base = 2, i = 0; base <= 36; ++base, ++i) {
+        std::cout << "Base " << std::setw(2) << base << ':';
+        for (auto digits : futures[i].get())
+            std::cout << ' ' << digits;
+        std::cout << '\n';
+    }
+}

Task/Repunit-primes/F-Sharp/repunit-primes.fs

@@ -0,0 +1,3 @@
+// Repunit primes. Nigel Galloway: January 24th., 2022
+let rUnitP(b:int)=let b=bigint b in primes32()|>Seq.takeWhile((>)1000)|>Seq.map(fun n->(n,((b**n)-1I)/(b-1I)))|>Seq.filter(fun(_,n)->Open.Numeric.Primes.MillerRabin.IsProbablePrime &n)|>Seq.map fst
+[2..16]|>List.iter(fun n->printf $"Base %d{n}: "; rUnitP(n)|>Seq.iter(printf "%d "); printfn "")

Task/Repunit-primes/Go/repunit-primes.go

@@ -0,0 +1,24 @@
+package main
+
+import (
+    "fmt"
+    big "github.com/ncw/gmp"
+    "rcu"
+    "strings"
+)
+
+func main() {
+    limit := 2700
+    primes := rcu.Primes(limit)
+    s := new(big.Int)
+    for b := 2; b <= 36; b++ {
+        var rPrimes []int
+        for _, p := range primes {
+            s.SetString(strings.Repeat("1", p), b)
+            if s.ProbablyPrime(15) {
+                rPrimes = append(rPrimes, p)
+            }
+        }
+        fmt.Printf("Base %2d: %v\n", b, rPrimes)
+    }
+}

Task/Repunit-primes/Julia/repunit-primes.julia

@@ -0,0 +1,7 @@
+using Primes
+
+repunitprimeinbase(n, base) = isprime(evalpoly(BigInt(base), [1 for _ in 1:n]))
+
+for b in 2:40
+    println(rpad("Base $b:", 9), filter(n -> repunitprimeinbase(n, b), 1:2700))
+end

Task/Repunit-primes/Mathematica/repunit-primes.math

@@ -0,0 +1,9 @@
+ClearAll[RepUnitPrimeQ]
+RepUnitPrimeQ[b_][n_] := PrimeQ[FromDigits[ConstantArray[1, n], b]]
+ClearAll[RepUnitPrimeQ]
+RepUnitPrimeQ[b_][n_] := PrimeQ[FromDigits[ConstantArray[1, n], b]]
+Do[
+ Print["Base ", b, ": ", Select[Range[2700], RepUnitPrimeQ[b]]]
+ ,
+ {b, 2, 16}
+]

Task/Repunit-primes/Nim/repunit-primes.nim

@@ -0,0 +1,14 @@
+import std/strformat
+import integers
+
+for base in 2..16:
+  stdout.write &"{base:>2}:"
+  var rep = ""
+  while true:
+    rep.add '1'
+    if rep.len > 2700: break
+    if not rep.len.isPrime: continue
+    let val = newInteger(rep, base)
+    if val.isPrime():
+      stdout.write ' ', rep.len
+  echo()

Task/Repunit-primes/Perl/repunit-primes.pl

@@ -0,0 +1,11 @@
+use strict;
+use warnings;
+use ntheory <is_prime fromdigits>;
+
+my $limit = 1000;
+
+print "Repunit prime digits (up to $limit) in:\n";
+
+for my $base (2..16) {
+    printf "Base %2d: %s\n", $base, join ' ', grep { is_prime $_ and is_prime fromdigits(('1'x$_), $base) and " $_" } 1..$limit
+}

Task/Repunit-primes/Phix/repunit-primes.phix

@@ -0,0 +1,32 @@
+(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: #008080;">procedure</span> <span style="color: #000000;">repunit</span><span style="color: #0000FF;">(</span><span style="color: #004080;">mpz</span> <span style="color: #000000;">z</span><span style="color: #0000FF;">,</span> <span style="color: #004080;">integer</span> <span style="color: #000000;">n</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">base</span><span style="color: #0000FF;">=</span><span style="color: #000000;">10</span><span style="color: #0000FF;">)</span>
+     <span style="color: #7060A8;">mpz_set_si</span><span style="color: #0000FF;">(</span><span style="color: #000000;">z</span><span style="color: #0000FF;">,</span><span style="color: #000000;">0</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;">1</span> <span style="color: #008080;">to</span> <span style="color: #000000;">n</span> <span style="color: #008080;">do</span>
+         <span style="color: #7060A8;">mpz_mul_si</span><span style="color: #0000FF;">(</span><span style="color: #000000;">z</span><span style="color: #0000FF;">,</span><span style="color: #000000;">z</span><span style="color: #0000FF;">,</span><span style="color: #000000;">base</span><span style="color: #0000FF;">)</span>
+         <span style="color: #7060A8;">mpz_add_si</span><span style="color: #0000FF;">(</span><span style="color: #000000;">z</span><span style="color: #0000FF;">,</span><span style="color: #000000;">z</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;">for</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;">constant</span> <span style="color: #0000FF;">{</span><span style="color: #000000;">limit</span><span style="color: #0000FF;">,</span><span style="color: #000000;">blimit</span><span style="color: #0000FF;">}</span> <span style="color: #0000FF;">=</span> <span style="color: #008080;">iff</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: #0000FF;">?{</span><span style="color: #000000;">400</span><span style="color: #0000FF;">,</span><span style="color: #000000;">16</span><span style="color: #0000FF;">}</span> <span style="color: #000080;font-style:italic;">-- 8.8s</span>
+                                            <span style="color: #0000FF;">:{</span><span style="color: #000000;">1000</span><span style="color: #0000FF;">,</span><span style="color: #000000;">16</span><span style="color: #0000FF;">})</span> <span style="color: #000080;font-style:italic;">-- 50.3s
+ --                                         :{1000,36}) -- 4 min 20s
+ --                                         :{2700,16}) -- 28 min 35s
+ --                                         :{2700,36}) -- &gt;patience</span>
+ <span style="color: #004080;">sequence</span> <span style="color: #000000;">primes</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">get_primes_le</span><span style="color: #0000FF;">(</span><span style="color: #000000;">limit</span><span style="color: #0000FF;">)</span>
+ <span style="color: #004080;">mpz</span> <span style="color: #000000;">z</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">mpz_init</span><span style="color: #0000FF;">()</span>
+ <span style="color: #008080;">for</span> <span style="color: #000000;">base</span><span style="color: #0000FF;">=</span><span style="color: #000000;">2</span> <span style="color: #008080;">to</span> <span style="color: #000000;">blimit</span> <span style="color: #008080;">do</span>
+     <span style="color: #004080;">sequence</span> <span style="color: #000000;">rprimes</span> <span style="color: #0000FF;">=</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;">1</span> <span style="color: #008080;">to</span> <span style="color: #7060A8;">length</span><span style="color: #0000FF;">(</span><span style="color: #000000;">primes</span><span style="color: #0000FF;">)</span> <span style="color: #008080;">do</span>
+         <span style="color: #004080;">integer</span> <span style="color: #000000;">p</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">primes</span><span style="color: #0000FF;">[</span><span style="color: #000000;">i</span><span style="color: #0000FF;">]</span>
+         <span style="color: #000000;">repunit</span><span style="color: #0000FF;">(</span><span style="color: #000000;">z</span><span style="color: #0000FF;">,</span><span style="color: #000000;">p</span><span style="color: #0000FF;">,</span><span style="color: #000000;">base</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;">z</span><span style="color: #0000FF;">)</span> <span style="color: #008080;">then</span>
+             <span style="color: #000000;">rprimes</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">append</span><span style="color: #0000FF;">(</span><span style="color: #000000;">rprimes</span><span style="color: #0000FF;">,</span><span style="color: #7060A8;">sprint</span><span style="color: #0000FF;">(</span><span style="color: #000000;">p</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: #7060A8;">printf</span><span style="color: #0000FF;">(</span><span style="color: #000000;">1</span><span style="color: #0000FF;">,</span><span style="color: #008000;">"Base %2d: %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: #7060A8;">join</span><span style="color: #0000FF;">(</span><span style="color: #000000;">rprimes</span><span style="color: #0000FF;">)})</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>
+<!--

Task/Repunit-primes/Python/repunit-primes.py

@@ -0,0 +1,3 @@
+from sympy import isprime
+for b in range(2, 17):
+    print(b, [n for n in range(2, 1001) if isprime(n) and isprime(int('1'*n, base=b))])

Task/Repunit-primes/Raku/repunit-primes.raku

@@ -0,0 +1,7 @@
+my $limit = 2700;
+
+say "Repunit prime digits (up to $limit) in:";
+
+.put for (2..16).hyper(:1batch).map: -> $base {
+    $base.fmt("Base %2d: ") ~ (1..$limit).grep(&is-prime).grep( (1 x *).parse-base($base).is-prime )
+}

Task/Repunit-primes/Ruby/repunit-primes.rb

@@ -0,0 +1,7 @@
+require 'prime'
+require 'gmp'
+
+(2..16).each do |base|
+  res = Prime.each(1000).select {|n| GMP::Z(("1" * n).to_i(base)).probab_prime? > 0}
+  puts "Base #{base}: #{res.join(" ")}"
+end

Task/Repunit-primes/Scheme/repunit-primes-1.ss

@@ -0,0 +1,42 @@
+; Test whether any integer is a probable prime.
+(define prime<probably>?
+  (lambda (n)
+    ; Fast modular exponentiation.
+    (define modexpt
+      (lambda (b e m)
+        (cond
+          ((zero? e) 1)
+          ((even? e) (modexpt (mod (* b b) m) (div e 2) m))
+          ((odd? e) (mod (* b (modexpt b (- e 1) m)) m)))))
+    ; Return multiple values s, d such that d is odd and 2^s * d = n.
+    (define split
+      (lambda (n)
+        (let recur ((s 0) (d n))
+          (if (odd? d)
+            (values s d)
+            (recur (+ s 1) (div d 2))))))
+    ; Test whether the number a proves that n is composite.
+    (define composite-witness?
+      (lambda (n a)
+        (let*-values (((s d) (split (- n 1)))
+                      ((x) (modexpt a d n)))
+          (and (not (= x 1))
+               (not (= x (- n 1)))
+               (let try ((r (- s 1)))
+                 (set! x (modexpt x 2 n))
+                 (or (zero? r)
+                     (= x 1)
+                     (and (not (= x (- n 1)))
+                          (try (- r 1)))))))))
+    ; Test whether n > 2 is a Miller-Rabin pseudoprime, k trials.
+    (define pseudoprime?
+      (lambda (n k)
+        (or (zero? k)
+            (let ((a (+ 2 (random (- n 2)))))
+              (and (not (composite-witness? n a))
+                   (pseudoprime? n (- k 1)))))))
+    ; Compute and return Probable Primality using the Miller-Rabin algorithm.
+    (and (> n 1)
+         (or (= n 2)
+             (and (odd? n)
+                  (pseudoprime? n 50))))))

Task/Repunit-primes/Scheme/repunit-primes-2.ss

@@ -0,0 +1,19 @@
+; Return list of the Repunit Primes in the given base up to the given limit.
+(define repunit_primes
+  (lambda (base limit)
+    (let loop ((count 2)
+               (value (1+ base)))
+      (cond ((> count limit)
+              '())
+            ((and (prime<probably>? count) (prime<probably>? value))
+              (cons count (loop (1+ count) (+ value (expt base count)))))
+            (else
+              (loop (1+ count) (+ value (expt base count))))))))
+
+; Show all the Repunit Primes up to 2700 digits for bases 2 through 16.
+(let ((max-base 16)
+      (max-digits 2700))
+  (printf "~%Repunit Primes up to ~d digits for bases 2 through ~d:~%" max-digits max-base)
+  (do ((base 2 (1+ base)))
+      ((> base max-base))
+    (printf "Base ~2d: ~a~%" base (repunit_primes base max-digits))))

Task/Repunit-primes/Sidef/repunit-primes.sidef

@@ -0,0 +1,8 @@
+var limit = 1000
+
+say "Repunit prime digits (up to #{limit}) in:"
+
+for n in (2..20) {
+    printf("Base %2d: %s\n", n,
+        {|k| is_prime((n**k - 1) / (n-1)) }.grep(1..limit))
+}

Task/Repunit-primes/Wren/repunit-primes.wren

@@ -0,0 +1,18 @@
+/* repunit_primes.wren */
+
+import "./gmp" for Mpz
+import "./math" for Int
+import "./fmt" for Fmt
+import "./str" for Str
+
+var limit = 2700
+var primes = Int.primeSieve(limit)
+
+for (b in 2..36) {
+    var rPrimes = []
+    for (p in primes) {
+        var s = Mpz.fromStr(Str.repeat("1", p), b)
+        if (s.probPrime(15) > 0) rPrimes.add(p)
+    }
+    Fmt.print("Base $2d: $n", b, rPrimes)
+}