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Revision as of 00:00, 28 September 2018
Mersenne primes:
Challenge:
Create code that will list (preferably calculate) all of the Mersenne primes until some limitation is reached. For information on what a Mersenne prime is, go to this link: [[1]]
AppleScript
<lang> on isPrime(integ) set isComposite to "" if (integ / 2) = (integ / 2 div 1) then log integ & " is composite because 2 is a factor" as string --buttons {"OK", "Cancel"} default button 1 cancel button 2
else set x to 2 set sqrtOfInteg to integ ^ 0.5 repeat until x = integ ^ 0.5 + 1 as integer if (integ / x) = integ / x div 1 then log integ & " is composite because " & x & " & " & (integ / x div 1) & " are factors" as string --buttons {"OK", "Cancel"} default button 1 cancel button 2 set isComposite to 1 set x to x + 1 else
set x to x + 1 end if
end repeat log integ & " is prime" as string --buttons {"OK", "Cancel"} default button 1 cancel button 2 if isComposite = 1 then log integ & "is composite" else display dialog integ end if end if
end isPrime set x to 2 repeat isPrime(((2 ^ x) - 1) div 1) set x to x + 1 end repeat </lang>
C#
Needs a better primality checking algorithm to do really large prime numbers. <lang csharp>using System; using System.Numerics;
namespace MersennePrimes {
class Program {
static BigInteger Sqrt(BigInteger x) {
if (x < 0) throw new ArgumentException("Negative argument.");
if (x < 2) return x;
BigInteger y = x / 2;
while (y > x / y) {
y = ((x / y) + y) / 2;
}
return y;
}
static bool IsPrime(BigInteger bi) {
if (bi < 2) return false;
if (bi % 2 == 0) return bi == 2;
if (bi % 3 == 0) return bi == 3;
if (bi % 5 == 0) return bi == 5;
if (bi % 7 == 0) return bi == 7;
if (bi % 11 == 0) return bi == 11;
if (bi % 13 == 0) return bi == 13;
if (bi % 17 == 0) return bi == 17;
if (bi % 19 == 0) return bi == 19;
BigInteger limit = Sqrt(bi);
BigInteger test = 23;
while (test < limit) {
if (bi % test == 0) return false;
test += 2;
if (bi % test == 0) return false;
test += 4;
}
return true;
}
static void Main(string[] args) {
const int MAX = 9;
int pow = 2;
int count = 0;
while (true) {
if (IsPrime(pow)) {
BigInteger p = BigInteger.Pow(2, pow) - 1;
if (IsPrime(p)) {
Console.WriteLine("2 ^ {0} - 1", pow);
if (++count >= MAX) {
break;
}
}
}
pow++;
}
}
}
}</lang>
- Output:
2 ^ 2 - 1 2 ^ 3 - 1 2 ^ 5 - 1 2 ^ 7 - 1 2 ^ 13 - 1 2 ^ 17 - 1 2 ^ 19 - 1 2 ^ 31 - 1 2 ^ 61 - 1
D
Simplest thing that could possibly work. Using better primality tests will allow for more results to be calculated in a reasonable amount of time. <lang D>import std.bigint; import std.stdio;
bool isPrime(BigInt bi) {
if (bi < 2) return false;
if (bi % 2 == 0) return bi == 2;
if (bi % 3 == 0) return bi == 3;
auto test = BigInt(5);
while (test * test < bi) {
if (bi % test == 0) return false;
test += 2;
if (bi % test == 0) return false;
test += 4;
}
return true;
}
void main() {
auto base = BigInt(2);
for (int pow=1; pow<32; pow++) {
if (isPrime(base-1)) {
writeln("2 ^ ", pow, " - 1");
}
base *= 2;
}
}</lang>
- Output:
2 ^ 2 - 1 2 ^ 3 - 1 2 ^ 5 - 1 2 ^ 7 - 1 2 ^ 13 - 1 2 ^ 17 - 1 2 ^ 19 - 1 2 ^ 31 - 1
Go
The github.com/ncw/gmp package is a drop-in replacement for Go's math/big package.
It's a CGo wrapper around the C GMP library and under these circumstances is two to four times as fast as the native Go package.
Editing just the import line you can use whichever is more convenient for you
(CGo has drawbacks, including limited portability).
Normally build tags would be used to control this instead of editing imports in the source, but this keeps the example simpler.
Note that the use of ProbablyPrime(0) requires Go 1.8 or later. When using the math/big package, passing a parameter of zero to this method forces it to apply only the Baillie-PSW test to check for primality. This is 100% accurate for numbers up to 2^64 and at the time of writing (June 2018) no known composite number above that bound passes the test.
<lang go>package main
import ( "fmt" "time"
// Use one or the other of these: "math/big" //big "github.com/ncw/gmp" )
func main() { start := time.Now() one := big.NewInt(1) mp := big.NewInt(0) bp := big.NewInt(0) const max = 22 for count, p := 0, uint(2); count < max; { mp.Lsh(one, p) mp.Sub(mp, one) if mp.ProbablyPrime(0) { elapsed := time.Since(start).Seconds() if elapsed >= 0.01 { fmt.Printf("2 ^ %-4d - 1 took %6.2f secs\n", p, elapsed) } else { fmt.Printf("2 ^ %-4d - 1\n", p) } count++ } for { if p > 2 { p += 2 } else { p = 3 } bp.SetUint64(uint64(p)) if bp.ProbablyPrime(0) { break } } } }</lang>
- Output using the GMP package on a 3.4 GHz Xeon E3-1245:
2 ^ 2 - 1 2 ^ 3 - 1 2 ^ 5 - 1 2 ^ 7 - 1 2 ^ 13 - 1 2 ^ 17 - 1 2 ^ 19 - 1 2 ^ 31 - 1 2 ^ 61 - 1 2 ^ 89 - 1 2 ^ 107 - 1 2 ^ 127 - 1 2 ^ 521 - 1 2 ^ 607 - 1 2 ^ 1279 - 1 took 0.05 secs 2 ^ 2203 - 1 took 0.38 secs 2 ^ 2281 - 1 took 0.44 secs 2 ^ 3217 - 1 took 1.53 secs 2 ^ 4253 - 1 took 4.39 secs 2 ^ 4423 - 1 took 5.02 secs 2 ^ 9689 - 1 took 73.78 secs 2 ^ 9941 - 1 took 81.24 secs
(A previous run on more modest hardware - Celeron N3050 @ 1.60GHz × 2 - was ~365 seconds for M9941.)
This can be sped up quite a bit for modern multi-core CPUs by some simple changes to use goroutines. <lang Go>package main
import ( "fmt" "runtime" "time"
// Use one or the other of these: "math/big" //big "github.com/ncw/gmp" )
func main() { start := time.Now()
nworkers := runtime.GOMAXPROCS(0) fmt.Println("Using", nworkers, "workers.") workC := make(chan uint, 1) resultC := make(chan uint, nworkers)
// Generate possible Mersenne exponents and send them to workC. go func() { workC <- 2 bp := big.NewInt(0) for p := uint(3); ; p += 2 { // Possible exponents must be prime. bp.SetUint64(uint64(p)) if bp.ProbablyPrime(0) { workC <- p } } }()
// Start up worker go routines, each takes // possible Mersenne exponents from workC as `p` // and if 2^p-1 is prime sends `p` to resultC. one := big.NewInt(1) for i := 0; i < nworkers; i++ { go func() { mp := big.NewInt(0) for p := range workC { mp.Lsh(one, p) mp.Sub(mp, one) if mp.ProbablyPrime(0) { resultC <- p } } }() }
// Receive some maximum number of Mersenne prime exponents // from resultC and show the Mersenne primes. const max = 24 for count := 0; count < max; count++ { // Note: these could come back out of order, although usually // only the first few. If that is an issue, correcting it is // left as an excercise to the reader :). p := <-resultC elapsed := time.Since(start).Seconds() if elapsed >= 0.01 { fmt.Printf("2 ^ %-5d - 1 took %6.2f secs\n", p, elapsed) } else { fmt.Printf("2 ^ %-5d - 1\n", p) } } }</lang>
- Output using the GMP package on the same 3.4 GHz Xeon E3-1245 (4 core × 2 SMT threads) as above:
Using 8 workers. 2 ^ 2 - 1 2 ^ 5 - 1 2 ^ 3 - 1 2 ^ 7 - 1 2 ^ 13 - 1 2 ^ 19 - 1 2 ^ 61 - 1 2 ^ 31 - 1 2 ^ 107 - 1 2 ^ 17 - 1 2 ^ 127 - 1 2 ^ 89 - 1 2 ^ 521 - 1 2 ^ 607 - 1 2 ^ 1279 - 1 took 0.01 secs 2 ^ 2203 - 1 took 0.09 secs 2 ^ 2281 - 1 took 0.12 secs 2 ^ 3217 - 1 took 0.36 secs 2 ^ 4253 - 1 took 0.94 secs 2 ^ 4423 - 1 took 1.06 secs 2 ^ 9689 - 1 took 16.28 secs 2 ^ 9941 - 1 took 18.02 secs 2 ^ 11213 - 1 took 26.76 secs 2 ^ 19937 - 1 took 194.16 secs
Using this approach, the Celeron machine (dual core) takes ~180 seconds to reach M9941 and ~270 seconds to reach M11213.
Java
<lang Java>import java.math.BigInteger;
public class MersennePrimes {
private static final int MAX = 20;
private static final BigInteger ONE = BigInteger.ONE; private static final BigInteger TWO = BigInteger.valueOf(2);
private static boolean isPrime(int n) {
if (n < 2) return false;
if (n % 2 == 0) return n == 2;
if (n % 3 == 0) return n == 3;
int d = 5;
while (d * d <= n) {
if (n % d == 0) return false;
d += 2;
if (n % d == 0) return false;
d += 4;
}
return true;
}
public static void main(String[] args) {
int count = 0;
int p = 2;
while (true) {
BigInteger m = TWO.shiftLeft(p - 1).subtract(ONE);
if (m.isProbablePrime(10)) {
System.out.printf("2 ^ %d - 1\n", p);
if (++count == MAX) break;
}
// obtain next prime, p
do {
p = (p > 2) ? p + 2 : 3;
} while (!isPrime(p));
}
}
}</lang>
- Output:
2 ^ 2 - 1 2 ^ 3 - 1 2 ^ 5 - 1 2 ^ 7 - 1 2 ^ 13 - 1 2 ^ 17 - 1 2 ^ 19 - 1 2 ^ 31 - 1 2 ^ 61 - 1 2 ^ 89 - 1 2 ^ 107 - 1 2 ^ 127 - 1 2 ^ 521 - 1 2 ^ 607 - 1 2 ^ 1279 - 1 2 ^ 2203 - 1 2 ^ 2281 - 1 2 ^ 3217 - 1 2 ^ 4253 - 1 2 ^ 4423 - 1
Julia
Julia module Primes uses Miller-Rabin primality test.
<lang julia>using Primes
mersenne(n::Integer) = convert(typeof(n), 2) ^ n - one(n) function main(nmax::Integer)
n = ith = zero(nmax)
while ith ≤ nmax
if isprime(mersenne(n))
println("M$n")
ith += 1
end
n += 1
end
end
main(big(20))</lang>
- Output:
M2 M3 M5 M7 M13 M17 M19 M31 M61 M89 M107 M127 M521 M607 M1279 M2203 M2281 M3217 M4253 M4423 M9689
Kotlin
This task is similar to the Lucas-Lehmer test task except that you can use whatever method you like to test the primality of the Mersenne numbers. Here, I've chosen to use the JDK's BigInteger.isProbablePrime(certainty) method. The exact algorithm is implementation dependent --- GNU classpath uses only Miller-Rabin, while Oracle JDK uses Miller-Rabin and sometimes adds a Lucas test (this is not the Lucas-Lehmer test).
A 'certainty' parameter of 10 is enough to find the first 20 Mersenne primes but as even this takes about 90 seconds on my modest machine I've not bothered going beyond that. <lang scala>// version 1.2.10
import java.math.BigInteger
const val MAX = 20
val bigOne = BigInteger.ONE val bigTwo = 2.toBigInteger()
/* for checking 'small' primes */ fun isPrime(n: Int): Boolean {
if (n < 2) return false
if (n % 2 == 0) return n == 2
if (n % 3 == 0) return n == 3
var d : Int = 5
while (d * d <= n) {
if (n % d == 0) return false
d += 2
if (n % d == 0) return false
d += 4
}
return true
}
fun main(args: Array<String>) {
var count = 0
var p = 2
while (true) {
val m = (bigTwo shl (p - 1)) - bigOne
if (m.isProbablePrime(10)) {
println("2 ^ $p - 1")
if (++count == MAX) break
}
// obtain next prime, p
while(true) {
p = if (p > 2) p + 2 else 3
if (isPrime(p)) break
}
}
}</lang>
- Output:
2 ^ 2 - 1 2 ^ 3 - 1 2 ^ 5 - 1 2 ^ 7 - 1 2 ^ 13 - 1 2 ^ 17 - 1 2 ^ 19 - 1 2 ^ 31 - 1 2 ^ 61 - 1 2 ^ 89 - 1 2 ^ 107 - 1 2 ^ 127 - 1 2 ^ 521 - 1 2 ^ 607 - 1 2 ^ 1279 - 1 2 ^ 2203 - 1 2 ^ 2281 - 1 2 ^ 3217 - 1 2 ^ 4253 - 1 2 ^ 4423 - 1
PARI/GP
<lang parigp>LL(p)={
my(m=Mod(4,1<<p-1)); for(i=3,p,m=m^2-2); m==0
}; forprime(p=2,, if(LL(p), print("2^"p"-1")))</lang>
Perl
Since GIMPS went to the trouble of dedicating thousands of CPU years to finding Mersenne primes, we should be kind enough to use the results. The ntheory module front end does this, so the results up to 43 million is extremely fast (4 seconds), and we can reduce this another 10x by only checking primes. After the GIMPS double-checked mark, a Lucas-Lehmer test is done using code similar to Rosetta Code Lucas-Lehmer in C+GMP.
If this is too contrived, we can use Math::Prime::Util::GMP::is_mersenne_prime instead, which will run the Lucas-Lehmer test on each input. The first 23 Mersenne primes are found in under 15 seconds.
<lang perl>use ntheory qw/forprimes is_mersenne_prime/; forprimes { is_mersenne_prime($_) && say } 1e9;</lang>
- Output:
2 3 5 7 13 17 19 31 61 ...
Perl 6
We already have a multitude of tasks that demonstrate how to find Mersenne primes; Prime decomposition, Primality by trial division, Trial factoring of a Mersenne number, Lucas-Lehmer test, Miller–Rabin primality_test, etc. that all have Perl 6 entries. I'm not sure what I could add here that would be useful.
Hmmm.
Create code that will list all of the Mersenne primes until some limitation is reached.
It doesn't specify to calculate them, only to list them; why throw away all of the computer millenia of processing power that the GIMPS has invested?
<lang perl6>use HTTP::UserAgent; use Gumbo;
my $table = parse-html(HTTP::UserAgent.new.get('https://www.mersenne.org/primes/').content, :TAG
); say 'All known Mersenne primes as of ', Date(now); say 'M', ++$, ": 2$_ - 1" for $table[1]».[*][0][*].comb(/'exp_lo='\d+/)».subst(/\D/, ,:g) .trans([<0123456789>.comb] => [<⁰¹²³⁴⁵⁶⁷⁸⁹>.comb]).words; </lang>- Output:
All known Mersenne primes as of 2018-01-27 M1: 2² - 1 M2: 2³ - 1 M3: 2⁵ - 1 M4: 2⁷ - 1 M5: 2¹³ - 1 M6: 2¹⁷ - 1 M7: 2¹⁹ - 1 M8: 2³¹ - 1 M9: 2⁶¹ - 1 M10: 2⁸⁹ - 1 M11: 2¹⁰⁷ - 1 M12: 2¹²⁷ - 1 M13: 2⁵²¹ - 1 M14: 2⁶⁰⁷ - 1 M15: 2¹²⁷⁹ - 1 M16: 2²²⁰³ - 1 M17: 2²²⁸¹ - 1 M18: 2³²¹⁷ - 1 M19: 2⁴²⁵³ - 1 M20: 2⁴⁴²³ - 1 M21: 2⁹⁶⁸⁹ - 1 M22: 2⁹⁹⁴¹ - 1 M23: 2¹¹²¹³ - 1 M24: 2¹⁹⁹³⁷ - 1 M25: 2²¹⁷⁰¹ - 1 M26: 2²³²⁰⁹ - 1 M27: 2⁴⁴⁴⁹⁷ - 1 M28: 2⁸⁶²⁴³ - 1 M29: 2¹¹⁰⁵⁰³ - 1 M30: 2¹³²⁰⁴⁹ - 1 M31: 2²¹⁶⁰⁹¹ - 1 M32: 2⁷⁵⁶⁸³⁹ - 1 M33: 2⁸⁵⁹⁴³³ - 1 M34: 2¹²⁵⁷⁷⁸⁷ - 1 M35: 2¹³⁹⁸²⁶⁹ - 1 M36: 2²⁹⁷⁶²²¹ - 1 M37: 2³⁰²¹³⁷⁷ - 1 M38: 2⁶⁹⁷²⁵⁹³ - 1 M39: 2¹³⁴⁶⁶⁹¹⁷ - 1 M40: 2²⁰⁹⁹⁶⁰¹¹ - 1 M41: 2²⁴⁰³⁶⁵⁸³ - 1 M42: 2²⁵⁹⁶⁴⁹⁵¹ - 1 M43: 2³⁰⁴⁰²⁴⁵⁷ - 1 M44: 2³²⁵⁸²⁶⁵⁷ - 1 M45: 2³⁷¹⁵⁶⁶⁶⁷ - 1 M46: 2⁴²⁶⁴³⁸⁰¹ - 1 M47: 2⁴³¹¹²⁶⁰⁹ - 1 M48: 2⁵⁷⁸⁸⁵¹⁶¹ - 1 M49: 2⁷⁴²⁰⁷²⁸¹ - 1 M50: 2⁷⁷²³²⁹¹⁷ - 1
Ring
<lang ring>
- Project : Mersenne primes
n = 0 while true
n = n +1
if isprime(pow(2,n)-1) = 1
see n + nl
ok
end
func isprime num
if (num <= 1) return 0 ok
if (num % 2 = 0) and num != 2 return 0 ok
for i = 3 to floor(num / 2) -1 step 2
if (num % i = 0) return 0 ok
next
return 1
</lang> Output:
2 3 5 7 13 17 19
Scala
<lang Scala>object MersennePrimes extends App {
import Stream._
def primeSieve(s: Stream[Int]): Stream[Int] =
s.head #:: primeSieve(s.tail filter { _ % s.head != 0 })
val primes = primeSieve(from(2))
def mersenne(p: Int): BigInt = (BigInt(2) pow p) - 1
def s(mp: BigInt, p: Int): BigInt = { if (p == 1) 4 else ((s(mp, p - 1) pow 2) - 2) % mp }
val upbPrime = 9941
println(s"Finding Mersenne primes in M[2..$upbPrime]")
((primes takeWhile (_ <= upbPrime)).par map { p => (p, mersenne(p)) }
map { p => if (p._1 == 2) (p, 0) else (p, s(p._2, p._1 - 1)) } filter { _._2 == 0 })
.foreach { p =>
println(s"prime M${(p._1)._1}: " +
{ if ((p._1)._1 < 200) (p._1)._2 else s"(${(p._1)._2.toString.size} digits)" })
}
println("That's All Folks!")
}</lang>
Sidef
Uses the is_mersenne_prime() function from Math::Prime::Util::GMP. <lang ruby>for p in (^Inf -> lazy.grep { .is_mersenne_prime }) {
say "2^#{p} - 1"
}</lang>
- Output:
2^2 - 1 2^3 - 1 2^5 - 1 2^7 - 1 2^13 - 1 2^17 - 1 2^19 - 1 2^31 - 1 2^61 - 1 2^89 - 1 2^107 - 1 2^127 - 1 2^521 - 1 2^607 - 1 2^1279 - 1 2^2203 - 1 2^2281 - 1 2^3217 - 1 2^4253 - 1 2^4423 - 1 2^9689 - 1 2^9941 - 1 ^C sidef mersenne.sf 12.47s user 0.02s system 99% cpu 12.495 total
zkl
Uses libGMP (GNU MP Bignum Library) and its Miller-Rabin probabilistic primality testing algorithm. <lang zkl>var [const] BN=Import("zklBigNum"); // libGMP fcn mprimes{
n,m := BN(2),0;
foreach e in ([2..]){
n,m = n.shiftLeft(1), n-1;
if(m.probablyPrime()) println("2^%d - 1".fmt(e));
}
}() // gets rather slow after M(4423)</lang>
- Output:
2^2 - 1 2^3 - 1 2^5 - 1 2^7 - 1 2^13 - 1 2^17 - 1 2^19 - 1 2^31 - 1 2^61 - 1 2^89 - 1 2^107 - 1 2^127 - 1 2^521 - 1 2^607 - 1 2^1279 - 1 2^2203 - 1 2^2281 - 1 2^3217 - 1 2^4253 - 1 2^4423 - 1 2^9689 - 1 2^9941 - 1 2^11213 - 1 ...