PROGRAMA PER TROBAR TOTS ELS NOMBRES PRIMERS DE MERSENNE FINS A N

Prova-ho a GfG Practice ' title=

El primer de Mersenne és un nombre primer que és un menys que una potència de dos. En altres paraules, qualsevol nombre primer és primer de Mersenne si és de la forma 2^k-1 on k és un nombre enter més gran o igual que 2. Els primers primers primers de Mersenne són 3 7 31 i 127.
La tasca és imprimir tots els primers de Mersenne més petits que un enter positiu d'entrada n.
Exemples:

    Input:    10  
    Output:    3 7  
3 and 7 are prime numbers smaller than or  
equal to 10 and are of the form 2^k-1  
  
    Input:    100  
    Output:    3 7 31

La idea és generar tots els nombres primers inferiors o iguals al nombre donat n utilitzant Sedós d'Eratostenes . Un cop hem generat tots aquests nombres primers, iterem per tots els nombres de la forma 2^k-1 i comproveu si són primers o no.
A continuació es mostra la implementació de la idea.

C++

// Program to generate mersenne primes #include   using namespace std; // Generate all prime numbers less than n. void SieveOfEratosthenes(int n bool prime[]) {  // Initialize all entries of boolean array  // as true. A value in prime[i] will finally  // be false if i is Not a prime else true  // bool prime[n+1];  for (int i=0; i<=n; i++)  prime[i] = true;  for (int p=2; p*p<=n; p++)  {  // If prime[p] is not changed then it  // is a prime  if (prime[p] == true)  {  // Update all multiples of p  for (int i=p*2; i<=n; i += p)  prime[i] = false;  }  } } // Function to generate mersenne primes less // than or equal to n void mersennePrimes(int n) {  // Create a boolean array 'prime[0..n]'  bool prime[n+1];  // Generating primes using Sieve  SieveOfEratosthenes(nprime);  // Generate all numbers of the form 2^k - 1  // and smaller than or equal to n.  for (int k=2; ((1<<k)-1) <= n; k++)  {  long long num = (1<<k) - 1;  // Checking whether number is prime and is  // one less than the power of 2  if (prime[num])  cout << num << ' ';  } } // Driven program int main() {  int n = 31;  cout << 'Mersenne prime numbers smaller '  << 'than or equal to ' << n << endl;  mersennePrimes(n);  return 0; }

Java

// Program to generate // mersenne primes import java.io.*; class GFG {    // Generate all prime numbers  // less than n.  static void SieveOfEratosthenes(int n  boolean prime[])  {  // Initialize all entries of   // boolean array as true. A   // value in prime[i] will finally  // be false if i is Not a prime   // else true bool prime[n+1];  for (int i = 0; i <= n; i++)  prime[i] = true;    for (int p = 2; p * p <= n; p++)  {  // If prime[p] is not changed  //  then it is a prime  if (prime[p] == true)  {  // Update all multiples of p  for (int i = p * 2; i <= n; i += p)  prime[i] = false;  }  }  }    // Function to generate mersenne  // primes lessthan or equal to n  static void mersennePrimes(int n)  {  // Create a boolean array  // 'prime[0..n]'  boolean prime[]=new boolean[n + 1];    // Generating primes   // using Sieve  SieveOfEratosthenes(n prime);    // Generate all numbers of  // the form 2^k - 1 and   // smaller than or equal to n.  for (int k = 2; (( 1 << k) - 1) <= n; k++)  {  long num = ( 1 << k) - 1;    // Checking whether number is   // prime and is one less than  // the power of 2  if (prime[(int)(num)])  System.out.print(num + ' ');  }  }    // Driven program  public static void main(String args[])  {  int n = 31;  System.out.println('Mersenne prime'+  'numbers smaller than'+  'or equal to '+n);    mersennePrimes(n);  } } // This code is contributed by Nikita Tiwari.

Python3

# Program to generate mersenne primes  # Generate all prime numbers # less than n.  def SieveOfEratosthenes(n prime): # Initialize all entries of boolean # array as true. A value in prime[i]  # will finally be false if i is Not  # a prime else true bool prime[n+1]  for i in range(0 n + 1) : prime[i] = True p = 2 while(p * p <= n): # If prime[p] is not changed  # then it is a prime  if (prime[p] == True) : # Update all multiples of p  for i in range(p * 2 n + 1 p): prime[i] = False p += 1 # Function to generate mersenne  # primes less than or equal to n  def mersennePrimes(n) : # Create a boolean array  # 'prime[0..n]'  prime = [0] * (n + 1) # Generating primes using Sieve  SieveOfEratosthenes(n prime) # Generate all numbers of the  # form 2^k - 1 and smaller # than or equal to n. k = 2 while(((1 << k) - 1) <= n ): num = (1 << k) - 1 # Checking whether number  # is prime and is one # less than the power of 2  if (prime[num]) : print(num end = ' ' ) k += 1 # Driver Code n = 31 print('Mersenne prime numbers smaller' 'than or equal to '  n ) mersennePrimes(n) # This code is contributed # by Smitha

// C# Program to generate mersenne primes using System; class GFG {    // Generate all prime numbers less than n.  static void SieveOfEratosthenes(int n  bool []prime)  {    // Initialize all entries of   // boolean array as true. A   // value in prime[i] will finally  // be false if i is Not a prime   // else true bool prime[n+1];  for (int i = 0; i <= n; i++)  prime[i] = true;    for (int p = 2; p * p <= n; p++)  {    // If prime[p] is not changed  // then it is a prime  if (prime[p] == true)  {    // Update all multiples of p  for (int i = p * 2; i <= n; i += p)  prime[i] = false;  }  }  }    // Function to generate mersenne  // primes lessthan or equal to n  static void mersennePrimes(int n)  {    // Create a boolean array  // 'prime[0..n]'  bool []prime = new bool[n + 1];    // Generating primes   // using Sieve  SieveOfEratosthenes(n prime);    // Generate all numbers of  // the form 2^k - 1 and   // smaller than or equal to n.  for (int k = 2; (( 1 << k) - 1) <= n; k++)  {  long num = ( 1 << k) - 1;    // Checking whether number is   // prime and is one less than  // the power of 2  if (prime[(int)(num)])  Console.Write(num + ' ');  }  }    // Driven program  public static void Main()  {  int n = 31;    Console.WriteLine('Mersenne prime numbers'  + ' smaller than or equal to ' + n);    mersennePrimes(n);  } } // This code is contributed by nitin mittal.

JavaScript

<script> // JavaScript to generate // mersenne primes   // Generate all prime numbers  // less than n.  function SieveOfEratosthenes( n  prime)  {  // Initialize all entries of   // boolean array as true. A   // value in prime[i] will finally  // be false if i is Not a prime   // else true bool prime[n+1];  for (let i = 0; i <= n; i++)  prime[i] = true;    for (let p = 2; p * p <= n; p++)  {  // If prime[p] is not changed  //  then it is a prime  if (prime[p] == true)  {  // Update all multiples of p  for (let i = p * 2; i <= n; i += p)  prime[i] = false;  }  }  }    // Function to generate mersenne  // primes lessthan or equal to n  function mersennePrimes(n)  {  // Create a boolean array  // 'prime[0..n]'  let prime=[];    // Generating primes   // using Sieve  SieveOfEratosthenes(n prime);    // Generate all numbers of  // the form 2^k - 1 and   // smaller than or equal to n.  for (let k = 2; (( 1 << k) - 1) <= n; k++)  {  let num = ( 1 << k) - 1;    // Checking whether number is   // prime and is one less than  // the power of 2  if (prime[(num)])  document.write(num + ' ');  }  } // Driver Code  let n = 31;  document.write('Mersenne prime'+  'numbers smaller than'+  'or equal to '+n + '  
');    mersennePrimes(n); // This code is contributed by code_hunt. </script>

PHP

 // Program to generate mersenne primes // Generate all prime numbers less than n. function SieveOf($n) { // Initialize all entries of boolean  // array as true. A value in prime[i] // will finally be false if i is Not // a prime else true $prime = array($n + 1); for ($i = 0; $i <= $n; $i++) $prime[$i] = true; for ($p = 2; $p * $p <= $n; $p++) { // If prime[p] is not changed  // then it is a prime if ($prime[$p] == true) { // Update all multiples of p for ($i = $p * 2; $i <= $n; $i += $p) $prime[$i] = false; } } return $prime; } // Function to generate mersenne  // primes less than or equal to n function mersennePrimes($n) { // Create a boolean array 'prime[0..n]' // bool prime[n+1]; // Generating primes using Sieve $prime = SieveOf($n); // Generate all numbers of the  // form 2^k - 1 and smaller  // than or equal to n. for ($k = 2; ((1 << $k) - 1) <= $n; $k++) { $num = (1 << $k) - 1; // Checking whether number is prime  // and is one less than the power of 2 if ($prime[$num]) echo $num . ' '; } } // Driver Code $n = 31; echo 'Mersenne prime numbers smaller ' . 'than or equal to $n ' . mersennePrimes($n); // This code is contributed by mits ?>

Sortida:

Mersenne prime numbers smaller than or equal to 31  
3 7 31

Complexitat temporal: O (n*log (logn))

Complexitat espacial: O(N)

Referències:
https://en.wikipedia.org/wiki/Mersenne_prime

Crea un qüestionari

TechCodeview

Programa per trobar tots els nombres primers de Mersenne fins a N