Given a number N, the task is to check whether the given number is Proth Prime or not.
A Proth prime is a Proth Number which is prime.
The first few Proth primes are –
3, 5, 13, 17, 41, 97, 113, 193, 241, 257, 353, 449, 577, 641, 673, 769, 929, 1153, 1217, …..
Examples:
Input: 41
Output: 41 is Proth PrimeInput: 19
Output: 19 is not a Proth Prime
Approach:
The idea is to find primes upto N using Sieve of Eratosthenes. Then check whether the given number is Proth Number or not. If number is a Proth Number and is also a prime number, then given number is Proth Prime.
Below is the implementation of the above algorithm:
C++
// C++ implementation of the above approach#include <bits/stdc++.h>using namespace std;int prime[1000000];// Calculate all primes upto n.void SieveOfEratosthenes(int n){ // Initialize all entries it as true. // A value in prime[i] will finally // false if i is Not a prime, else true. for (int i = 1; i <= n + 1; i++) prime[i] = true; prime[1] = false; 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 // greater than or equal to // the square of it numbers // which are multiple of p and are // less than p^2 are already been marked. for (int i = p * p; i <= n; i += p) prime[i] = false; } }}// Utility function to check power of twobool isPowerOfTwo(int n){ return (n && !(n & (n - 1)));}// Function to check if the Given// number is Proth number or notbool isProthNumber(int n){ int k = 1; while (k < (n / k)) { // check if k divides n or not if (n % k == 0) { // Check if n/k is power of 2 or not if (isPowerOfTwo(n / k)) return true; } // update k to next odd number k = k + 2; } // If we reach here means there // exists no value of K such // that k is odd number and n/k // is a power of 2 greater than k return false;}// Function to check whether the given// number is Proth Prime or Not.bool isProthPrime(int n){ // Check n for Proth Number if (isProthNumber(n - 1)) { // if number is prime, return true if (prime[n]) return true; else return false; } else return false;}// Driver Codeint main(){ int n = 41; // if number is proth number, // calculate primes upto n SieveOfEratosthenes(n); for (int i = 1; i <= n; i++) // Check n for Proth Prime if (isProthPrime(i)) cout << i << endl; return 0;} |
Java
// Java implementation of the above approachimport java.util.*;class GFG{static boolean[] prime = new boolean[1000000];// Calculate all primes upto n.static void SieveOfEratosthenes(int n){ // Initialize all entries it as true. // A value in prime[i] will finally // false if i is Not a prime, else true. for (int i = 1; i <= n + 1; i++) prime[i] = true; prime[1] = false; 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 // greater than or equal to // the square of it numbers // which are multiple of p and are // less than p^2 are already been marked. for (int i = p * p; i <= n; i += p) prime[i] = false; } }}// Utility function to check power of twostatic boolean isPowerOfTwo(int n){ return (n > 0 && (n & (n - 1)) == 0);}// Function to check if the Given// number is Proth number or notstatic boolean isProthNumber(int n){ int k = 1; while (k < (int)(n / k)) { // check if k divides n or not if (n % k == 0) { // Check if n/k is power of 2 or not if (isPowerOfTwo((int)(n / k))) return true; } // update k to next odd number k = k + 2; } // If we reach here means there // exists no value of K such // that k is odd number and n/k // is a power of 2 greater than k return false;}// Function to check whether the given// number is Proth Prime or Not.static boolean isProthPrime(int n){ // Check n for Proth Number if (isProthNumber(n - 1)) { // if number is prime, return true if (prime[n]) return true; else return false; } else return false;}// Driver Codepublic static void main(String args[]){ int n = 41; // if number is proth number, // calculate primes upto n SieveOfEratosthenes(n); for (int i = 1; i <= n; i++) // Check n for Proth Prime if (isProthPrime(i)) System.out.println(i);}}// This code is contributed by// Surendra_Gangwar |
Python3
# Python3 implementation of the# above approachimport math as mtprime = [0 for i in range(1000000)]# Calculate all primes upto n.def SieveOfEratosthenes(n): # Initialize all entries it as true. # A value in prime[i] will finally # false if i is Not a prime, else true. for i in range(1, n + 2): prime[i] = True prime[1] = False for p in range(2, mt.ceil(n**(0.5))): # If prime[p] is not changed, # then it is a prime if (prime[p] == True): # Update all multiples of p # greater than or equal to # the square of it numbers # which are multiple of p and are # less than p^2 are already been marked. for i in range(p * p, n + 1, p): prime[i] = False# Utility function to check power of twodef isPowerOfTwo(n): return (n and (n & (n - 1)) == False)# Function to check if the Given# number is Proth number or notdef isProthNumber(n): k = 1 while (k < (n // k)): # check if k divides n or not if (n % k == 0): # Check if n/k is power of 2 or not if (isPowerOfTwo(n // k)): return True # update k to next odd number k = k + 2 # If we reach here means there # exists no value of K such # that k is odd number and n/k # is a power of 2 greater than k return False# Function to check whether the given# number is Proth Prime or Not.def isProthPrime(n): # Check n for Proth Number if (isProthNumber(n - 1)): # if number is prime, return true if (prime[n]): return True else: return False else: return False# Driver Coden = 41# if number is proth number,# calculate primes upto nSieveOfEratosthenes(n)for i in range(1, n + 1): # Check n for Proth Prime if isProthPrime(i) == True: print(i) # This code is contributed by # Mohit kumar 29 |
C#
// C# implementation of the above approachusing System;class GFG{static Boolean[] prime = new Boolean[1000000];// Calculate all primes upto n.static void SieveOfEratosthenes(int n){ // Initialize all entries it as true. // A value in prime[i] will finally // false if i is Not a prime, else true. for (int i = 1; i <= n + 1; i++) prime[i] = true; prime[1] = false; 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 // greater than or equal to // the square of it numbers // which are multiple of p and are // less than p^2 are already been marked. for (int i = p * p; i <= n; i += p) prime[i] = false; } }}// Utility function to check power of twostatic Boolean isPowerOfTwo(int n){ return (n > 0 && (n & (n - 1)) == 0);}// Function to check if the Given// number is Proth number or notstatic Boolean isProthNumber(int n){ int k = 1; while (k < (int)(n / k)) { // check if k divides n or not if (n % k == 0) { // Check if n/k is power of 2 or not if (isPowerOfTwo((int)(n / k))) return true; } // update k to next odd number k = k + 2; } // If we reach here means there // exists no value of K such // that k is odd number and n/k // is a power of 2 greater than k return false;}// Function to check whether the given// number is Proth Prime or Not.static Boolean isProthPrime(int n){ // Check n for Proth Number if (isProthNumber(n - 1)) { // if number is prime, return true if (prime[n]) return true; else return false; } else return false;}// Driver Codestatic public void Main(String []args){ int n = 41; // if number is proth number, // calculate primes upto n SieveOfEratosthenes(n); for (int i = 1; i <= n; i++) // Check n for Proth Prime if (isProthPrime(i)) Console.WriteLine(i);}}// This code is contributed by Arnab Kundu |
PHP
<?php// PHP implementation of the above approach $GLOBALS['prime'] = array();// Calculate all primes upto n. function SieveOfEratosthenes($n) { // Initialize all entries it as true. // A value in prime[i] will finally // false if i is Not a prime, else true. for ($i = 1; $i <= $n + 1; $i++) $GLOBALS['prime'][$i] = true; $GLOBALS['prime'][1] = false; for ($p = 2; $p * $p <= $n; $p++) { // If prime[p] is not changed, // then it is a prime if ($GLOBALS['prime'][$p] == true) { // Update all multiples of p greater // than or equal to the square of it // numbers which are multiple of p and are // less than p^2 are already been marked. for ($i = $p * $p; $i <= $n; $i += $p) $GLOBALS['prime'][$i] = false; } } } // Utility function to check power of two function isPowerOfTwo($n) { return ($n && !($n & ($n - 1))); } // Function to check if the Given // number is Proth number or not function isProthNumber($n) { $k = 1; while ($k < ($n / $k)) { // check if k divides n or not if ($n % $k == 0) { // Check if n/k is power of 2 or not if (isPowerOfTwo($n / $k)) return true; } // update k to next odd number $k = $k + 2; } // If we reach here means there // exists no value of K such // that k is odd number and n/k // is a power of 2 greater than k return false; } // Function to check whether the given // number is Proth Prime or Not. function isProthPrime($n) { // Check n for Proth Number if (isProthNumber($n - 1)) { // if number is prime, return true if ($GLOBALS['prime'][$n]) return true; else return false; } else return false; } // Driver Code $n = 41; // if number is proth number, // calculate primes upto n SieveOfEratosthenes($n); for ($i = 1; $i <= $n; $i++) // Check n for Proth Prime if (isProthPrime($i) == true) echo $i, "\n";// This code is contributed by Ryuga?> |
Javascript
<script>// Javascript implementation of the above approachlet prime = new Array();// Calculate all primes upto n.function SieveOfEratosthenes(n){ // Initialize all entries it as true. // A value in prime[i] will finally // false if i is Not a prime, else true. for (let i = 1; i <= n + 1; i++) prime[i] = true; prime[1] = false; 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 greater // than or equal to the square of it // numbers which are multiple of p and are // less than p^2 are already been marked. for (let i = p * p; i <= n; i += p) prime[i] = false; } }}// Utility function to check power of twofunction isPowerOfTwo(n){ return (n && !(n & (n - 1)));}// Function to check if the Given// number is Proth number or notfunction isProthNumber(n){ let k = 1; while (k < (n / k)) { // check if k divides n or not if (n % k == 0) { // Check if n/k is power of 2 or not if (isPowerOfTwo(n / k)) return true; } // update k to next odd number k = k + 2; } // If we reach here means there // exists no value of K such // that k is odd number and n/k // is a power of 2 greater than k return false;}// Function to check whether the given// number is Proth Prime or Not.function isProthPrime(n){ // Check n for Proth Number if (isProthNumber(n - 1)) { // if number is prime, return true if (prime[n]) return true; else return false; } else return false;}// Driver Codelet n = 41;// if number is proth number,// calculate primes upto nSieveOfEratosthenes(n);for (let i = 1; i <= n; i++) // Check n for Proth Prime if (isProthPrime(i) == true) document.write(i + "<br>");// This code is contributed by gfgking</script> |
Output:
3 5 13 17 41
Time Complexity: O(n*log(log(n)))
Auxiliary Space: O(1), constant extra space is required as the size of the prime array is constant.
References:
- https://en.wikipedia.org/wiki/Proth_number#Proth_primes
- https://www.neveropen.co.uk/program-to-check-whether-a-number-is-proth-number-or-not/
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