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# Print all Proth primes up to N

• Last Updated : 20 May, 2021

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 Prime

Input: 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 ``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 two``bool` `isPowerOfTwo(``int` `n)``{``    ``return` `(n && !(n & (n - 1)));``}` `// Function to check if the Given``// number is Proth number or not``bool` `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 Code``int` `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 approach``import` `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 two``static` `boolean` `isPowerOfTwo(``int` `n)``{``    ``return` `(n > ``0` `&& (n & (n - ``1``)) == ``0``);``}` `// Function to check if the Given``// number is Proth number or not``static` `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 Code``public` `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 approach``import` `math as mt` `prime ``=` `[``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 two``def` `isPowerOfTwo(n):``    ``return` `(n ``and` `(n & (n ``-` `1``)) ``=``=` `False``)` `# Function to check if the Given``# number is Proth number or not``def` `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 Code``n ``=` `41` `# if number is proth number,``# calculate primes upto n``SieveOfEratosthenes(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 approach``using` `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 two``static` `Boolean isPowerOfTwo(``int` `n)``{``    ``return` `(n > 0 && (n & (n - 1)) == 0);``}` `// Function to check if the Given``// number is Proth number or not``static` `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 Code``static` `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

 ``

## Javascript

 ``
Output:
```3
5
13
17
41```

Time Complexity: O(n*log(log(n)))
References:

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