Count of interesting primes upto N

Given a number N, the task is to find the number of interesting primes less than equal to N.

An interesting prime is any prime number which can be written as a² + b⁴, where a and b are positive integers. For e.g. The smallest interesting prime number is 2 = 1² + 1⁴.

Examples:

Input: N = 10
Output: 2
2 = 1² + 1⁴
5 = 2² + 1⁴
Both are interesting primes less than equal to 10

Input: N = 1000
Output: 28

Recommended: Please try your approach on {IDE} first, before moving on to the solution.

Naive Approach:

Iterate through all numbers from 1 to N.
For each number, check whether its prime or not.
If it is prime, then check whether it can be represented as a² + b⁴ by:
- Iterate through all possible values of b from 1 to N^1/4.
- For each value of b, check whether N – b⁴ is a perfect square or not (i.e it can be a² or not).

Below is the implementation of the above approach:

CPP

// C++ program to find the number 
// of interesting primes up to N 
  
#include <bits/stdc++.h> 
using namespace std; 
  
// Function to check if a number 
// is prime or not 
bool isPrime(int n) 
{ 
  
    int flag = 1; 
  
    // If n is divisible by any 
    // number between 2 and sqrt(n), 
    // it is not prime 
    for (int i = 2; i * i <= n; i++) { 
        if (n % i == 0) { 
            flag = 0; 
            break; 
        } 
    } 
  
    return (flag == 1 ? true : false); 
} 
  
// Function to check if a number 
// is perfet square or not 
bool isPerfectSquare(int x) 
{ 
    // Find floating point value of 
    // square root of x. 
    long double sr = sqrt(x); 
  
    // If square root is an integer 
    return ((sr - floor(sr)) == 0); 
} 
  
// Function to find the number of interesting 
// primes less than equal to N. 
int countInterestingPrimes(int n) 
{ 
  
    int answer = 0; 
    for (int i = 2; i <= n; i++) { 
  
        // Check whether the number 
        // is prime or not 
        if (isPrime(i)) { 
  
            // Iterate for values of b 
            for (int j = 1; 
                 j * j * j * j <= i; 
                 j++) { 
  
                // Check condition for a 
                if ( 
                    isPerfectSquare( 
                        i - j * j * j * j)) { 
                    answer++; 
                    break; 
                } 
            } 
        } 
    } 
  
    // Return the required answer 
    return answer; 
} 
  
// Driver code 
int main() 
{ 
    int N = 10; 
  
    cout << countInterestingPrimes(N); 
  
    return 0; 
} 

Java

// Java program to find the number 
// of interesting primes up to N 
class GFG{ 
   
// Function to check if a number 
// is prime or not 
static boolean isPrime(int n) 
{ 
   
    int flag = 1; 
   
    // If n is divisible by any 
    // number between 2 and Math.sqrt(n), 
    // it is not prime 
    for (int i = 2; i * i <= n; i++) { 
        if (n % i == 0) { 
            flag = 0; 
            break; 
        } 
    } 
   
    return (flag == 1 ? true : false); 
} 
   
// Function to check if a number 
// is perfet square or not 
static boolean isPerfectSquare(int x) 
{ 
    // Find floating point value of 
    // square root of x. 
    double sr = Math.sqrt(x); 
   
    // If square root is an integer 
    return ((sr - Math.floor(sr)) == 0); 
} 
   
// Function to find the number of interesting 
// primes less than equal to N. 
static int countInterestingPrimes(int n) 
{ 
   
    int answer = 0; 
    for (int i = 2; i <= n; i++) { 
   
        // Check whether the number 
        // is prime or not 
        if (isPrime(i)) { 
   
            // Iterate for values of b 
            for (int j = 1; 
                 j * j * j * j <= i; 
                 j++) { 
   
                // Check condition for a 
                if ( 
                    isPerfectSquare( 
                        i - j * j * j * j)) { 
                    answer++; 
                    break; 
                } 
            } 
        } 
    } 
   
    // Return the required answer 
    return answer; 
} 
   
// Driver code 
public static void main(String[] args) 
{ 
    int N = 10; 
   
    System.out.print(countInterestingPrimes(N)); 
} 
} 
  
// This code is contributed by Princi Singh 

Python3

# Python3 program to find the number 
# of interesting primes up to N 
import math 
  
# Function to check if a number 
# is prime or not 
def isPrime(n): 
  
    flag = 1
  
    # If n is divisible by any 
    # number between 2 and sqrt(n), 
    # it is not prime 
    i = 2
    while(i * i <= n): 
        if (n % i == 0): 
            flag = 0
            break
        i += 1
      
    return (True if flag == 1 else False) 
  
# Function to check if a number 
# is perfet square or not 
def isPerfectSquare(x): 
  
    # Find floating povalue of 
    # square root of x. 
    sr = math.sqrt(x) 
  
    # If square root is an integer 
    return ((sr - math.floor(sr)) == 0) 
  
# Function to find the number of interesting 
# primes less than equal to N. 
def countInterestingPrimes(n): 
  
    answer = 0
    for i in range(2, n): 
  
        # Check whether the number 
        # is prime or not 
        if (isPrime(i) != None) : 
  
            # Iterate for values of b 
            j = 1
            while(j * j * j * j <= i): 
  
                # Check condition for a 
                if (isPerfectSquare(i - j * j * j * j)) : 
                    answer += 1
                    break
                j += 1
  
    # Return the required answer 
    return answer 
  
# Driver code 
if __name__=='__main__':  
  
    N = 10
  
    print(countInterestingPrimes(N)) 
  
# This code is contributed by AbhiThakur 

C#

// C# program to find the number 
// of interesting primes up to N 
using System; 
using System.Collections.Generic; 
  
class GFG{ 
    
// Function to check if a number 
// is prime or not 
static bool isPrime(int n) 
{ 
    
    int flag = 1; 
    
    // If n is divisible by any 
    // number between 2 and Math.Sqrt(n), 
    // it is not prime 
    for (int i = 2; i * i <= n; i++) { 
        if (n % i == 0) { 
            flag = 0; 
            break; 
        } 
    } 
    
    return (flag == 1 ? true : false); 
} 
    
// Function to check if a number 
// is perfet square or not 
static bool isPerfectSquare(int x) 
{ 
    // Find floating point value of 
    // square root of x. 
    double sr = Math.Sqrt(x); 
    
    // If square root is an integer 
    return ((sr - Math.Floor(sr)) == 0); 
} 
    
// Function to find the number of interesting 
// primes less than equal to N. 
static int countInterestingPrimes(int n) 
{ 
    
    int answer = 0; 
    for (int i = 2; i <= n; i++) { 
    
        // Check whether the number 
        // is prime or not 
        if (isPrime(i)) { 
    
            // Iterate for values of b 
            for (int j = 1; 
                 j * j * j * j <= i; 
                 j++) { 
    
                // Check condition for a 
                if ( 
                    isPerfectSquare( 
                        i - j * j * j * j)) { 
                    answer++; 
                    break; 
                } 
            } 
        } 
    } 
    
    // Return the required answer 
    return answer; 
} 
    
// Driver code 
public static void Main(String[] args) 
{ 
    int N = 10; 
    
    Console.Write(countInterestingPrimes(N)); 
} 
} 
  
// This code is contributed by Rajput-Ji 

Output:

Time Complexity: O(N)

Efficient Approach:

If we store all perfect squares and perfect quadruples up to N, then we can iterate through all the pairs and check whether the result is prime or not.
To further optimise we can store all primes till N using sieve of eratosthenes and do the primality check in O(1).

Below is the implementation of the above approach:

C++

// C++ program to find the number 
// of interesting primes up to N. 
  
#include <bits/stdc++.h> 
using namespace std; 
  
// Function to find all prime numbers 
void SieveOfEratosthenes( 
    int n, 
    unordered_set<int>& allPrimes) 
{ 
    // Create a boolean array "prime[0..n]" 
    // and initialize all entries as true. 
    // A value in prime[i] will finally 
    // be false if i is Not a prime. 
    bool prime[n + 1]; 
    memset(prime, true, sizeof(prime)); 
  
    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 
            for (int i = p * p; i <= n; i += p) 
                prime[i] = false; 
        } 
    } 
  
    // Store all prime numbers 
    for (int p = 2; p <= n; p++) 
        if (prime[p]) 
            allPrimes.insert(p); 
} 
  
// Function to check if a number 
// is perfet square or not 
int countInterestingPrimes(int n) 
{ 
    // To store all primes 
    unordered_set<int> allPrimes; 
  
    SieveOfEratosthenes(n, allPrimes); 
  
    // To store all interseting primes 
    unordered_set<int> intersetingPrimes; 
  
    vector<int> squares, quadruples; 
  
    // Store all perfect squares 
    for (int i = 1; i * i <= n; i++) { 
        squares.push_back(i * i); 
    } 
  
    // Store all perfect quadruples 
    for (int i = 1; i * i * i * i <= n; i++) { 
        quadruples.push_back(i * i * i * i); 
    } 
  
    // Store all interseting primes 
    for (auto a : squares) { 
        for (auto b : quadruples) { 
            if (allPrimes.count(a + b)) 
                intersetingPrimes.insert(a + b); 
        } 
    } 
  
    // Return count of interseting primes 
    return intersetingPrimes.size(); 
} 
  
// Driver code 
int main() 
{ 
    int N = 10; 
  
    cout << countInterestingPrimes(N); 
  
    return 0; 
} 

Java

// Java program to find the number 
// of interesting primes up to N. 
import java.util.*; 
  
class GFG{ 
   
// Function to find all prime numbers 
static void SieveOfEratosthenes( 
    int n, HashSet<Integer> allPrimes) 
{ 
    // Create a boolean array "prime[0..n]" 
    // and initialize all entries as true. 
    // A value in prime[i] will finally 
    // be false if i is Not a prime. 
    boolean []prime = new boolean[n + 1]; 
    Arrays.fill(prime, 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 
            // greater than or equal to 
            // the square of it 
            for (int i = p * p; i <= n; i += p) 
                prime[i] = false; 
        } 
    } 
   
    // Store all prime numbers 
    for (int p = 2; p <= n; p++) 
        if (prime[p]) 
            allPrimes.add(p); 
} 
   
// Function to check if a number 
// is perfet square or not 
static int countInterestingPrimes(int n) 
{ 
    // To store all primes 
    HashSet<Integer> allPrimes = new HashSet<Integer>(); 
   
    SieveOfEratosthenes(n, allPrimes); 
   
    // To store all interseting primes 
    HashSet<Integer> intersetingPrimes = new HashSet<Integer>(); 
   
    Vector<Integer> squares = new Vector<Integer>() 
            , quadruples = new Vector<Integer>(); 
   
    // Store all perfect squares 
    for (int i = 1; i * i <= n; i++) { 
        squares.add(i * i); 
    } 
   
    // Store all perfect quadruples 
    for (int i = 1; i * i * i * i <= n; i++) { 
        quadruples.add(i * i * i * i); 
    } 
   
    // Store all interseting primes 
    for (int a : squares) { 
        for (int b : quadruples) { 
            if (allPrimes.contains(a + b)) 
                intersetingPrimes.add(a + b); 
        } 
    } 
   
    // Return count of interseting primes 
    return intersetingPrimes.size(); 
} 
   
// Driver code 
public static void main(String[] args) 
{ 
    int N = 10; 
   
    System.out.print(countInterestingPrimes(N)); 
} 
} 
  
// This code is contributed by 29AjayKumar 

C#

// C# program to find the number 
// of interesting primes up to N. 
using System; 
using System.Collections.Generic; 
  
class GFG{ 
    
// Function to find all prime numbers 
static void SieveOfEratosthenes( 
    int n, HashSet<int> allPrimes) 
{ 
    // Create a bool array "prime[0..n]" 
    // and initialize all entries as true. 
    // A value in prime[i] will finally 
    // be false if i is Not a prime. 
    bool []prime = new bool[n + 1]; 
    for(int i = 0; i < n + 1; 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 
            // greater than or equal to 
            // the square of it 
            for (int i = p * p; i <= n; i += p) 
                prime[i] = false; 
        } 
    } 
    
    // Store all prime numbers 
    for (int p = 2; p <= n; p++) 
        if (prime[p]) 
            allPrimes.Add(p); 
} 
    
// Function to check if a number 
// is perfet square or not 
static int countInterestingPrimes(int n) 
{ 
    // To store all primes 
    HashSet<int> allPrimes = new HashSet<int>(); 
    
    SieveOfEratosthenes(n, allPrimes); 
    
    // To store all interseting primes 
    HashSet<int> intersetingPrimes = new HashSet<int>(); 
    
    List<int> squares = new List<int>() 
            , quadruples = new List<int>(); 
    
    // Store all perfect squares 
    for (int i = 1; i * i <= n; i++) { 
        squares.Add(i * i); 
    } 
    
    // Store all perfect quadruples 
    for (int i = 1; i * i * i * i <= n; i++) { 
        quadruples.Add(i * i * i * i); 
    } 
    
    // Store all interseting primes 
    foreach (int a in squares) { 
        foreach (int b in quadruples) { 
            if (allPrimes.Contains(a + b)) 
                intersetingPrimes.Add(a + b); 
        } 
    } 
    
    // Return count of interseting primes 
    return intersetingPrimes.Count; 
} 
    
// Driver code 
public static void Main(String[] args) 
{ 
    int N = 10; 
    
    Console.Write(countInterestingPrimes(N)); 
} 
} 
   
// This code is contributed by Rajput-Ji 

Output:

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My Personal Notes

Recommended: Please try your approach on {IDE} first, before moving on to the solution.

CPP

Java

Python3

C#

C++

Java

C#

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