Count of interesting primes upto N

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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

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:

C++

// 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 perfect 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 perfect 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 perfect 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)):
 
            # 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 perfect 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

Javascript

<script>
// Java  script program to find the number
// of interesting primes up to N
 
// Function to check if a number
// is prime or not
function isPrime( n)
{
 
    let flag = 1;
 
    // If n is divisible by any
    // number between 2 and Math.sqrt(n),
    // it is not prime
    for (let 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 perfect square or not
function isPerfectSquare( x)
{
    // Find floating point value of
    // square root of x.
    let 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.
function countInterestingPrimes( n)
{
 
    let answer = 0;
    for (let i = 2; i <= n; i++) {
 
        // Check whether the number
        // is prime or not
        if (isPrime(i)) {
 
            // Iterate for values of b
            for (let 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
 
    let N = 10;
 
    document.write(countInterestingPrimes(N));
 
 
 
// This code is contributed by Bobby
</script>

Output:

Time Complexity: O(N)

Auxiliary Space: O(1)

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 perfect 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 perfect 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

Python3

# Python3 program to find the number 
# of interesting primes up to N. 
 
# Function to find all prime numbers 
def SieveOfEratosthenes(n, 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. 
    prime = [True] * (n + 1) 
     
    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 
            # greater than or equal to 
            # the square of it 
            for i in range(p * p, n + 1, p): 
                prime[i] = False
        p += 1
     
    # Store all prime numbers 
    for p in range(2, n + 1): 
        if prime[p]: 
            allPrimes.add(p) 
 
# Function to check if a number 
# is perfect square or not 
def countInterestingPrimes(n): 
     
    # To store all primes 
    allPrimes = set() 
     
    # To store all interseting primes 
    SieveOfEratosthenes(n, allPrimes) 
     
    # To store all interseting primes 
    interestingPrimes = set() 
     
    squares, quadruples = [], [] 
     
    # Store all perfect squares 
    i = 1
    while i * i <= n: 
        squares.append(i * i) 
        i += 1
     
    # Store all perfect quadruples 
    i = 1
    while i * i * i * i <= n: 
        quadruples.append(i * i * i * i) 
        i += 1
     
    # Store all interseting primes 
    for a in squares: 
        for b in quadruples: 
            if a + b in allPrimes: 
                interestingPrimes.add(a + b) 
                 
    # Return count of interseting primes 
    return len(interestingPrimes) 
 
# Driver code 
N = 10
print(countInterestingPrimes(N)) 
 
# This code is contributed by Shivam Singh 

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 perfect 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

Javascript

// Function to find all prime numbers
function SieveOfEratosthenes(n, 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.
  let prime = new Array(n + 1).fill(true);
 
  for (let p = 2; p * p <= n; p++) 
  {
   
    // If prime[p] is not changed,
    // then it is a prime
    if (prime[p])
    {
     
      // Update all multiples of p
      // greater than or equal to
      // the square of it
      for (let i = p * p; i <= n; i += p)
        prime[i] = false;
    }
  }
 
  // Store all prime numbers
  for (let p = 2; p <= n; p++)
    if (prime[p])
      allPrimes.add(p);
}
 
// Function to check if a number
// is perfect square or not
function countInterestingPrimes(n)
{
 
  // To store all primes
  let allPrimes = new Set();
 
  SieveOfEratosthenes(n, allPrimes);
 
  // To store all interseting primes
  let intersetingPrimes = new Set();
 
  let squares = [];
  let quadruples = [];
 
  // Store all perfect squares
  for (let i = 1; i * i <= n; i++) {
    squares.push(i * i);
  }
 
  // Store all perfect quadruples
  for (let i = 1; i * i * i * i <= n; i++) {
    quadruples.push(i * i * i * i);
  }
 
  // Store all interseting primes
  for (let a of squares) {
    for (let b of quadruples) {
      if (allPrimes.has(a + b))
        intersetingPrimes.add(a + b);
    }
  }
 
  // Return count of interseting primes
  return intersetingPrimes.size;
}
 
// Driver code
let N = 10;
console.log(countInterestingPrimes(N));

Output:

Time Complexity: O(N)

Auxiliary Space: O(N)

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