Count of primes in a given range that can be expressed as sum of perfect squares

Last Updated : 15 Jul, 2025

Given two integers L and R, the task is to find the number of prime numbers in the range [L, R] that can be represented by the sum of two squares of two numbers.
Examples:

Input: L = 1, R = 5
Output: 1
Explanation:
Only prime number that can be expressed as sum of two perfect squares in the given range is 5 (2² + 1²)
Input: L = 7, R = 42
Output: 5
Explanation:
The prime numbers in the given range that can be expressed as sum of two perfect squares are:
13 = 2² + 3²
17 = 1² + 4²
29 = 5² + 2²
37 = 1² + 6²
41 = 5² + 4²

Approach:
The given problem can be solved using Fermat's Little theorem, which states that a prime number p can be expressed as the sum of two squares if p satisfies the following equation:

(p - 1) % 4 == 0

Follow the steps below to solve the problem:

Traverse the range [L, R].
For every number, check if it is a prime number of not.
If found to be so, check if the prime number is of the form 4K + 1. If sp, increase count.
After traversing the complete range, print count.

Below is the implementation of the above approach:

C++

// C++ Program to implement
// the above approach

#include <bits/stdc++.h>
using namespace std;

// Function to check if a prime number
// satisfies the condition to be
// expressed as sum of two perfect squares
bool sumSquare(int p)
{
    return (p - 1) % 4 == 0;
}

// Function to check if a
// number is prime or not
bool isPrime(int n)
{
    // Corner cases
    if (n <= 1)
        return false;

    if (n <= 3)
        return true;

    if (n % 2 == 0 || n % 3 == 0)
        return false;

    for (int i = 5; i * i <= n; i = i + 6)
        if (n % i == 0 || n % (i + 2) == 0)
            return false;

    return true;
}

// Function to return the count of primes
// in the range which can be expressed as
// the sum of two squares
int countOfPrimes(int L, int R)
{
    int count = 0;
    for (int i = L; i <= R; i++) {

        // If i is a prime
        if (isPrime(i)) {

            // If i can be expressed
            // as the sum of two squares
            if (sumSquare(i))
                count++;
        }
    }

    // Return the count
    return count;
}

// Driver Code
int main()
{
    int L = 5, R = 41;
    cout << countOfPrimes(L, R);
}

Java

// Java program to implement
// the above approach
import java.util.*;

class GFG{

// Function to check if a prime number
// satisfies the condition to be 
// expressed as sum of two perfect
// squares
static boolean sumSquare(int p)
{
    return (p - 1) % 4 == 0;
}

// Function to check if a
// number is prime or not
static boolean isPrime(int n)
{
    
    // Corner cases
    if (n <= 1)
        return false;

    if (n <= 3)
        return true;

    if (n % 2 == 0 || n % 3 == 0)
        return false;

    for(int i = 5; i * i <= n; i = i + 6)
        if (n % i == 0 || n % (i + 2) == 0)
            return false;

    return true;
}

// Function to return the count of primes
// in the range which can be expressed as
// the sum of two squares
static int countOfPrimes(int L, int R)
{
    int count = 0;
    for(int i = L; i <= R; i++) 
    {

        // If i is a prime
        if (isPrime(i)) 
        {

            // If i can be expressed
            // as the sum of two squares
            if (sumSquare(i))
                count++;
        }
    }
    
    // Return the count
    return count;
}

// Driver code
public static void main(String[] args)
{
    int L = 5, R = 41;

    System.out.println(countOfPrimes(L, R));
}
}

// This code is contributed by offbeat

Python3

# Python3 program for the
# above approach

# Function to check if a prime number
# satisfies the condition to be 
# expressed as sum of two perfect
# squares 
def sumsquare(p):
  
  return (p - 1) % 4 == 0

# Function to check if a 
# number is prime or not 
def isprime(n):
  
  # Corner cases 
  if n <= 1:
    return False
  if n <= 3:
    return True
  if (n % 2 == 0) or (n % 3 == 0):
    return False
  
  i = 5
  while (i * i <= n):
    if ((n % i == 0) or 
        (n % (i + 2) == 0)):
      return False
    i += 6
    
  return True

# Function to return the count of primes 
# in the range which can be expressed as 
# the sum of two squares 
def countOfPrimes(L, R):
  
  count = 0
  for i in range(L, R + 1):
    
    # If i is a prime 
    if (isprime(i)):
      
      # If i can be expressed 
      # as the sum of two squares 
      if sumsquare(i):
        count += 1

  # Return the count 
  return count

# Driver code
if __name__=='__main__':
  
  L = 5
  R = 41
  print(countOfPrimes(L, R))
    
# This code is contributed by virusbuddah_

// C# program to implement
// the above approach
using System;

class GFG{

// Function to check if a prime number
// satisfies the condition to be 
// expressed as sum of two perfect
// squares
static bool sumSquare(int p)
{
    return (p - 1) % 4 == 0;
}

// Function to check if a
// number is prime or not
static bool isPrime(int n)
{
    
    // Corner cases
    if (n <= 1)
        return false;

    if (n <= 3)
        return true;

    if (n % 2 == 0 || n % 3 == 0)
        return false;

    for(int i = 5; i * i <= n; i = i + 6)
        if (n % i == 0 || n % (i + 2) == 0)
            return false;

    return true;
}

// Function to return the count of primes
// in the range which can be expressed as
// the sum of two squares
static int countOfPrimes(int L, int R)
{
    int count = 0;
    for(int i = L; i <= R; i++) 
    {

        // If i is a prime
        if (isPrime(i)) 
        {

            // If i can be expressed
            // as the sum of two squares
            if (sumSquare(i))
                count++;
        }
    }
    
    // Return the count
    return count;
}

// Driver code
public static void Main(String[] args)
{
    int L = 5, R = 41;

    Console.WriteLine(countOfPrimes(L, R));
}
}

// This code is contributed by Rajput-Ji

JavaScript

<script>
// javascript program to implement
// the above approach

    // Function to check if a prime number
    // satisfies the condition to be
    // expressed as sum of two perfect
    // squares
    function sumSquare(p) {
        return (p - 1) % 4 == 0;
    }

    // Function to check if a
    // number is prime or not
    function isPrime(n) {

        // Corner cases
        if (n <= 1)
            return false;

        if (n <= 3)
            return true;

        if (n % 2 == 0 || n % 3 == 0)
            return false;

        for (i = 5; i * i <= n; i = i + 6)
            if (n % i == 0 || n % (i + 2) == 0)
                return false;

        return true;
    }

    // Function to return the count of primes
    // in the range which can be expressed as
    // the sum of two squares
    function countOfPrimes(L , R) {
        var count = 0;
        for (var i = L; i <= R; i++) {

            // If i is a prime
            if (isPrime(i)) {

                // If i can be expressed
                // as the sum of two squares
                if (sumSquare(i))
                    count++;
            }
        }

        // Return the count
        return count;
    }

    // Driver code
        var L = 5, R = 41;
        document.write(countOfPrimes(L, R));

// This code is contributed by todaysgaurav 
</script>

Output:

Time Complexity: O(N^3/2)
Auxiliary Space: O(1)

Sum of all Perfect Squares lying in the range [L, R] for Q queries

spp____

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Count of primes in a given range that can be expressed as sum of perfect squares

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