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Nicomachus's Theorem

Last Updated : 11 Jul, 2025
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Nicomachus's Theorem states that sum of cubes of first n natural numbers is equal to squares of natural number sum.
1^{3}+2^{3}+3^{3}+\cdots +n^{3}=\left(1+2+3+\cdots +n\right)^{2}      
In other words
1^{3}+2^{3}+3^{3}+\cdots +n^{3}=\left(n*(n+1)/2)^{2}      
Or we can say that the sum is equal to square of n-th triangular number.
Mathematical Induction based proof can be found here. 
 

C++ 
// CPP program to verify Nicomachus's Theorem
#include <bits/stdc++.h>
using namespace std;

void NicomachusTheorem_sum(int n)
{
   // Compute sum of cubes
   int sum = 0;
   for (int k=1; k<=n; k++)
      sum += k*k*k;
   
   // Check if sum is equal to
   // given formula.
   int triNo = n*(n+1)/2;
   if (sum == triNo * triNo)
     cout << "Yes";
   else
     cout << "No";
}

// driver function
int main()
{
    int n = 5;
    NicomachuTheorem_sum(n);
    return 0;
}
Java 
// Java program to verify Nicomachus's Theorem
import java.io.*;

class GFG {

    static void NicomachuTheorem_sum(int n)
    {
        
        // Compute sum of cubes
        int sum = 0;
        
        for (int k = 1; k <= n; k++)
            sum += k * k * k;
            
        // Check if sum is equal to
        // given formula.
        int triNo = n * (n + 1) / 2;
        
        if (sum == triNo * triNo)
            System.out.println("Yes");
        else
            System.out.println("No");
    }
    
    // driver function
    public static void main (String[] args)
    {
        int n = 5;
        NicomachuTheorem_sum(n);
    }
}

// This code is contributed by anuj_67.
Python3 
# Python3 program to verify
# Nicomachus's Theorem

def NicomachuTheorem_sum(n):
    
    # Compute sum of cubes
    sum = 0;
    for k in range(1, n + 1):
        sum += k * k * k;
        
    # Check if sum is equal to
    # given formula.
    triNo = n * (n + 1) / 2;
    if (sum == triNo * triNo):
        print("Yes");
    else:
        print("No");

# Driver Code
n = 5;
NicomachuTheorem_sum(n);

# This code is contributed 
# by mits
C# 
// C# program to verify
// Nicomachus's Theorem
using System;
 
class GFG {
 
    static void NicomachuTheorem_sum(int n)
    {
         
        // Compute sum of cubes
        int sum = 0;
         
        for (int k = 1; k <= n; k++)
            sum += k * k * k;
             
        // Check if sum is equal to
        // given formula.
        int triNo = n * (n + 1) / 2;
         
        if (sum == triNo * triNo)
            Console.WriteLine("Yes");
        else
            Console.WriteLine("No");
    }
     
    // Driver Code
    public static void Main ()
    {
        int n = 5;
        NicomachuTheorem_sum(n);
    }
}
 
// This code is contributed by anuj_67
PHP 
<?php
// PHP program to verify
// Nicomachus's Theorem

function NicomachuTheorem_sum($n)
{
    
    // Compute sum of cubes
    $sum = 0;
    for ($k = 1; $k <= $n; $k++)
        $sum += $k * $k * $k;
        
    // Check if sum is equal to
    // given formula.
    $triNo = $n * ($n + 1) / 2;
    if ($sum == $triNo * $triNo)
        echo "Yes";
    else
        echo "No";
}

    // Driver Code
    $n = 5;
    NicomachuTheorem_sum($n);

// This code is contributed by anuj_67.
?>
JavaScript 
<script>

// JavaScript program to verify Nicomachus's Theorem

    function NicomachuTheorem_sum(n)
    {
          
        // Compute sum of cubes
        let sum = 0;
          
        for (let k = 1; k <= n; k++)
            sum += k * k * k;
              
        // Check if sum is equal to
        // given formula.
        let triNo = n * (n + 1) / 2;
          
        if (sum == triNo * triNo)
            document.write("Yes");
        else
            document.write("No");
    }
      

// Driver code

        let n = 5;
        NicomachuTheorem_sum(n);
          
          // This code is contributed by souravghosh0416.
</script>

Output: 
Yes

 

Time complexity : O(n) 
Auxiliary Space : O(1)


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Shashank_Pathak
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