Program to check if N is a Pentagonal Number

Given a number (N), check if it is pentagonal or not.
Examples :

Input: 12 
Output: Yes
Explanation: 12 is the third pentagonal number

Input: 19
Output: No
Explanation: The third pentagonal number is 12
while the fourth pentagonal number is 22.
Hence 19 is not a pentagonal number.

Pentagonal numbers are numbers which can be arranged to form a pentagon. If N is a pentagonal number then we can use N dots or points to generate a regular pentagon (Please see figure below).

The first few pentagonal numbers are 1, 5, 12, 22, 35, 51, 70, …

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

Image Source: Wiki

Method I (Iterative)

We begin by noting that the nth Pentagonal Number is given by
$P_n = \frac{3*n^2-n}{2}$

Follow an iterative process. Consecutively substitute n = 1, 2, 3 … into the formula and store the result in some variable M. Stop, if M >= N. After iteration if M equals N then N must be a pentagonal number. Else if M exceeds N then N cannot be a pentagonal number.

Algorithm

function isPentagonal(N) 
    Set i = 1
    do 
        M = (3*i*i - i)/2
        i += 1
    while M < N
    
    if M == N
        print Yes
    else
        print No

Below is the implementation of the algorithm

C++

// C++ program to check 
// pentagonal numbers. 
#include <iostream> 
using namespace std; 
  
// Function to determine 
// if N is pentagonal or not. 
bool isPentagonal(int N)  
{ 
    int i = 1, M; 
      
    do { 
  
        // Substitute values of i 
        // in the formula. 
        M = (3*i*i - i)/2; 
        i += 1; 
    } 
    while ( M < N ); 
      
    return (M == N); 
} 
  
// Driver Code 
int main() 
{ 
    int N = 12; 
      
    if (isPentagonal(N))  
        cout << N << " is pentagonal " << endl;     
    else
        cout << N << " is not pentagonal" << endl; 
      
    return 0; 
}  

Java

// Java program to check 
// pentagonal numbers. 
import java.io.*; 
  
class GFG { 
      
// Function to determine 
// if N is pentagonal or not. 
static Boolean isPentagonal(int N)  
{ 
    int i = 1, M; 
       
    do { 
   
        // Substitute values of 
        // i in the formula. 
        M = (3*i*i - i)/2; 
        i += 1; 
    } 
    while ( M < N ); 
       
    return (M == N); 
} 
    public static void main (String[] args) { 
    int N = 12; 
       
    if (isPentagonal(N))  
        System.out.println( N + " is pentagonal " );     
    else
        System.out.println( N + " is not pentagonal"); 
  
    } 
} 
  
// This code is contributed by Gitanjali. 

Python3

# python3 program to check 
# pentagonal numbers. 
import math  
  
# Function to determine if 
# N is pentagonal or not. 
def isPentagonal( N ) : 
  
    i = 1
    while True: 
  
        # Substitute values of i 
        # in the formula. 
        M = (3 * i * i - i) / 2
        i += 1
      
        if ( M >= N ): 
            break
      
    return (M == N) 
      
# Driver method 
N = 12
if (isPentagonal(N)): 
    print(N , end = ' ') 
    print ("is pentagonal " )  
else: 
    print (N , end = ' ') 
    print ("is not pentagonal") 
  
# This code is contributed by Gitanjali. 

C#

// C# program to check pentagonal numbers. 
using System; 
  
class GFG { 
      
// Function to determine 
// if N is pentagonal or not. 
static bool isPentagonal(int N)  
{ 
    int i = 1, M; 
      
    do { 
  
        // Substitute values of 
        // i in the formula. 
        M = (3 * i * i - i) / 2; 
        i += 1; 
    } 
    while ( M < N ); 
      
    return (M == N); 
} 
  
// Driver Code 
public static void Main ()  
{ 
    int N = 12; 
      
    if (isPentagonal(N))  
    Console.Write( N + " is pentagonal " );  
    else
    Console.Write( N + " is not pentagonal"); 
  
} 
} 
  
// This code is contributed by vt_m. 

PHP

<?php 
// PHP program to check 
// pentagonal numbers. 
  
// Function to determine 
// if N is pentagonal or not. 
function isPentagonal(int $N)  
{ 
    $i = 1;  
    $M; 
      
    do { 
  
        // Substitute values of i 
        // in the formula. 
        $M = (3 * $i * $i - $i) / 2; 
        $i += 1; 
    } 
    while ($M < $N); 
      
    return ($M == $N); 
} 
  
    // Driver Code 
    $N = 12; 
      
    if (isPentagonal($N))  
        echo $N , " is pentagonal " ;  
    else
        echo $N ," is not pentagonal" ; 
      
// This code is contributed by anuj_67. 
?> 

Output:

12 is pentagonal

The Time complexity of this method is O(n) since we need to compute successive values of pentagonal numbers up to N.

Method 2 (Efficient)
The formula indicates that the n-th pentagonal number depends quadratically on n. Therefore, try to find the positive integral root of N = P(n) equation.

P(n) = nth pentagonal number
N = Given Number

Solve for n:
P(n) = N
or (3*n*n – n)/2 = N
or 3*n*n – n – 2*N = 0 … (i)

The positive root of equation (i)
n = (1 + sqrt(24N+1))/6

After obtaining n, check if it is an integer or not. n is an integer if n – floor(n) is 0.

Implementation of the method is given below :

C++

// C++ Program to check a 
// pentagonal number 
#include <bits/stdc++.h> 
using namespace std; 
  
// Function to determine if 
// N is pentagonal or not. 
bool isPentagonal(int N) 
{     
    // Get positive root of 
    // equation P(n) = N. 
    float n = (1 + sqrt(24*N + 1))/6; 
      
    // Check if n is an integral 
    // value of not. To get the 
    // floor of n, type cast to int. 
    return (n - (int) n) == 0; 
} 
  
// Driver Code 
int main() 
{ 
    int N = 19;     
    if (isPentagonal(N))  
        cout << N << " is pentagonal " << endl;     
    else
        cout << N << " is not pentagonal" << endl;     
    return 0; 
} 

Java

// Java program to check 
// pentagonal numbers. 
import java.io.*; 
  
class GFG { 
      
// Function to determine if 
// N is pentagonal or not. 
static Boolean isPentagonal(int N)  
{ 
        // Get positive root of 
    // equation P(n) = N. 
    double n = (1 + Math.sqrt(24*N + 1))/6; 
      
    // Check if n is an integral 
    // value of not. To get the 
    // floor of n, type cast to int. 
    return (n - (int) n) == 0; 
} 
    public static void main (String[] args) { 
    int N = 19; 
       
    if (isPentagonal(N))  
        System.out.println( N + " is pentagonal " );     
    else
        System.out.println( N + " is not pentagonal"); 
  
    } 
} 
  
// This code is contributed by Gitanjali. 

Python3

# Python3 code Program to   
# check a pentagonal number 
  
# Import math library 
import math as m 
  
# Function to determine if 
# N is pentagonal or not 
def isPentagonal( n ): 
      
    # Get positive root of 
    # equation P(n) = N. 
    n = (1 + m.sqrt(24 * N + 1)) / 6
      
  
    # Check if n is an integral 
    # value of not. To get the 
    # floor of n, type cast to int 
    return( (n - int (n)) == 0) 
  
# Driver Code 
N = 19
  
if (isPentagonal(N)): 
    print ( N, " is pentagonal " )  
else: 
    print ( N, " is not pentagonal" )  
  
# This code is contributed by 'saloni1297' 

C#

// C# program to check pentagonal numbers. 
using System; 
  
class GFG { 
  
    // Function to determine if 
    // N is pentagonal or not. 
    static bool isPentagonal(int N) 
    { 
        // Get positive root of 
        // equation P(n) = N. 
        double n = (1 + Math.Sqrt(24 * N + 1)) / 6; 
  
        // Check if n is an integral 
        // value of not. To get the 
        // floor of n, type cast to int. 
        return (n - (int)n) == 0; 
    } 
      
    // Driver Code 
    public static void Main() 
    { 
        int N = 19; 
  
        if (isPentagonal(N)) 
            Console.Write(N + " is pentagonal "); 
        else
            Console.Write(N + " is not pentagonal"); 
    } 
} 
  
// This code is contributed by vt_m. 

PHP

<?php 
// PHP Program to check 
// a pentagonal number 
  
// Function to determine if 
// N is pentagonal or not. 
function isPentagonal($N) 
{  
    // Get positive root of 
    // equation P(n) = N. 
    $n = (1 + sqrt(24 * $N + 1)) / 6; 
      
    // Check if n is an integral 
    // value of not. To get the 
    // floor of n, type cast to int. 
    return ($n - (int) $n) == 0; 
} 
  
// Driver Code 
$N = 19;  
if (isPentagonal($N))  
    echo $N . " is pentagonal ";  
else
    echo $N . " is not pentagonal"; 
  
// This code is contributed by mits. 
?> 

Output :

19 is not pentagonal

Time and Space Complexities of this method are both O(1).

References :
1) Wikipedia – Pentagonal Numbers
2) Wolfram Alpha – Pentagonal Numbers

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My Personal Notes

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

C++

Java

Python3

C#

PHP

C++

Java

Python3

C#

PHP

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