Program to check if N is a Pentagonal Number
Last Updated :
06 Dec, 2023
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, ...
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 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 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# 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.
<script>
// javascript program to check
// pentagonal numbers.
// Function to determine
// if N is pentagonal or not.
function isPentagonal(N)
{
var 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);
}
var N = 12;
if (isPentagonal(N))
document.write( N + " is pentagonal " );
else
document.write( N + " is not pentagonal");
// This code is contributed by Amit Katiyar
</script>
<?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.
?>
Time Complexity: O(n), since we need to compute successive values of pentagonal numbers up to N.
Auxiliary Space: O(1) because it is using constant space for variables
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 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 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# 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.
<script>
// javascript program to check
// pentagonal numbers.
// Function to determine if
// N is pentagonal or not.
function isPentagonal(N)
{
// Get positive root of
// equation P(n) = N.
var 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 - parseInt( n) == 0);
}
var N = 19;
if (isPentagonal(N))
document.write( N + " is pentagonal " );
else
document.write( N + " is not pentagonal");
// This code is contributed by Amit Katiyar
</script>
<?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.
?>
Output19 is not pentagonal
Time complexity: O(log N) for given n, as it is using inbuilt sqrt function
Auxiliary Space: O(1)
References :
1) Wikipedia - Pentagonal Numbers
2) Wolfram Alpha - Pentagonal Numbers
Approach#3: Using binary search
This approach checks if a given number is a pentagonal number using binary search. It calculates the maximum value of n for the given number, creates a list of pentagonal numbers up to that limit, and searches for the given number in the list using binary search. If the number is found, it returns "Yes"; otherwise, it returns "No"
Algorithm
1. Use the formula to calculate the maximum possible value of n for a given number.
2. Use binary search to find the position of the given number in the list of pentagonal numbers from 1 to n.
3. If the value at the calculated position is equal to the given number, return "Yes". Otherwise, return "No".
C++
#include <iostream>
#include <vector>
#include <cmath>
using namespace std;
// Function to check if a number is a pentagonal number
string isPentagonal(int num) {
// Calculate the maximum 'n' for pentagonal numbers
int maxN = static_cast<int>((sqrt(24 * num + 1) + 1) / 6);
// Generate a list of pentagonal numbers up to maxN
vector<int> pentagonalList;
for (int n = 1; n <= maxN; n++) {
// Calculate the nth pentagonal number using the formula
int pentagonal = n * (3 * n - 1) / 2;
pentagonalList.push_back(pentagonal);
}
// Binary search to check if 'num' is a pentagonal number
int left = 0;
int right = pentagonalList.size() - 1;
while (left <= right) {
int mid = (left + right) / 2;
if (pentagonalList[mid] == num) {
// 'num' is found in the list, so it's a pentagonal number
return "Yes, it is a pentagonal number";
} else if (pentagonalList[mid] < num) {
// 'num' is larger, so search the right half of the list
left = mid + 1;
} else {
// 'num' is smaller, so search the left half of the list
right = mid - 1;
}
}
// If 'num' is not found in the list, it's not a pentagonal number
return "Not a pentagonal number";
}
int main() {
int num = 19;
// Call the isPentagonal function and print the result
cout << isPentagonal(num) << endl;
return 0;
}
import java.util.ArrayList;
public class PentagonalChecker {
// Function to check if a number is a pentagonal number
static String isPentagonal(int num)
{
// Calculate the maximum 'n' for pentagonal numbers
int maxN = (int)((Math.sqrt(24 * num + 1) + 1) / 6);
// Generate a list of pentagonal numbers up to maxN
ArrayList<Integer> pentagonalList
= new ArrayList<>();
for (int n = 1; n <= maxN; n++) {
// Calculate the nth pentagonal number using the
// formula
int pentagonal = n * (3 * n - 1) / 2;
pentagonalList.add(pentagonal);
}
// Binary search to check if 'num' is a pentagonal
// number
int left = 0;
int right = pentagonalList.size() - 1;
while (left <= right) {
int mid = (left + right) / 2;
if (pentagonalList.get(mid) == num) {
// 'num' is found in the list, so it's a
// pentagonal number
return "Yes, it is a pentagonal number";
}
else if (pentagonalList.get(mid) < num) {
// 'num' is larger, so search the right half
// of the list
left = mid + 1;
}
else {
// 'num' is smaller, so search the left half
// of the list
right = mid - 1;
}
}
// If 'num' is not found in the list, it's not a
// pentagonal number
return "Not a pentagonal number";
}
public static void main(String[] args)
{
int num = 19;
// Call the isPentagonal function and print the
// result
System.out.println(isPentagonal(num));
}
}
import math
def is_pentagonal(num):
max_n = int((math.sqrt(24*num + 1) + 1) / 6)
pentagonal_list = [(n * (3*n - 1) // 2) for n in range(1, max_n+1)]
left = 0
right = len(pentagonal_list) - 1
while left <= right:
mid = (left + right) // 2
if pentagonal_list[mid] == num:
return "Yes, it is pentagonal number"
elif pentagonal_list[mid] < num:
left = mid + 1
else:
right = mid - 1
return "Not a pentagonal number"
num=19
print(is_pentagonal(num))
using System;
using System.Collections.Generic;
class Program
{
// Function to check if a number is a pentagonal number
static string IsPentagonal(int num)
{
// Calculate the maximum 'n' for pentagonal numbers
int maxN = (int)((Math.Sqrt(24 * num + 1) + 1) / 6);
// Generate a list of pentagonal numbers up to maxN
List<int> pentagonalList = new List<int>();
for (int n = 1; n <= maxN; n++)
{
// Calculate the nth pentagonal number using the formula
int pentagonal = n * (3 * n - 1) / 2;
pentagonalList.Add(pentagonal);
}
// Binary search to check if 'num' is a pentagonal number
int left = 0;
int right = pentagonalList.Count - 1;
while (left <= right)
{
int mid = (left + right) / 2;
if (pentagonalList[mid] == num)
{
// 'num' is found in the list, so it's a pentagonal number
return "Yes, it is a pentagonal number";
}
else if (pentagonalList[mid] < num)
{
// 'num' is larger, so search the right half of the list
left = mid + 1;
}
else
{
// 'num' is smaller, so search the left half of the list
right = mid - 1;
}
}
// If 'num' is not found in the list, it's not a pentagonal number
return "Not a pentagonal number";
}
static void Main()
{
int num = 19;
// Call the IsPentagonal function and print the result
Console.WriteLine(IsPentagonal(num));
}
}
// Function to check if a number is a pentagonal number
function isPentagonal(num) {
// Calculate the maximum 'n' for pentagonal numbers
const maxN = Math.floor((Math.sqrt(24 * num + 1) + 1) / 6);
// Check if the number is pentagonal using the inverse formula
const checkPentagonal = (Math.sqrt(1 + 24 * num) + 1) / 6;
// If the result is an integer and within the valid range, it's pentagonal
if (Number.isInteger(checkPentagonal) && checkPentagonal <= maxN) {
return "Yes, it is a pentagonal number";
} else {
return "Not a pentagonal number";
}
}
// Driver code
const num = 19;
// Call the isPentagonal function and print the result
console.log(isPentagonal(num));
OutputNot a pentagonal number
Time complexity: O(log n)
Space complexity: O(sqrt(n))
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