What is an Automorphic Number?
An Automorphic number is a special kind of number whose square ends with the same digits as the original number.
From number theory enthusiasts to coding learners, automorphic numbers offer an interesting concept that blends logic, patterns, and fun. In simple words, when you square an automorphic number, the last digits of the result are the number itself.
Mathematical Definition:
A number n is said to be automorphic if:
For example, 5² = 25 (ends with 5) and 76² = 5776 (ends with 76). Automorphic numbers can be single-digit or multiple-digit. Some numbers, like 25 and 76, appear frequently in Automorphic sequences.
Properties of Automorphic Numbers
- Self-Ending Pattern: The square of the number ends with the same number.
- Non-Negative Integers: Mathematicians typically consider automorphic numbers within the set of non-negative integers.
- Rare Sequence: Automorphic numbers are rare to find. They form a unique and finite set within a given number system.
- Base-Dependent: Automorphic properties change with number systems (binary, decimal, etc.). This article focuses on the decimal system.
Why Are Automorphic Numbers Important?
- Mathematical Curiosity: Automorphic numbers illustrate fascinating properties of numbers and modular arithmetic.
- Coding Challenges: They serve as common programming problems to test understanding of string manipulation, modulo operations, and loops.
- Cryptography and Number Theory: Understanding such number patterns can help in studying the properties of numbers and modular congruences, which form the foundations of cryptography.
Examples and Data
Below are the first 10 automorphic numbers in base 10, along with their squares:
| Automorphic Number | Square | Last Digits Match |
| 1 | 1 | 1 |
| 5 | 25 | 5 |
| 6 | 36 | 6 |
| 25 | 625 | 25 |
| 76 | 5776 | 76 |
| 376 | 141376 | 376 |
| 625 | 390625 | 625 |
| 9376 | 87909376 | 9376 |
| 90625 | 8212890625 | 90625 |
| 109376 | 11963109376 | 109376 |
How to Check if a Number is Automorphic?
- Find the square of the number. (62 = 36)
- Compare the last digits of the square with the original number. (62 = 36)
- If they match, the number is automorphic. (6)
Sample Code to Identify Automorphic Numbers
1. Using a Python Program
def is_automorphic(n):
square = n ** 2
num_digits = len(str(n))
last_digits = square % (10 ** num_digits)
return last_digits == n
numbers = [5, 6, 13, 25, 76, 376, 625, 890, 9376]
for num in numbers:
if is_automorphic(num):
print(f"{num} is an Automorphic Number.")
else:
print(f"{num} is NOT an Automorphic Number.")Output:
2. Using C++ Snippet
#include <iostream>
#include <cmath>
using namespace std;
bool isAutomorphic(long long n) {
int k = floor(log10(n)) + 1;
long long mod = 1;
for (int i = 0; i < k; i++) mod *= 10;
return (n * n) % mod == n;
}
int main() {
for (int i = 0; i < 10000; i++) {
if (isAutomorphic(i))
cout << i << " ";
}
return 0;
}Output:
3. Using JavaScript Code
function isAutomorphic(n) {
let square = (n * n).toString();
let numStr = n.toString();
return square.endsWith(numStr);
}
// Test the function
const testNumbers = [5, 6, 25, 76, 376, 625, 9376, 13];
testNumbers.forEach(num => {
if (isAutomorphic(num)) {
console.log(`${num} is an Automorphic number.`);
} else {
console.log(`${num} is NOT an Automorphic number.`);
}
});Output:
Automorphic Numbers and Their Applications
While automorphic numbers are mostly recreational, they have some interesting applications:
- Mathematical puzzles and recreational math: Great for math competitions and quizzes.
- Computer science: Understanding modular arithmetic and number properties.
- Cryptography: Although people do not directly use automorphic numbers, they rely on modular arithmetic, which forms the foundation for many mathematical applications.
- Digital signal processing: Developers explore a similar number of properties to design efficient algorithms.
Final Thoughts
Automorphic numbers blend elegant mathematical theory with practical coding challenges. From the simplicity of 5 and 6 to larger multi-digit marvels like 9376 and beyond, these numbers provide excellent exercises in modular arithmetic, algorithm design, and performance analysis. Whether you are a student honing your coding skills or a math enthusiast exploring number patterns, automorphic numbers are sure to captivate and educate.
Frequently Asked Questions (FAQs)
Q1. Are all square numbers automorphic?
Answer: No, not all square numbers are automorphic. Only specific numbers whose squares end with the same digits as the original number itself qualify as automorphic numbers.
Q2. Is there a largest automorphic number?
Answer: No, there is no largest automorphic number. In theory, automorphic numbers can extend infinitely, though they become extremely rare and difficult to find as the numbers grow larger.
Q3. Can negative numbers be automorphic?
Answer: Generally, the concept of automorphic numbers applies only to non-negative integers. Mathematicians focus on this idea in the relationship between a number and the ending digits of its square, and they typically do not extend it to negative numbers.
Recommended Articles
We hope this article on automorphic numbers helps you understand this fascinating mathematical concept with clear examples. Check out these recommended articles for more fun facts, number patterns, and interesting topics in mathematics.
