Last Updated on July 4, 2023 by Mayank Dham
Imagine a number that exhibits an extraordinary characteristic: its square ends with the same digits as the number itself. These numbers, known as automorphic numbers, hold a remarkable ability to preserve their identity even after undergoing a mathematical transformation. They form a captivating realm where the digits embrace their own mirrored existence, creating an intriguing puzzle waiting to be unraveled.
In this article, we embark on a journey into the enchanting world of automorphic numbers. We will explore their definition, properties, and uncover the underlying principles that make them unique. From their ancient origins to their modernday applications, we will delve into the diverse facets of automorphic numbers and witness the fascinating interplay between mathematics and number theory.
What are Automorphic Numbers?
To comprehend automorphic numbers, we must first grasp their fundamental definition. An automorphic number is a nonzero positive integer that, when squared, produces a result in which its original number appears as the ending digits. In other words, an automorphic number "reflects" itself within its square. For example, let’s consider the automorphic number 5. When we square it, the result is 25, with the digit 5 appearing at the end.
Properties and Characteristics:
Automorphic numbers possess a set of unique properties that distinguish them from other numbers. Here are some key characteristics of automorphic numbers:

SelfReflection: As mentioned earlier, the defining trait of automorphic numbers is their ability to preserve their identity within their square. This selfreflective property gives them a sense of symmetry and adds an element of intrigue to their nature.

Terminal Digits: Automorphic numbers are primarily identified by their terminal (or ending) digits. These digits are crucial in determining if a number qualifies as automorphic, as they must match the corresponding digits at the end of its square.

Digit Placement: The position of the automorphic digits within the square depends on the number of digits in the original number. For example, a singledigit automorphic number will have its digit at the units place in its square, while a twodigit automorphic number will have its digits at the units and tens places, and so on.
Examples of Automorphic Numbers:
Let’s explore a few examples to illustrate the concept of automorphic numbers:
The number 5 is an automorphic number since its square, 25, ends with the digit 5.
Similarly, the number 76 is an automorphic number, as its square, 5776, ends with the same two digits.
The number 376 is not an automorphic number, as its square, 141,376, does not end with the original digits.
Finding whether a Number is an Automorphic Number or Not
You are given the number N, and your task is to determine whether or not it is an automorphic number. In our introduction, we discussed what an automorphic number is and provided an example. Now, letâ€™s quickly discuss how we can approach this problem.
Approach 1 : Finding whether a Number is an Automorphic Number or Not
 Keep track of the square of the given number.
 Loop until N equals 0 because we need to match all digits with their squares.
 Check whether (n%10 == sq%10), that is, whether the last digit of the number is the last digit of the square, is true. If not equal, return false.
 Otherwise, proceed by reducing the number and square, i.e., n = n/10 and sq = sq/10;
 If all of the digits match, return true.
Code Implementation
C++ Code: https://ideone.com/VakVKL
Python Code: https://ideone.com/wu54gB
Java Code: https://ideone.com/denhm4
Output:
Automorphic
Time Complexity: O(log10N)
Approach 2 : Finding whether a Number is an Automorphic Number or Not
 Check to see if the number is negative, then make it positive.
 Save the numberâ€™s square.
 Find the count of the digit of the number so that you can find the count of the digit of the last number of the square of the number equal to the number; note that this does not imply that if the count of the last number of the square is equal to the number, they will be equal.
 After counting the digits of the number, perform: squareNum%power(10, count)
 Finally, check to see if the numberâ€™s last square is equal to the number.
Code Implementation
C++ Code: https://ideone.com/tFCssX
Python Code: https://ideone.com/xiFK9B
Java Code: https://ideone.com/ZZlJVX
Output:
Not Automorphic
Time Complexity: O(log10N)
Applications and Significance:
Automorphic numbers, while captivating in their own right, also find practical applications in various fields. Some areas where automorphic numbers have proven useful include:

Mathematics and Number Theory: The study of automorphic numbers deepens our understanding of number theory and serves as a rich source of mathematical exploration. They offer insights into number patterns, sequences, and relationships between numbers.

Cryptography: Automorphic numbers can play a role in certain cryptographic algorithms. Their unique properties can be leveraged to enhance encryption techniques and ensure secure communication.

Digital Security: Automorphic numbers contribute to the development of secure systems and protocols by adding an additional layer of complexity and unpredictability to cryptographic algorithms.
Conclusion
In conclusion, the exploration of automorphic numbers takes us on a captivating journey through the realm of selfreflecting digits. These numbers, with their ability to preserve their identity within their squares, offer a glimpse into the intricate patterns and symmetries that lie within the world of mathematics. From their definition and properties to their applications in cryptography and digital security, automorphic numbers have proven to be both intellectually stimulating and practically relevant.
By unraveling the secrets of automorphic numbers, we deepen our understanding of number theory, uncovering hidden connections and shedding light on the fascinating interplay between digits and their squares. The study of automorphic numbers not only enriches mathematical knowledge but also contributes to the development of secure systems and cryptographic algorithms, bolstering digital security in an increasingly interconnected world.
As we conclude this exploration, we invite you to embrace the beauty of automorphic numbers and continue your journey into the vast landscape of mathematics. The realm of numbers is filled with mysteries yet to be unraveled, and automorphic numbers serve as a reminder of the captivating and everevolving nature of mathematical discovery.
FAQs (Frequently Asked Questions):
Q1: Are automorphic numbers limited to specific digits?
A: No, automorphic numbers can consist of any positive integer digits. They can range from singledigit numbers to multidigit numbers, depending on their properties.
Q2: Are there infinitely many automorphic numbers?
A: Yes, there are infinitely many automorphic numbers. The existence of automorphic numbers can be proven through mathematical reasoning and exploration.
Q3: Can automorphic numbers have more than one digit at the end of their squares?
A: Yes, automorphic numbers can have multiple digits at the end of their squares. The number of digits at the end of the square depends on the number of digits in the original number.
Q4: Can automorphic numbers be negative or fractional?
A: No, automorphic numbers are defined as nonzero positive integers. Negative numbers and fractions do not fall into the category of automorphic numbers.
Q5: How are automorphic numbers relevant in cryptography?
A: Automorphic numbers can be utilized in cryptographic algorithms to enhance encryption techniques and ensure secure communication. Their unique properties add complexity and contribute to the overall security of cryptographic systems.
Q6: Can automorphic numbers be used in other branches of mathematics?
A: Absolutely! Automorphic numbers have applications in various mathematical fields, including number theory, sequences, and patterns. They offer valuable insights into the underlying principles and relationships within the realm of mathematics.