Discovering the Magic of Numbers: The Infinite World of Automorphic Numbers

Mathematics is like an adventure, filled with patterns waiting to be uncovered. One such magical journey leads us to a peculiar set of numbers known as automorphic numbers. Let’s embark on this mathematical quest together!

Automorphic numbers are like mathematical superheroes that have unique abilities. Imagine squaring a number and noticing something surprising – the last digit (or digits) of the result is the same as the number itself! For example, take the number 5; when you square it, you get 25, ending with 5 again. Fascinating, isn’t it?

Now, let’s meet another hero, the number 6. When you square it, you get 36, and although 6 doesn’t appear in the result, the last digit is always 6. These special numbers are called automorphic.

The most exciting hero in our mathematical adventure is 5. Not only is 5 automorphic, but its square and the square of its square are also automorphic! This leads us to a thrilling question: does this sequence of automorphic numbers go on forever when we keep squaring 5?

Surprisingly, the answer is no! After the third square, the pattern seems to collapse. However, if we look closely, at least the last five digits of the number appear in the square. And when we square this part, we discover a new automorphic number! This process can continue infinitely, creating a sequence of ever-growing automorphic numbers.

But here’s where it gets even more enchanting. Mathematicians have a secret code, a new number system called p-adic numbers, where p is a prime number. These numbers allow us to play with infinity in a magical way. We can create infinitely large automorphic numbers and even find a mysterious “fixed point” in infinity.

Imagine having a number so big that it is completely automorphic, meaning its square corresponds to itself! Although we can’t write down this infinite number, we know its last digits are …918,212,890,625.

The journey into automorphic numbers doesn’t stop there. Mathematicians use p-adic numbers to explore these magical realms, where numbers behave in ways that might seem strange at first but become a fascinating part of our mathematical adventure.