Cyclic Patterns in Numbers

Number Theory · Axiom Academy

INTRO Cyclic Patterns in Numbers Discover how numbers repeat themselves in mysterious and beautiful ways. Let's explore what happens when we raise a number to increasing powers. Use the slider to change the base number and watch the pattern in the last digits. Different numbers have different cycle lengths for their last digits. Try different bases and observe how long it takes before the pattern repeats. The famous Fibonacci sequence (1, 1, 2, 3, 5, 8, 13...) also has cyclic patterns! But instead of looking at last digits, let's look at remainders when dividing by different numbers. Choose a modulus to explore. The Power of Modular Arithmetic All these patterns come from modular arithmetic - arithmetic with remainders. When we work "modulo n", we only care about remainders when dividing by n. Click to see different multiplication tables modulo various numbers. When we work modulo n, we're confined to a finite set of remainders 0, 1, 2, ..., n-1 . Any infinite sequence in a finite space must eventually repeat. From last digits of powers to Fibonacci sequences to multiplication tables - cyclic patterns appear throughout mathematics when we use modular arithmetic. These patterns aren't just beautiful - they're useful! Cryptography, computer science, music theory, and calendar calculations all rely on modular arithmetic and cyclic patterns.

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