The digraphs of commutative rings under modular arithmetic reveal intriguing cycle patterns, many of which have yet to be explained. To help illuminate these patterns, we establish a set of new theorems. Rings with relatively prime moduli a and b are used to predict cycles in the digraph of the ring with modulus ab. Rings that use Pythagorean primes as their modulus are shown to always have a cycle in common. Rings with perfect square moduli have cycles that relate to their square root.

Author Bio

Morgan Bounds, double majoring in mathematics and humanities, is a senior at Indiana Wesleyan University. When he's not trying to prove the Riemann Hypothesis or engage in lively discussions about panopticism, Morgan enjoys playing pickup basketball and performing live music at local coffee shops. After graduation, he plans to marry his lovely fiancé Alyssa and pursue a PhD in mathematics.