The Prime Hexagon

Presenter: Kaustubh Shivdikar

Research Category: Computer and Information Sciences
Student Type: Graduate
Additional Authors: Julian Gutierrez, Nicholas Velcea, Samuel Smucny, Morgan Rosenkranz
PI: Kaustubh Shivdikar
Faculty Advisor: David Kaeli
Award Winner Category: Computer and Information Sciences

The distribution of integer prime numbers remains a mystery today. There has been no mathematical equation derived that explains which numbers will be prime. Ulam’s Spiral and Primal Chaos Theory are just a few of the many attempts made in the past to unravel this mystery.

In this project, we leverage a novel approach to visualize prime numbers using overlapping hexagons that can bound all prime numbers. The Prime Hexagon is a mathematical structure developed by mathematician T. Gallion.  A Prime Hexagon is formed when integers are sequentially added to a field of tessellating equilateral triangles, where the path of the integers is changed whenever a prime number is encountered. Since prime numbers are never multiples of two or three, all numbers from “2” to infinity are confined within a 24-cell hexagon.

We color-code the six hexagons, identifying patterns in key number sequences, including the Fibonacci sequence, powers of two and three, and power of pi. For the series of consecutive powers of pi, we have found that no two fall within the same six-cell hexagon. We have computed this for pi^32, which has less than a 1/400 chance of occurring randomly.

We have used a distributed implementation of the CUDA Sieve and Hex Sieve algorithms to compute pi, enabling parallel execution on a GPU-based cluster. We use four NVIDIA V100 GPUs and four AMD EPYC 7551 CPUs, totaling 20,480 CUDA cores and 256 CPU cores, achieving a 9X speedup versus serial computation.