The reason that your emails to bank details can be kept private and safe is modern cryptography systems. These systems rely on extremely large prime numbers, numbers that can only be divided by themselves and one. Luckily, there exist plenty of so-called primality tests to determine if a number is prime or not. We decided to study one of these tests, called the Fermat primality test, which sometimes determines a number to be prime even though it is not. It was examined how often this happened and how reliable the test is.

Having written a program that ran all numbers between 1 and 5000 through the Fermat test, graphs indicating the reliability of the test for each number were generated. Surprisingly, distinct visual patterns emerged. Although we have been able to prove some of the patterns by finding similarities in structural properties of numbers, some aspects of the patterns still remain a mystery.

The results from our investigation indicate that the Fermat test works well as a primality test.

However, there are a few exceptions. The already discovered Carmichael numbers will always give false positives in the Fermat test, but our study shows that there are other numbers with a relatively high risk of giving false positives. Although these numbers do not pose a threat to today’s cryptography system, they may be interesting from a purely theoretical perspective since in the pursuit to explain the visual patterns, links to abstract algebra were found.