Extension and Generalization of Fermat’s Little Theorem to the Gaussian Integers

It can . . . come as a bit of a shock to meet your first non-obvious theorem, which will typically be Fermat’s Little Theorem. — Dominic Yeo [10] The non-obviousness of Fermat’s Little Theorem is the most interesting part of any introductory number theory course. We are therefore motivated to determine if Fermat’s Little Theorem can be extended to the Gaussian integers, as many other useful properties of the integers can. After proving an extension of Fermat’s Little Theorem to the Gaussian integers, we generalize by extending Euler’s φ function and product formula and Euler’s Theorem to the Gaussian integers.