Gaussian Integers

 

Title: Gaussian Integers
Author: Manuel Eberl
Submission date: 2020-04-24
Abstract:

The Gaussian integers are the subring ℤ[i] of the complex numbers, i. e. the ring of all complex numbers with integral real and imaginary part. This article provides a definition of this ring as well as proofs of various basic properties, such as that they form a Euclidean ring and a full classification of their primes. An executable (albeit not very efficient) factorisation algorithm is also provided.

Lastly, this Gaussian integer formalisation is used in two short applications:

  1. The characterisation of all positive integers that can be written as sums of two squares
  2. Euclid's formula for primitive Pythagorean triples

While elementary proofs for both of these are already available in the AFP, the theory of Gaussian integers provides more concise proofs and a more high-level view.

BibTeX:
@article{Gaussian_Integers-AFP,
  author  = {Manuel Eberl},
  title   = {Gaussian Integers},
  journal = {Archive of Formal Proofs},
  month   = apr,
  year    = 2020,
  note    = {\url{http://isa-afp.org/entries/Gaussian_Integers.html},
            Formal proof development},
  ISSN    = {2150-914x},
}
License: BSD License
Depends on: Polynomial_Factorization