Perfect Gaussian integers

On the Form of Odd Perfect Gaussian Integers

Let Z[i] = {a + bi : a, b ∈ Z} be the ring of Gaussian integers. All Gaussian integers will be represented by Greek letters and rational integers by ordinary Latin letters. Primes will be denoted by π and p respectively. Units will be denoted by ε = ±1,±i and 1 respectively. In 1961 Robert Spira [3] defined the sum-of-divisors function on Z[i] as follows. Let η = εΠπi i be a Gaussian integer. T...

Gaussian Integers

We will investigate the ring of ”Gaussian integers” Z[i] = {a + bi | a, b ∈ Z}. First we will show that this ring shares an important property with the ring of integers: every element can be factored into a product of finitely many ”primes”. This result is the key to all the remaining concepts in this paper, which includes the ring Z[i]/αZ[i], analogous statements of famous theorems in Z, and q...

Gaussian Integers

Gaussian integer is one of basic algebraic integers. In this article we formalize some definitions about Gaussian integers [27]. We also formalize ring (called Gaussian integer ring), Z-module and Z-algebra generated by Gaussian integer mentioned above. Moreover, we formalize some definitions about Gaussian rational numbers and Gaussian rational number field. Then we prove that the Gaussian rat...

The Gaussian Integers

Since the work of Gauss, number theorists have been interested in analogues of Z where concepts from arithmetic can also be developed. The example we will look at in this handout is the Gaussian integers: Z[i] = {a + bi : a, b ∈ Z}. Excluding the last two sections of the handout, the topics we will study are extensions of common properties of the integers. Here is what we will cover in each sec...

Quadratic Reciprocity for the Rational Integers and the Gaussian Integers