Gaussian Integers

1 Introduction

The ring of Gaussian integers is given by ℤ[i] = {a + ib∣a,b ∈ ℤ}. The units of ℤ[i] are {1,-1,i,-i}. The irreducible elements of ℤ[i] are the associates of

• i + 1,
• prime numbers p of ℤ such that p ≡ 3 mod 4 and
• a + ib where p = a² + b² ≡ 1 mod 4 is prime in ℤ.

Consider the factors of integers in ℤ[i].

Integer n	Factors with positive real part	s(n)
1	1	1
2	1, 1 + i , 1 - i , 2	5
3	1, 3	4
4	1, 1 - i, 1 + i, 2 - i, 2 + i, 4	13
5	1, 1 - 2i, 1 + 2i, 2 - i, 2 + i, 5,	12

The function s(n) is the sum of the positive real parts of the factors.

Proposition 1 Let k ≥ 1. Then s(2^k) = 5 ⋅ 2^k-1 + 3 ⋅ (2^k-1 - 1).

Proposition 2 Let k ≥ 1 and p ≡ 3 mod 4 be prime. Then s(p^k) = .

Proposition 3 Let k ≥ 0, p_i ≡ 3 mod 4 be primes and a_i ≥ 0 for i = 1, 2,…,r. Then s(2^k ∏ _i=1^rp _i^a_i) = s(2^k) ∏ _i=1^rs(p _i^a_i).

Proposition 4 Let p_i ≡ 3 mod 4, q_j ≡ 1 mod 4 be primes and a_i,b_j ≥ 0 for i = 1, 2,…,r and j = 1, 2,…,s. Then

s(∏ _i=1^rp _i^a_i ∏ _j=1^sq _j^b_j) = ∏ _i=1^rs(p _i^a_i) ⋅ s(∏ _j=1^sq _j^b_j).

Proposition 5 Suppose p ≡ 1 mod 4 such that p = σσ^* where σ = a+ib ∈ ℤ[i]. Then s(p) = 1 + p + 2(a + b).