Hatcher-Topology-of-Numbers · Problem 8.1.3

Quadratic Fields

Mathematics — Algebraic Topology · prime-factorization-quadratic-fields

Problem

There are four different ways to write the number

1105 = 5 \cdot 13 \cdot 17

as a sum of two squares. Find these four ways using the factorization of

1105

into primes in

\mathbb{Z}[i]

. Here we are not counting

5^2 + 2^2

and

2^2 + 5^2

as different ways of expressing

29

as the sum of two squares. Note that an equation

n = a^2 + b^2

is equivalent to an equation

n = (a+bi)(a-bi)