Some people were wondering about this in a recent topic, so I'll post a discussion of it here.

Problem: Generate an infinite number of integer solutions to



For a squarefree integer.

Solution: We will employ the ring . This is the set of all numbers of the form .

Definition: For , let .

Conjugation is multiplicative; in other words, .

Definition: Let the norm .

The norm is also multiplicative, which is the real gem of using this particular approach. We want to be a square, so if we take an arbitrary element (which has some integer norm) and square it, we will get an element with a square norm. In other words, let



Then



Which has norm



Therefore we have generated the solutions

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It turns out that this approach generates all primitive solutions for a few cases; most famously, is the familiar generation of Pythagorean triples, where is the Gaussian integers, a tool I have used before, and I recently saw another topic that asked for the case .

Both of these cases have the identifying common trait that is a Unique Factorization Domain (which means unique prime factorization exists), which is not generally true. It fails for , for example, although it holds for .

The question of what produces a UFD is open, as far as I know. Gauss conjectured a long and fascinating list that I don't remember.

This tool is extremely useful, and solves another kind of Diophantine equation with which you may be familiar.

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Practice Problem: Find an infinite number of solutions to



For a squarefree integer .