Solutions of Pell's equation are powers of the fundamental solution

Builds on Pell's equation Fundamental solution of Pell's equation

Statement

Let $d$ be a positive integer that is not a perfect square, and let $(x_1, y_1)$ be the fundamental solution of $x^2 - d y^2 = 1$ . Let $x$ and $y$ be integers with $x \ge 1$ , $y \ge 0$ and $x^2 - d y^2 = 1$ . Then there is a unique natural number $n$ with $x + y \sqrt{d} = (x_1 + y_1 \sqrt{d})^n$ .

Remarks

Conversely, each power $(x_1 + y_1 \sqrt{d})^n = x_n + y_n \sqrt{d}$ gives a solution $(x_n, y_n)$ . The integer solutions are exactly the pairs with $x + y \sqrt{d} = \pm (x_1 + y_1 \sqrt{d})^n$ for $n \in \mathbb{Z}$ . For $d = 2$ : $(3 + 2\sqrt{2})^2 = 17 + 12 \sqrt{2}$ , so $(17, 12)$ is the next solution after $(3, 2)$ . In algebraic terms, the units of norm $1$ in $\mathbb{Z}[\sqrt{d}]$ form the group $\{\pm 1\} \times \mathbb{Z}$ .

Reading path

To understand this page from the ground up, read these 5 pages in order. The foundations come first.

Show the path as a graph

Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.

no evidence level

Diophantine equation concept
Pell's equation definition
Existence of nontrivial solutions of Pell's equation statement
Fundamental solution of Pell's equation definition
Solutions of Pell's equation are powers of the fundamental solution statement

Tag	`0J1TT091TN`
URI	`m2://0J1TT091TN`
Kind	statement, theorem
Writing	W0 stub
MSC	11D09
Status	This copy of the site has no build data. Run `m2 status 0J1TT091TN`.
Uses	Pell's equation, Fundamental solution of Pell's equation

Solutions of Pell's equation are powers of the fundamental solution

Statement

Remarks

Reading path

Further reading

What this page needs for W1

What links here