Pellian equations

Hi guys, this is my first note on this account. Pellian equations are of the form

$x^2+ny^2 = 1$

How do you solve such equations? Or possibly, what are the "solvable" and "unsolvable" cases?

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Math	Appears as
Remember to wrap math in `$` ... `$` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$

Comments

Pell equations are equations of the form x^2 - ny^2 = 1 (this is one type) where n is not a perfect square. To solve the equation with a norm 1, find first the least ordered non-trivial pair of solutions to the equation (The trivial solution is always (-1, 0) and (1, 0).). From this, you can use the assertion that all solutions (x, y) are of the form (x + y sqrt(n))^d for any positive integer d. And you are done!

If the norm is -1, find the least ordered non-trivial pair of solutions to the equation. From this, you can use the assertion that all solutions (x, y) are of the form (x - y sqrt(n))^d for any positive odd integer d. And you are done! But for norms other than -1 and 1, there is a need of understanding of factorization in number fields, etc.

John Ashley Capellan - 6 years, 10 months ago

See Calvin's post here for how to solve Pell equations with the norm 3.

mathh mathh - 6 years, 10 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$