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?

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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TopNewestPell 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 · 2 years, 9 months ago

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See Calvin's post here for how to solve Pell equations with the norm 3. – Mathh Mathh · 2 years, 9 months ago

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