Saved on this device

Follow your curiosity

What discovery has been shared with you?

FACTREE gives you something interesting to discover every day — and something worth talking about together.

Start with one fact. Explore it, go deeper, then follow whichever branch catches your imagination.

Choose subjects for a surprise

Exploring any topic

Begin your discovery

Your next discovery is one click away.

Choose one or more subjects above, or leave Any Topic selected and let curiosity decide.

Mathematics

The Pell Equation and Its Solution via Continued Fractions

Quick fact

For any non-square positive integer D, the Pell equation x² – Dy² = 1 always has infinitely many integer solutions, and they are generated from the continued fraction expansion of √D. The smallest non-trivial solution for D=2 is (3,2), for D=3 it's (2,1), and for D=60 it's (31,4).

Why this is interesting

You know how to solve equations like x² – 2y² = 1? It looks simple, but the solutions are surprisingly large and tricky to find. Can you guess the next one after (3, 2) and (17, 12)?