Pell's equation x²−Dy²=1: for non-square D>0, infinitely many solutions exist. Fundamental solution (x₁,y₁) from continued fraction of √D. All solutions: xₙ₊₁=x₁xₙ+Dy₁yₙ, yₙ₊₁=x₁yₙ+y₁xₙ. Equivalently (xₙ+yₙ√D) = (x₁+y₁√D)ⁿ.
How to Use
Enter D:
- D: Non-square positive integer
- (x₁,y₁): Fundamental solution
- More: Generated solutions
Method
Expand √D as continued fraction [a₀; a₁,a₂,...,aₖ] (periodic). Convergent pₖ/qₖ at end of period gives fundamental solution (x₁,y₁) = (pₖ,qₖ). If period length is odd, use (p₂ₖ,q₂ₖ).
History
Misnamed after John Pell. Actually studied by Brahmagupta (628 CE), Bhaskara II (1150), Fermat, and Euler. Lagrange proved all non-square D have solutions.
Step-by-Step Instructions
- 1Enter D.
- 2Get fundamental solution.
- 3Generate more.
- 4Verify x²−Dy²=1.
- 5View CF expansion.