Pell Equation

1. Definition:

• Proper Pell equation: x^2 - D y^2 = 1, D > 0, not square
• Trivial solution: (x, y) = (1, 0). 
• Fundamental solution: the smallest non trivial solution.
• Brahmagupta–Fibonacci identity
• Uniqueness of fundamental solution

• Generalized Pell equation: x^2 - D y^2 = N
• Equivalent fundamental solutions
• A set of fundamental solutions

2. Algorithm:

• Online solver: http://www.numbertheory.org/php/main_pell.html
• PQA algorithm 
• LMM algorithm: N^2 > D.

3. Reference:

• http://www.jpr2718.org/pell.pdf
• http://www.numbertheory.org/pdfs/talk_2004.pdf
• http://mathafou.free.fr/themes_en/kpell.html