Lethbridge Number Theory and Combinatorics Seminar: Amir Akbary

It is known that if the negative Pell equation X2−DY2=−1 is solvable (in integers), and if (x,y) is its solution with smallest positive x and y, then all of its solutions (xn,yn) are given by the formula

          xn + yn√D = ±(x + y√D) 2n+1

for n∈Z. Furthermore, a theorem of Walker from 1967 states that if the equation aX2−bY2=±1 is solvable, and if (x,y) is its solution with smallest positive x and y, then all of its solutions (xn,yn) are given by

          xn√a + yn√b = ±(x√a + y√b)2n+1

for n∈Z. We describe a unifying theorem that includes both of these results as special cases. The key observation is a structural theorem for the non-trivial ambiguous classes of the solutions of Pell equations X2−DY2=±N. This talk is based on the work of Forrest Francis in an NSERC USRA project in summer 2015.