Simultaneous Pellian equations with a single or no solution
by
Alain Togbe
Purdue University North Central
Coauthors: Bo He

Let m and b > 1 be positive integers with b not a perfect square, d = ±1, ±2, ±4. In this talk, we will discuss how we show that the system of simultaneous Pellian equations

(m+d)x2 - my2 = d,     y2 - bz2 = 1

has at most one positive integer solution (x, y, z), extending a result due to Li, Xia and Yuan. Then we generalize a theorem of Walsh on the equations x²-dy²=1, z²-2dy²=1, obtained in 1997. Moreover, we will consider the particular case b=b'|4m/d+ 4 |,   b' ∈ {1, 2, p}, where p is an odd prime and give all the solutions.

Date received: June 11, 2008