New assistance for solving the formula axx + 1 = yy
This describes Euler's solution of the Pell equation without the use of continued fractions. It opens with a big one: if a=61, then x=226,153,980 and y=1,766,319,049. Euler includes the following problems: Knowing a solution to app−1=qq, find numbers x and y that make axx+1=yy; i.e., find a relationship between the positive equation and the negative equation. Then, knowing app−2=qq, solve axx+1=yy.
Original Source Citation
Opuscula analytica, Volume 1, pp. 329-344.
Opera Omnia Citation
Series 1, Volume 4, pp.76-90.