Continued fractions expansion of √D and Pell equation x2 - Dy2 = 1

Abstract

Abstract. Let D = 1 be a positive non-square integer. In the first section, we give some preliminaries from Pell equations and simple continued fraction expansion. In the second section, we give a formula for the continued fraction expansion of √ D for some specific values of D and then we consider the integer solutions of Pell equations x 2 −Dy 2 = 1 for these values of D including recurrence relations on the integer solutions of it.

On the positive integral solutions to the Diophantine equation of n-variables (x_1+x_2+x_3+⋯+x_n )^2 = x_1 x_2 x_3…x_n

“…There are eight (a 1 , a 2 , a 3 , a 4 ) up to permutations satisfying the above conditions: By using the formula (15) we obtain the following sequences of solutions to equation (18):…”

“…Integer solutions to equation of three or more variables are given in various parametric forms see [4] the reference sited there in. A special class of Diophantine Equation known as Pell's equation is an active of area of research for long time and many researchers have been investigated the solvability problem through many different methods see [3], [5]- [9], [11]- [18], the reference sited there in. In [10], it is proved that the Diophantine equation x + y + z = xyz has solutions to the quadratic field Q( √ d) if and only if d = −1, 2 or 5 and in these cases all solutions are given.…”

“…The following table contains the general Pell's equation(18) corresponding to the above (a 1 , a 2 , a 3 , a 4 ), their Pell's resolvent, both equations with their fundamental solutions. , a 2 , a 3 , a 4 ) General Pell's Equation(18) and Pell's resolvent equation and…”

Simultaneous Pell equations

“…They also deduced several relations on the integer solutions . Tekcan [2] provided a formula for the continued fraction expansion of for some specific values of with and then considered the integer solutions of the Pell equation . Abu Muriefah and Al Rashed [4] proved that set {1, 5, 442} cannot be extended and it is equivalent to solve the system of Pell equations and .…”