Solving Diophantine Equations via Lucas-Lehmer Theory
نویسنده
چکیده
In this work we look at an approach to solving Pell’s equation using continued fractions and fundamental units in real quadratic orders. We demonstrate that there is an underlying general approach using Lucas-Lehmer methods for solving Pell and other quadratic Diophantine equations that is often overlooked in the literature. Mathematics Subject Classification: Primary: 11D09; 11A55; Secondary: 11R11; 11R29
منابع مشابه
Diophantine Equations Related with Linear Binary Recurrences
In this paper we find all solutions of four kinds of the Diophantine equations begin{equation*} ~x^{2}pm V_{t}xy-y^{2}pm x=0text{ and}~x^{2}pm V_{t}xy-y^{2}pm y=0, end{equation*}% for an odd number $t$, and, begin{equation*} ~x^{2}pm V_{t}xy+y^{2}-x=0text{ and}text{ }x^{2}pm V_{t}xy+y^{2}-y=0, end{equation*}% for an even number $t$, where $V_{n}$ is a generalized Lucas number. This pape...
متن کاملInteger Solutions of Some Diophantine Equations via Fibonacci and Lucas Numbers
We study the problem of finding all integer solutions of the Diophantine equations x2 − 5Fnxy − 5 (−1) y2 = ±Ln, x2 − Lnxy + (−1) y2 = ±5F 2 n , and x2 − Lnxy + (−1) y2 = ±F 2 n . Using these equations, we also explore all integer solutions of some other Diophantine equations.
متن کاملTernary Diophantine Equations via Galois Representations and Modular Forms
In this paper, we develop techniques for solving ternary Diophantine equations of the shape Axn + Byn = Cz2 , based upon the theory of Galois representations and modular forms. We subsequently utilize these methods to completely solve such equations for various choices of the parameters A, B and C . We conclude with an application of our results to certain classical polynomial-exponential equat...
متن کاملOn the Diophantine Equation
In this paper, we study the Diophantine equation x2 + C = 2yn in positive integers x, y with gcd(x, y) = 1, where n ≥ 3 and C is a positive integer. If C ≡ 1 (mod 4) we give a very sharp bound for prime values of the exponent n; our main tool here is the result on existence of primitive divisors in Lehmer sequence due Bilu, Hanrot and Voutier. When C 6≡ 1 (mod 4) we explain how the equation can...
متن کاملOn integral points on biquadratic curves and near multiples of squares in Lucas sequences
We describe an algorithmic reduction of the search for integral points on a curve y2 = ax4 + bx2 + c with ac(b2 − 4ac) 6= 0 to solving a finite number of Thue equations. While existence of such reduction is anticipated from arguments of algebraic number theory, our algorithm is elementary and to best of our knowledge is the first published algorithm of this kind. In combination with other metho...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
دوره شماره
صفحات -
تاریخ انتشار 2010