Lebesgue–nagell Equation
نویسندگان
چکیده
We discuss the feasibility of an elementary solution to the Diophantine equation of the 15 form x 2 + D = y n , where D > 1, n ≥ 3 and x > 0, called the Lebesgue–Nagell equation, which has recently been solved for 1 ≤ D ≤ 100 in [1].
منابع مشابه
On generalized Lebesgue-Ramanujan-Nagell equations
We give a brief survey on some classical and recent results concerning the generalized Lebesgue-Ramanujan-Nagell equation. Moreover, we solve completely the equation x + 1117 = y in nonnegative integer unknowns with n ≥ 3 and gcd(x, y) = 1. 1 Generalized Ramanujan-Nagell equations Mixed polynomial-exponential equations are of classical and recent interest. One of the most famous equation of thi...
متن کاملClassical and Modular Approaches to Exponential Diophantine Equations Ii. the Lebesgue–nagell Equation
This is the second in a series of papers where we combine the classical approach to exponential Diophantine equations (linear forms in logarithms, Thue equations, etc.) with a modular approach based on some of the ideas of the proof of Fermat’s Last Theorem. In this paper we use a general and powerful new lower bound for linear forms in three logarithms, together with a combination of classical...
متن کاملSolutions of some generalized Ramanujan – Nagell equations
in positive integers x, y,D and n > 2 with gcd(x, y)= 1. When D = 1, the equation has no solution by an old result of Lebesgue [14]. We assume from now on that D > 1. Eq. (1) has been extensively studied by many authors, in particular, by Cohn and Le. See [8,10–13] for several results. We also refer to [8] for a survey. The equation is referred as the generalized Ramanujan–Nagell equation who p...
متن کاملRamanujan-nagell Cubics
A well-known result of Beukers [3] on the generalized Ramanujan-Nagell equation has, at its heart, a lower bound on the quantity |x2 − 2n|. In this paper, we derive an inequality of the shape |x3 − 2n| ≥ x4/3, valid provided x3 6= 2n and (x, n) 6= (5, 7), and then discuss its implications for a variety of Diophantine problems.
متن کاملProducts of Prime Powers in Binary Recurrence Sequences Part I: The Hyperbolic Case, with an Application to the Generalized Ramanujan-Nagell Equation
We show how the Gelfond-Baker theory and diophantine approximation techniques can be applied to solve explicitly the diophantine equation C„ = wpTM< ■ ■ ■ p"< (where {G„}^_o is a binary recurrence sequence with positive discriminant), for arbitrary values of the parameters. We apply this to the equation x2 + k = p\' ■ ■ ■ pf', which is a generalization of the Ramanujan-Nagell equation x2 + 7 = ...
متن کامل