Let D1 and D2 be coprime positive integers and let k be an odd positive integer coprime with D1D2. We consider the Diophantine equation D1x2 + D2 = kn in the unknowns x≥1, n≥1. We give a necessary and sufficient condition on D1, D2 and k under which this equation has at most 2ω(k)−1 solutions where ω(k) denoted the number of distinct prime divisors of k. Thus, under a necessary and sufficient conditon, the equation has at most one solution whenever k is a prime. We also consider some related equations and we prove that the Diophantine equation x2+7=4yn has no solution in integers x≥1, y > 2 and n >1.

On the number of solutions of the generalized Ramanujan-Nagell equation

In this paper, we refine work of Beukers, applying results from the theory of Pade approximation to (1 − z)1/2 to the problem of restricted rational approximation to quadratic irrationals. As a result, we derive effective lower bounds for rational approximation to $\sqrt m$ (where m is a positive nonsquare integer) by rationals of certain types. Forexample, we have

$$\left| {\sqrt 2 - \frac{p}{q}} \right| \gg q^{ - 1.47} {\text{ and }}\left| {\sqrt 3 - \frac{p}{q}} \right| \gg q^{ - 1.62}$$

provided q is a power of 2 or 3, respectively. We then use this approach to obtain sharp bounds for the number of solutions to certain families of polynomial-exponential Diophantine equations. In particular, we answer a question of Beukers on the maximal number of solutions of the equation x2 + D = pn where D is a nonzero integer and p is an odd rational prime, coprime to D.

/pdf/applications-of-the-hypergeometric-method-to-the-generalized-2yhx09ebhg.pdf

Applications of the Hypergeometric Method to the Generalized Ramanujan-Nagell Equation

The generalized Lebesgue-Ramanujan-Nagell equation is an important type of polynomial-exponential Diophantine equation in number theory. In this survey, the recent results and some unsolved problems of this equation are given.

A brief survey on the generalized Lebesgue-Ramanujan-Nagell equation

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 2 + 11 a 17 b = y n in nonnegative integer unknowns with n 3 and gcd(x;y) = 1.

/pdf/on-generalized-lebesgue-ramanujan-nagell-equations-3fbi4wlt9j.pdf

On generalized Lebesgue-Ramanujan-Nagell equations

On the Generalized Ramanujan–Nagell Equation x2+D=pz

Let D be a positive integer, and let p be an odd prime with p|D. In this Note we prove that if max (D,p≥10 10193 and (D,p)¬=(3s 2 +1,4s 2 +1), then the equation x 2 +D=p n has at most one positive integer solution (x, n)

On the number of solutions of the diophantine equation x2+D=pn

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On the number of solutions of the diophantine equation x2+D=pn