Practical solution of the Diophantine equation  $y^2 = x(x+2^ap^b)(x-2^ap^b)$

Draziotis, Konstantinos A.; Poulakis, Dimitrios

doi:10.1090/s0025-5718-06-01841-2

Cited by 6 publications

(14 citation statements)

References 19 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…It follows that 4M 4 − pN 2 = 1. By Theorem 4, we have (p, z) = (3, 2), (7,8). If p | z and 2 z, then z = pM 2 and z 2 − 1 = 2N 2 , where M, N ∈ Z, and so p 2 M 4 − 2N 2 = 1.…”

Section: Moreover the Index Form Equation I Kmentioning

confidence: 95%

“…In this case, we developed in [7], an algorithm for the calculation of the integer solutions of (1). Here, we give an alternative method using the above ideas.…”

Section: Moreover the Index Form Equation I Kmentioning

confidence: 99%

“…By Theorem 5 and Proposition 2, we have Proof. This equation has been solved already in [7]. We shall solve it using Proposition 6.…”

Section: Moreover the Index Form Equation I Kmentioning

confidence: 99%

“…Thus, we cannot deduce that the associated index form equation has no solution and so, we continue to the next steps of our algorithm. We use the computational system KANT 2.5 and we obtain the following integral basis for the field K : [10,9,15,7]), ([2, 3 where θ = √ 3 + √ 5. We represent an algebraic integer of K , z = 3 (2,5,11,1), (11, 10, 1, 1), (11, 5, 2, 0) and (10, 11, 1, 0).…”

Section: Remark 2 In Step 6 Ifmentioning

confidence: 99%

“…When n = 2 a p b where p is an odd prime and a > 0, b > 0, an algorithm is given in [7] for the calculation of the integer solutions of (1) which does not use the structure of the group E n (Q). The aim of this paper is to generalize the approach of [7] and develop an algorithm for the determination of the integer solutions of (1), for any n. We use the "multiplication by 2" on E n and we reduce the problem of the determination of integer solutions of (1) to the solution of a finite number of unit equations over some bicyclic biquadratic fields.…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

Solving the Diophantine equationy2=x(x2−n2)

Draziotis¹,

Poulakis²

2009

Journal of Number Theory

Self Cite

View full text Add to dashboard Cite

show abstract

“…It follows that 4M 4 − pN 2 = 1. By Theorem 4, we have (p, z) = (3, 2), (7,8). If p | z and 2 z, then z = pM 2 and z 2 − 1 = 2N 2 , where M, N ∈ Z, and so p 2 M 4 − 2N 2 = 1.…”

Section: Moreover the Index Form Equation I Kmentioning

confidence: 95%

“…In this case, we developed in [7], an algorithm for the calculation of the integer solutions of (1). Here, we give an alternative method using the above ideas.…”

Section: Moreover the Index Form Equation I Kmentioning

confidence: 99%

“…By Theorem 5 and Proposition 2, we have Proof. This equation has been solved already in [7]. We shall solve it using Proposition 6.…”

Section: Moreover the Index Form Equation I Kmentioning

confidence: 99%

Section: Remark 2 In Step 6 Ifmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Solving the Diophantine equationy2=x(x2−n2)

Draziotis¹,

Poulakis²

2009

Journal of Number Theory

Self Cite

View full text Add to dashboard Cite

show abstract

Perfect powers in elliptic divisibility sequences

Reynolds

2012

Journal of Number Theory

View full text Add to dashboard Cite

It is shown that there are finitely many perfect powers in an elliptic divisibility sequence whose first term is divisible by 2 or 3. For Mordell curves the same conclusion is shown to hold if the first term is greater than 1. Examples of Mordell curves and families of congruent number curves are given with corresponding elliptic divisibility sequences having no perfect power terms. The proofs combine primitive divisor results with modular methods for Diophantine equations. 1 2 JONATHAN REYNOLDS singular point and so not define an elliptic curve. Via the work of Everest [21, 23], Ingram [29], Silverman [31, 41] et al, it has now become conventional to use the following Definition 1.1. Let E/Q be an elliptic curve and (1.1) a Weierstrass equation for E. Let P ∈ E(Q) be a non-torsion point. For m ∈ N denote B mP , as in (1.2), by B m . The sequence (B m ) is an elliptic divisibility sequence.In the current paper, we are interested in analogues for elliptic divisibility sequences (in the sense of Definition 1.1) to the result for Fibonacci numbers. There are certainly perfect powers in some elliptic divisibility sequences. For example, 3) gives B m = 1 for m = 1, 2, 3, 4, 7 and B 12 = 2 7 . However, the following theorem shows that one can often prove that there are only finitely many perfect powers in such sequences.Theorem 1.2. Let (B m ) be an elliptic divisibility sequence whose first term is divisible by 2 or 3. There are finitely many perfect powers in (B m ). Moreover, if B m = z l for some integer z and prime l then l can be bounded explicitly in terms of E and P.The proof of Theorem 1.2 combines a recent Frey-Hellegouarch construction for Klein forms by Bennett and Dahmen [3] with a primitive divisor result due to Silverman [41]. The method of proof is so flexible that it also allows one in certain concrete cases to completely determine the set of all perfect power terms, as was done for the Fibonacci sequence (see Proposition 1.5 and Example 1.9 below). The condition that only 2 or 3 divides the first term is because higher primes, such as 5, do not give a Klein form as in Defintion 3.7.Let E/Q be an elliptic curve and (1.1) a Weierstrass equation for E. Siegel [40] proved that there are finitely many P ∈ E(Q) with B P = 1. In [24] it is shown that for fixed l > 1, there are finitely many P ∈ E(Q) with B P = z l for some z ∈ Z. Since their denominator is a perfect power, perhaps it is reasonable to give the following Definition 1.3. Call P ∈ E(Q) power integral if B P (as in (1.2)) is equal to a perfect power.Note that 1 is a perfect power and so power integral points can be thought of as a generalization of the integral points. A lot of work has been done to make Siegel's theorem effective [2,11,25,28,34] and there are many techniques which can find all of the integral points for large classes of elliptic curves [26,35,46, 47]. For certain curves we are now able to find all of the power integral points.1.1. Mordell curves. Theorem 1.2 can be strengthened considerably for Mordell curves.Theorem 1.4. Let D ...

show abstract

On certain Diophantine equations of the form z2=f(x)2±g(y)2

Tengely

Ulas

2017

Journal of Number Theory

View full text Add to dashboard Cite

Practical solution of the Diophantine equation $y^2 = x(x+2^ap^b)(x-2^ap^b)$

Cited by 6 publications

References 19 publications

Solving the Diophantine equationy2=x(x2−n2)

Solving the Diophantine equationy2=x(x2−n2)

Perfect powers in elliptic divisibility sequences

On certain Diophantine equations of the form z2=f(x)2±g(y)2

Contact Info

Product

Resources

About