## 17 Citations

### On Solutions of the Equations x 2 ± y 8 = ±z 6

- Mathematics
- 2012

In this article, we determine all solutions to the equation x a +y b = z c , (a,b,c) ∈{ (2,8,6),(2,6,8),(8,6,2)} in coprime integers x,y,z. In this case we get a set of curves by doing some…

### Explicit Chabauty over number fields

- Mathematics
- 2010

Let C be a smooth projective absolutely irreducible curve of genus g ≥ 2 over a number field K of degree d, and denote its Jacobian by J . Denote the Mordell–Weil rank of J(K) by r. We give an…

### Generalized Fermat equations: A miscellany

- Mathematics
- 2015

This paper is devoted to the generalized Fermat equation xp + yq = zr, where p, q and r are integers, and x, y and z are nonzero coprime integers. We begin by surveying the exponent triples (p, q,…

### A refined modular approach to the Diophantine equation $x^2+y^{2n}=z^3$

- Mathematics
- 2010

Let $n$ be a positive integer and consider the Diophantine equation of generalized Fermat type $x^2+y^{2n}=z^3$ in nonzero coprime integer unknowns $x,y,z$. Using methods of modular forms and Galois…

### On Some Diophantine Equations of the Form hX+Y= Z

- Mathematics
- 2017

We show the impossibility of primitive non-zero solutions to the title equation if h = 3 and n ∈ {3, 4, 5, 6} and if h ∈ {11, 19, 43, 67, 163} and n ∈ {4, 5, }. The proofs are based solely on…

### The generalized Fermat equation with exponents 2, 3, $n$

- MathematicsCompositio Mathematica
- 2019

We study the generalized Fermat equation $x^{2}+y^{3}=z^{p}$ , to be solved in coprime integers, where $p\geqslant 7$ is prime. Modularity and level-lowering techniques reduce the problem to the…

### On ternary Diophantine equations of signature(p,p,2)over number fields

- MathematicsTURKISH JOURNAL OF MATHEMATICS
- 2020

Let $K$ be a totally real number field with narrow class number one and $O_K$ be its ring of integers. We prove that there is a constant $B_K$ depending only on $K$ such that for any prime exponent…

### The generalised Fermat equation x2 + y3 = z15

- Mathematics
- 2013

We determine the set of primitive integral solutions to the generalised Fermat equation x2 + y3 = z15. As expected, the only solutions are the trivial ones with xyz = 0 and the non-trivial one (x, y,…

### On The Diophantine Equation x y z

- Mathematics
- 2011

In this article, we proved that an integral solution (a, b, c) to the Diophantine equation x y z 3 3 2 + = is of the , , a rs b rt c r m 3 1 2 = = =^ h for any two positive integers s, t.

### Lucas sequences whose 8th term is a square

- Mathematics
- 2004

Let P and Q be non-zero integers. The Lucas sequence U_n(P,Q), n=0,1,2,... is defined by U_0=0, U_1=1, U_n= P U_{n-1}-Q U_{n-2} for n>1. For each positive integer n<8 we describe all Lucas sequences…

## References

SHOWING 1-10 OF 20 REFERENCES

### The Diophantine Equations x2± y4=±z6 and x2+y8= z3

- MathematicsCompositio Mathematica
- 1999

In this article we determine all solutions to the equationxp+yq=zr, (p,q,r)∈(2,4,6), (2,6,4), (4,6,2), (2,8,3) in coprime integers x,y,z. First we determine a set of curves of genus 2, such that…

### The Diophantine Equations x 2 ± y 4 = ± z 6 and x 2 + y 8 = z 3

- Mathematics
- 1999

In this article we determine all solutions to the equation xp + yq = zr , (p, q, r) ∈ {(2,4,6), (2,6, 4), (4,6, 2), (2, 8,3)} in coprime integersx, y, z. First we determine a set of curves of genus…

### On the Equations Z

- Mathematics
- 1995

We investigate integer solutions of the superelliptic equation (1) z = F (x, y), where F is a homogenous polynomial with integer coefficients, and of the generalized Fermat equation (2) Ax + By = Cz,…

### The arithmetic of elliptic curves

- Mathematics, Computer ScienceGraduate texts in mathematics
- 1986

It is shown here how Elliptic Curves over Finite Fields, Local Fields, and Global Fields affect the geometry of the elliptic curves.

### Diophantine Approximations and Diophantine Equations

- Mathematics
- 1991

Siegel's lemma and heights.- Diophantine approximation.- The thue equation.- S-unit equations and hyperelliptic equations.- Diophantine equations in more than two variables.