# The primitive solutions to x3+y9=z2

```@article{Bruin2005ThePS,
title={The primitive solutions to x3+y9=z2},
author={Nils Bruin},
journal={Journal of Number Theory},
year={2005},
volume={111},
pages={179-189}
}```
• N. Bruin
• Published 31 October 2003
• Mathematics
• Journal of Number Theory
17 Citations

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• 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

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### Generalized Fermat equations: A miscellany

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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

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\$

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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

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TURKISH JOURNAL OF MATHEMATICS
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### 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

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### 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