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SOME QUARTIC DIOPHANTINE EQUATIONS IN THE GAUSSIAN INTEGERS
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- Journal:
- Bulletin of the Australian Mathematical Society / Volume 92 / Issue 2 / October 2015
- Published online by Cambridge University Press:
- 16 June 2015, pp. 187-194
- Print publication:
- October 2015
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In this paper we examine solutions in the Gaussian integers to the Diophantine equation
$ax^{4}+by^{4}=cz^{2}$ for different choices of 
$a,b$ and 
$c$. Elliptic curve methods are used to show that these equations have a finite number of solutions or have no solution.
A problem of Ozanam
- Part of
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- Journal:
- Proceedings of the Edinburgh Mathematical Society / Volume 52 / Issue 1 / February 2009
- Published online by Cambridge University Press:
- 02 February 2009, pp. 37-44
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RATIONAL POINTS ON CERTAIN DEL PEZZO SURFACES OF DEGREE ONE
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- Journal:
- Glasgow Mathematical Journal / Volume 50 / Issue 3 / September 2008
- Published online by Cambridge University Press:
- 01 September 2008, pp. 557-564
- Print publication:
- September 2008
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Let
and let us consider a del Pezzo surface of degree one given by the equation 
. In this paper we prove that if the set of rational points on the curve Ea,b : Y2 = X3 + 135(2a−15)X−1350(5a + 2b − 26) is infinite then the set of rational points on the surface ϵf is dense in the Zariski topology.
and let us consider a del Pezzo surface of degree one given by the equation 
. In this paper we prove that if the set of rational points on the curve 