2 results
THE DIOPHANTINE EQUATION
$\boldsymbol{x}^{\boldsymbol{4}} \boldsymbol{+} \boldsymbol{2}^{\boldsymbol{n}}\boldsymbol{y}^{\boldsymbol{4}} \boldsymbol{=} \boldsymbol{1}$ IN QUADRATIC NUMBER FIELDS
- Part of
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- Journal:
- Bulletin of the Australian Mathematical Society / Volume 104 / Issue 1 / August 2021
- Published online by Cambridge University Press:
- 06 November 2020, pp. 21-28
- Print publication:
- August 2021
-
- Article
- Export citation
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Aigner showed in 1934 that nontrivial quadratic solutions to
$x^4 + y^4 = 1$
exist only in
$\mathbb Q(\sqrt {-7})$
. Following a method of Mordell, we show that nontrivial quadratic solutions to
$x^4 + 2^ny^4 = 1$
arise from integer solutions to the equations
$X^4 \pm 2^nY^4 = Z^2$
investigated in 1853 by V. A. Lebesgue.
SOME QUARTIC DIOPHANTINE EQUATIONS IN THE GAUSSIAN INTEGERS
- Part of
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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
-
- Article
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- You have access
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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.
