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ON FINDING SOLUTIONS TO EXPONENTIAL CONGRUENCES
- Part of
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- Journal:
- Bulletin of the Australian Mathematical Society / Volume 99 / Issue 3 / June 2019
- Published online by Cambridge University Press:
- 27 December 2018, pp. 388-391
- Print publication:
- June 2019
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- Article
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We improve some previously known deterministic algorithms for finding integer solutions
$x,y$ to the exponential equation of the form 
$af^{x}+bg^{y}=c$ over finite fields.
PERFECT POWERS THAT ARE SUMS OF TWO POWERS OF FIBONACCI NUMBERS
- Part of
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- Journal:
- Bulletin of the Australian Mathematical Society / Volume 99 / Issue 1 / February 2019
- Published online by Cambridge University Press:
- 30 August 2018, pp. 34-41
- Print publication:
- February 2019
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- Article
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- You have access
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In this paper, we consider the Diophantine equations
with positive integers
$$\begin{eqnarray}\displaystyle F_{n}^{q}\pm F_{m}^{q}=y^{p} & & \displaystyle \nonumber\end{eqnarray}$$
$q,p\geq 2$ and 
$\gcd (F_{n},F_{m})=1$, where 
$F_{k}$ is a Fibonacci number. We obtain results for 
$q=2$ or 
$q$ an odd prime with 
$q\equiv 3\;(\text{mod}\;4),3<q<1087$, and complete solutions for 
$q=3$.
