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2 results

ALTERNATING CIRCULAR SUMS OF BINOMIAL COEFFICIENTS
Part of
- Combinatorics
- Sequences and sets
WENCHANG CHU
Journal:

Bulletin of the Australian Mathematical Society / Volume 106 / Issue 3 / December 2022

Published online by Cambridge University Press:

13 April 2022, pp. 385-395

Print publication:

December 2022
- Article
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By combining the generating function approach with the Lagrange expansion formula, we evaluate, in closed form, two multiple alternating sums of binomial coefficients, which can be regarded as alternating counterparts of the circular sum evaluation discovered by Carlitz [‘The characteristic polynomial of a certain matrix of binomial coefficients’, Fibonacci Quart. 3(2) (1965), 81–89].

PERFECT POWERS THAT ARE SUMS OF TWO POWERS OF FIBONACCI NUMBERS
Part of
- Diophantine equations
ZHONGFENG ZHANG, ALAIN TOGBÉ
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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In this paper, we consider the Diophantine equations
$$\begin{eqnarray}\displaystyle F_{n}^{q}\pm F_{m}^{q}=y^{p} & & \displaystyle \nonumber\end{eqnarray}$$
with positive integers $q,p\geq 2$ 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 $q=3$.