The Horadam sequence $\{W_n\}$ is defined by the Binet formula [Using notation from here]:
$$W_n=\frac{A\alpha^n-B\beta^n}{\alpha-\beta}$$
where,
$$A\ =\ b\ -\ a\beta \text{ and } B\ =\ b\ -a\alpha$$
$$\alpha = {p + d \over 2}, \beta = {p - d \over 2}, d = \sqrt {p^2 - 4q}$$
Questions:
Take all $a,b,\alpha,\beta \in \mathbb{Z}$.
- Does $W_n$ ever equal a perfect power? Since the Horadam sequence is a generalization of Fibonnacci, Lucas sequences, there are a finite set of perfect power values assumed by $W_n$. Can we say anything about perfect powers in general for $A, B, \alpha, \beta$ that defines $W_n$?
- If we take $A = \alpha, B = \beta$ and $GCD(\alpha, \beta) = 1$, is $W_n$ ever a power of $\alpha - \beta$? [Note: This is a special case of Beal's conjecture. So, the answer is likely to be $W_n$ is never a power of $\alpha - \beta$.]
For the second question, I tried $\beta = 1$ and set $W_n = y^m$ and obtained the Nagell-Ljunggren equation:
$$y^m = {\alpha^{n+1} - \beta^{n+1} \over \alpha - \beta} = {\alpha^{n+1} - 1 \over \alpha - 1}$$
with solutions $(\alpha, n+1, y, m) \in \{(3,5,11,2), (7,4,20,2), (18,3,7,3)\}$
What can we say about $\alpha \ne 1, \beta \ne 1$?
If $A=\alpha$ and $B=\beta$ then $$W_n=\frac{\alpha^{n+1}-\beta^{n+1}}{\alpha-\beta}=\sum_{i=0}^n\alpha^i\beta^{n-i}.$$ In particular, for $n=1$ this becomes $W_1=\alpha+\beta$, which is clearly a perfect power for plenty of choices of $\alpha$ and $\beta$. For $n=2$ this becomes $$W_2=\alpha^2+\alpha\beta+\beta^2,$$ and for example for $\alpha=8$ and $\beta=7$ this becomes $$W_2=\alpha^2+\alpha\beta+\beta^2=8^2+8\times7+7^2=13^2.$$