I want to prove that $5^{2n}-2^{3n}$ is divisible by 17 for all positive integers $n$. I know this can be done by induction (sketch proof shown below) but want to know if:
Are there any alternative proof methods that do not use induction?
How many different induction proof approaches are possible for a question such as this? (By different I mean different groupings of terms or perhaps adding and subtracting a new term).
$P(1): 5^{2}-2^{3}=17$ which is divisible by 17, so $P(1)$ is true.
Now, we need to show that the truth of $P(k)$ implies the truth of $P(k+1)$ which means showing that if $5^{2k}-2^{3l}$ is divisible by 17 for some positive integer $k$ then $5^{2(k+1)}-2^{3(k+1)}$ is also divisible by 17.
$5^{2(k+1)}-2^{3(k+1)}=25\times 5^{2k}-8\times 2^{3k}$
$=17\times 5^{2k}+8\times 5^{2k}-8\times 2^{3k}$
$=17\times 5^{2k}+8(5^{2k}-2^{3k})$
Then by $P(1)$, $5^{2k}-2^{3k}$ is divisible by 17 so we can write $5^{2k}-2^{3k}=17s$
$P(k+1)$ then becomes $17\times 5^{2k}+8\times 17s$ from which the result follows.
What other approaches (if any) are possible and how are they different?
