I have a question about very early argument in the proof of Thereom 1.1.
Theorem 1.1 of On Some Exponential Equations of S.S. Pillai states that if $a,b,c$ are nonzero integers with $a,b \ge 2$, then the equation:
$$a^x - b^y = c$$
has at most two solutions in positive integers $x$ and $y$.
The first step in the proof is to assume that three solutions ($x_i,y_i$) exist in positive integers where:
$$x_1 < x_2 < x_3$$ and $$y_1 < y_2 < y_3$$
The part that I am having trouble understanding relates to what should be a simple argument by contradiction:
- Assume that gcd($a,b$) $> 1$
- Then, there exists a prime $p$ such that $p \mid a$ and $p \mid b$
Now here's is the step that I am misunderstanding:
- The claim is made that $p$ has ord$_p{a} = \alpha \ge 1$ and $p$ has ord$_p{b} = \beta \ge 1$ since:
$$a^{x_i}(a^{x_{i+1}-x_i} - 1) = b^{y_i}(b^{y_{i+1}-y_i}-1)$$
But I thought that ord only applied when gcd($p,a$)$=1$ and gcd($p,b$)$=1$. So, I am confused by the statement.
Then, based on the above, it is concluded for $i=1,2$ that:
$$\alpha{x_i} = \beta{y_i}$$
If someone could explain what ord means in this circumstance and how it leads to the conlusion of $\alpha{x_i} = \beta{y_i}$, I would greatly appreciate it.
Thanks,
-Larry
This is the other most common meaning, the $p$-adic valuation. It, $\operatorname{ord}_p a,$ is just the highest exponent, $e,$ of $p$ such that $$ p^e | a.$$ In Gouvea's book this is written $\nu_p a,$ in other books $v_p a,$ the letter $v$ being convenient for the English word valuation. Alright, holding Gouvea's book, page 25, he also uses letter $v.$