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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

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  • $\begingroup$ Larry, could you please email me something, the gmail address is most convenient, see ams.org/cml if you don't have it. I cannot seem to find an address for you. The short version is that Gerhard would like to get in touch with you. $\endgroup$ – Will Jagy Jul 9 '14 at 21:18
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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 $v.$

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  • $\begingroup$ Thanks very much! I thought that there had to be another meaning of the term. $\endgroup$ – Larry Freeman Apr 7 '14 at 3:03

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