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$\DeclareMathOperator{\Ord}{Ord}$When does $q^b = (p^a - 1)/(p - 1)$ for $p, q$ odd primes and $a, b$ odd integers $> 1$ ? If no examples are possible, please give a simple proof.

A proof of this for $q=3, p=5$ might be

Assume $3^b = (5^a - 1)/4$; then \begin{equation} 5^a - 4 \cdot 3^b = 1. \tag{1} \end{equation} We show that the exponents $a,b$ in (1) are even.

We see that $3^b \mid 5^a - 1$ $\Rightarrow$ $5^a \equiv 1 \bmod 3$. Also $5 \mid 4\cdot3^b+1$ or $4 \cdot 3^b \equiv -1 \bmod 5$ so that $3^b \equiv 1 \bmod 5$.

In short: \[ 3^b\equiv1 \bmod 5, \quad 5^a\equiv1 \bmod 3. \]

By a well known theorem: if $X^c \equiv 1 \bmod p$, then $\Ord(c,p) \mid c$. By inspection, $\Ord(3,5) = 4$ so that $4 \mid b$. Similarly, $\Ord(5,3) = 2$ so that $2 \mid a$.

This violates our assumption that $a,b$ are odd. Proof complete.

However, moving further:

As $a,b$ are both even we can write (1) as $1 = 5^{2A} - 4\cdot 3^{2B} = (5 - 2\cdot3^B)(5 + 2\cdot3^B)$. Thus, $5 + 2\cdot3^B = 1$ but there is no positive value of $B$ satisfying this. Therefore (1) is not true.

Note: I want to be clear that the above proof, if it is correct, is not mine but was given to me by someone I am not free to name. If it is incorrect, the fault is entirely mine.

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    I TeX'ed the question because I wanted to read it. If you plan to ask more questions here, it would be worth learning the (basic!) commands involved in typesetting something like this. Take a look at the new source, and let me know if you have any questions.2012-06-20
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    Mr. Moreland. I appreciate your effort very much. It looks GREAT! I have tried to LaTex my questions but it ends up totally scrambled. It may be due to my Web browser which is very restricted by my work. Or, it could just be incompetence on my part. You are not the first to complain so I will probably just stop bothering folks with my posts. I'm not very good at math anyway. Just curious about it. Again. thanks so much2012-06-20
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    @Nick: LaTeX becomes easy with a little practice. Just try putting simple equations between dollar signs, they immediately become nicer. (Also the preview function is very helpful at showing you how it will look.) If you don't know how to create special symbols, you can look at the source of any equation on the site by right-clicking on it and choosing "Show Math As"/"TeX Commands". (At least that's how it seems to work with Firefox.) You will probably get a feel for it very quickly.2012-06-20
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    @Nick I can't speak for everyone, but I'd rather that you continued to post your questions.2012-06-20
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    @Nick: I forgot to mention this in the above comment, but I too would love to see more of your questions. At least to me, they seem quite interesting.2012-06-20
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    Nick, please don't stop posting. It's better to have the question unLaTeXed, than to have no question at all. There are loads of people around here who will be happy to tidy up the formatting of your questions, as Dylan did on this occasion.2012-06-20

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