The conjecture is wrong.
David BroadHurst has found counter-examples.
The terrible "law of small numbers" has struck again...




(but the numbers were not so small...).
I've updated the paper and just conjectured that the highest power of 3 that divides the order of 3 mod M_q is 2. But it is not so much interesting...
Never mind, we learn by knowing what's false too.
I've updated the
paper.
Sorry, the way David found the counter-examples was not so difficult...
Tony