[math-fun] order of 3 in Mersenne-prime group

Math-funners: I observe that, for many Mersenne primes Mp > 3, the co-order* of 3 (in Mp's multiplicative group) is a power of 3. Indeed, it's true for any such Mp-group, whose order (Mp - 1) has been fully factored. Does anyone know a related theorem? *co-order: group order divided by element order -- Don Reble djr@nk.ca -- This message has been scanned for viruses and dangerous content by MailScanner, and is believed to be clean.