Merely expressing an opinion as to whether or not you think there are an
infinite or finite number of Mersenne primes doesn't add anything to the
discussion unless you can furnish some argument one way or the other. As
with many issues in pure mathematics, it is unlikely (but not impossible)
that a proof will be found one way or the other.
I acknowledge Euclid's proof of an infinity of primes, cited by Michael
Clark, but do not see any compelling evidence in this pointing towards an
infinity of Mersenne primes. The rarity of these numbers is part of what
leads me to suppose that they are finite in number, but I concede that
numerical evidence does not carry much weight. For example, if I were to
claim that all integers are below 2 raised to the power of M37, I could
exhibit a huge number of examples of integers supporting this point of view,
but anybody could furnish as many counterexamples as needed to refute this
extraordinary claim. As we know, one counterexample suffices.
Nevertheless, my view is that the rarity of Mersenne primes points towards
finiteness.
If we look at Mersenne primes as a subset of the integers, we have a
fraction which is truly minuscule. If we look at them as a fraction of all
primes, we're not much better off. Progressively considering them as a
proportion of { 2^n - 1 } and { 2^p - 1 } they start to become slightly
noticeable. I suppose that the view of finite v infinite is partly related
to what one sees Mersenne primes as a subset or special case of.
Brian Beesley wrote intelligently on the subject: I thank him for conceding
that this is probably a matter for viewpoint rather than proof at the
moment. He is citing circumstantial evidence just as much as I am, however.
Certainly the earlier correspondent seemed to have eccentric ideas in many
areas of mathematics. Infinity is hard for us to understand because we are,
in some ways, only finite ourselves.
I am aware that the chances of a false positive are extremely small and look
forward with eagerness to somebody being able to tell us the value of M38. I
mentioned the possibility only to pre-empt any claim "there are only 37 of
the things - you haven't proved M38 yet". I don't see why anybody would try
to nobble the server either, since such a person would indeed be found out.
Personally, I participate in the search for mathematical reasons and not for
the money. Winning the Mersenne prize offers much less money than the
average state lottery, but more long-term acknowledgement. The average
reader of this list can probably name several Mersenne discoverers but
probably not the winners of recent lotteries. I've been looking for Mersenne
primes through GIMPS since before there was any mention of prizes, recalling
the "good old days" when I checked the whole range of 928,000 to 929,000,
mailed the results to George and visited a website at
ourworld.compuserve.com, the full URL of which nobody ever seemed to
remember. Now I'm in the same boat as S Gunderson, who has switched to
double checking because he doesn't like having to wait months for a result.
It's encouraging to know I'm not the only one who thinks like that.
Regards,
Ian
Ian W Halliday, BA Hons, MIMIS, ANZCS, CTM
P O Box 5472, Wellington 6040, New Zealand
________________________________________________________________
Unsubscribe & list info -- http://www.scruz.net/~luke/signup.htm
