I may be a little obtuse here (and spelling, expression of ideas may be
inadequate) but
A Mersenne number's prime divisors are unique to that number. Letting a and
b be primes, 2^a - 1 and 2^b - 1 have completely different factors. So we
can make a table (database) with
p1 divides M(q1)
p2 divides M(q2)
p3 cannot divide any Mersenne number
p4 divides M(q3)
p5 cannot divide any Mersenne number
p6 divides M(q2)
etc.
where p1 = 3, p2 = 5 and so on for all the primes in sequence.
The q1, q2 will also be primes but not necessarily in sequence and may occur
several times (as it does for p6). I will use the language that "pn belongs
to qm" to mean that the nth prime is a divisor of the Mersenne number q
sub-m (2^qm - 1). I can then say that pn (the nth prime) either fits
condition A (it divides M(qm) where qm is known) or that pn fits condition B
(it cannot divide any Mersenne number).
ALERT: Am I right about conditions A and B or does condition B work for
specific Mersenne numbers only?
I suspect that if someone looked at the primes up to p200, the 200th prime,
all have an associated Mersenne number M(qm) (M of q sub-m).
Lucas Wiman, for example, states that he has found 1868 new factors in the
range of Brian's 10,000,000+ digits, which I take to mean that 1,868 new
primes are known to belong to a specific qm.
Why test factor for primes in the range 2^1 to 2^10? If someone made the
table I described, it is possible that all primes less than 2^10 are in the
table I have described because they are known divisors of a Mersenne number
OR are not candidates for dividing any Mersenne number by other conditions
(subject to the ALERT above).
Therefore factoring could start at 2^10 + 1 and continue up to 2^x (where x
is a suitable integer that GIMPS choses).
Eventually, all primes in the range 2^10 to 2^11 will be covered (not a
candidate for a divisor of a Mersenne number or the prime is known to belong
to qm for some m). Then 2^11 from 2^12 will be covered, etc.
There is a key idea here, that each pn will eventually belong to some qm,
and that the smallest pn without a qm will rise over time due to the effort
of factorers such as Lucas Wiman. Would we be able to try brute force
factoring at some point for the Mersenne number M(s) where s 10,000,000
since there are so many primes taken out of the potential list?
ALERT: I said "each pn will eventually belong to some qm". Is it possible
that p1234 (the 1,234th prime) is never a divisor of a Mersenne number?
O.K. Ken, Luke, Brian and you other smarter-than-me guys and good
mathematicians, fix this up, shoot it down, use it or whatever else is
possible.
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