Matthew Lewis wrote:
> this is a new preprint on the los alamos preprint server that may
> interest some of you.
>
> it deals with the sequence of perfect numbers (and hence Mersenne
> primes) and whether or not it is finite...
I glanced at Simon Davis' article last night. I can't comment on the
superstring stuff (my eyes tend to glaze over when people start to talk
about homotopy type, fiber bundles, and other heavy-duty differential-
geometric topics), but here's my take on the Mersenne-prime-related part:
On page 2 the author gives a lemma of de la Rosa (1978), which says that
a positive integer is a prime or a power of 2 iff it cannot be expressed
as the sum of at least three consecutive positive integers.
{NB to our non-math-major readers: when one uses the phrase "if and only if,"
commonly abbreviated as "iff," it is meant that the converse of the statement
also holds.}
Then there are several pages of stuff about triangular numbers and a certain
kind of geometrical representation of Mersenne numbers, followed at the top
of page 5 by the much-ballyhooed (pg. 4, penultimate paraqraph) "technique
[which] has been developed for determining whether a Mersenne number is prime."
The paragraph at top of pg. 5 summarizes the technique: a Mersenne number
is prime iff it can be expressed as the sum of just 2 consecutive positive
integers, not more. Of course this also follows directly from the above lemma.
The issue of novelty aside, we ask, is this useful? I note that the author
makes no attempt to analyze the complexity of any algorithm based on this
result.
Some small examples may be useful at this point:
N = 2^5-1 = 31 is prime since 31 = 15+16, but cannot be expressed as a sum
of consecutive naturals any other way.
On the other hand, N = 2^4-1 = 15 is not prime since 15 = 7+8, but also
15=4+5+6 and 15 = 1+2+3+4+5.
Note that besides the length-2 sum, 15 can also be expressed via a length-3
and length-5 sum, these latter lengths simply being the factors of 15. The
following result follows immediately:
If an odd number N has a factor F < N, then N can be expressed as the sum of F
consecutive integers, as follows: Let Q = N/F and D = (F-1)/2, then
N = (Q-D) + (Q-D+1) + ... + Q + ... + (Q+D-1) + (Q+D).
So, the upshot is that Davis' criterion for determining whether a Mersenne
(or in fact any other odd) number is prime amounts to determining whether
the number has an odd factor > 1.
As the mathematicians say, "big whup."
{NB to our non-native-english-speaking readers: when one uses the phrase
"big whup," sometimes pronounced "whoop dee doo" or "I hate to pee on
your parade, but...", it is meant that the statement in question is trivial ;}
In other words, don't stop any Lucas-Lehmer tests you might be running.
Cheers,
-Ernst
