Mersenne Digest Sunday, 14 March 1999 Volume 01 : Number 530
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From: Henk Stokhorst <[EMAIL PROTECTED]>
Date: Sat, 13 Mar 1999 19:26:36 +0100
Subject: Re: Mersenne: VME claim
L.S.,
M(727) is not prime. VME made the claim that they could compute the first
prime following M(727) in two seconds. Just want to know someone who can do
the same trick and what software it takes.
It took less than 2 seconds to find the next sequential prime :
7060034896770543742372772105511569658378384779628943811708504827156734575902
9962497646848024880749924272446637457099914453082421646959773690663827212173
6526607699022870679030143158018123175881930939339869708632591433883
YotN,
Henk Stokhorst
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From: Gordon Bower <[EMAIL PROTECTED]>
Date: Sat, 13 Mar 1999 12:15:50 -0900 (AKST)
Subject: Re: Mersenne: free sphere is half deflated
>
> From: The thrill of minimalism <[EMAIL PROTECTED]>
> Date: Fri, 12 Mar 1999 22:35:53 +0000
> Subject: Mersenne: free sphere is half deflated
>
> "Ernst W. Mayer" wrote:
> >
> > Joth Tupper writes:
> >
> > >the Bolzano-Tarski theorem proved (what, back in the 1920's?)
> > >that you could cut a solid 3D sphere into finitely many chunks,
> > >then rearrange the chunks to make another solid (no holes or gaps)
> > >3D sphere with _twice_ the volume. Pretty spooky, I always felt.
> >
> > I don't know about spooky, but 'twould seem to violate conservation
> > of mass (or mass/energy, if you're a postmodern relativist :),
> > 'twouldn't it?
Well... no one said how easy it was to actually take hold of the pieces
and rearrange them. We mathematicians tend to think of our objects as
continuous, infinitely divisible, lumps, but that doesn't work so well
with matter. Quarks, for instance, come in pairs and triplets, never
alone. And if you try to tear a quark pair apart --- you put so much
energy into the system pulling on them that you create two new quarks in
the process. I wonder if anyone has tried to reconcile Banach-Tarski with
physical reality by saying "yes, you can do that, but the amount of energy
it takes to pull the pieces apart and reassemble them turns out to be
enough energy to create the additional mater needed"...
>
> You can map a line segment of "length" 1 onto two line segments of
> length 1 with a simple mapping function f(x)=2x+1 for instance.
Banach-Tarski doesn't do any stretching, just rotating, translating,
cutting.
- ---
Gordon Bower
PS: Those were SPHERICAL fishes and loaves of bread that Jesus broke to
feed the thousands of people on the shores of the Sea of Galilee. :)
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From: Joth Tupper <[EMAIL PROTECTED]>
Date: Sat, 13 Mar 1999 17:29:00 -0500
Subject: Mersenne: Banach-Tarski and Spheres
I am beginning to feel sorry for starting these threads, especially as they
are off topic. I was just
pointing out that there are good quality mathematicians (like R.Robinson,
Goro Shimura, J-P Serre,
Andre Weil and many, many others) whose "stamp of approval" would carry
almost instant acceptance
while other mathematicians (and I would include myself as a sort of applied
mathematician) really carry
no credibility to speak of.
I picked the Banach-Tarski Theorem (BTT) on spheres because it is so
mind-numbingly wierd that Robinson's work
makes it seem almost certainly unbelievable.
The result is that everyone commenting on the problem thinks that
"stretching" or "mapping" is what the BTT talks about.
No, no, no. The BTT is not done with mirrors, bent or otherwise. This is
an honest-to-set-theory result
about solid spheres in 3-space. This is not about the physical universe,
it is about mathematics.
Now, I am not a set theorist or analyst. However, I have one [readily
accessible] reference which mentions BTT:
"The World of Mathematics" by James E. Newman, Simon and Schuster, 1956.
See volume 3, page 1944-1945.
This 4 volume work has been printed numerous times in hardcover and
paperback and is quite a nice read for
high school students (and others of us) interested in mathematics.
The description of BTT given in [Newman] is pretty much as follows:
Take two solid 3 dimensional spheres S1 and S2 of unequal volume (i.e.,
different radii). The volumes can be as different as you wish -- Newman
suggests taking one the size of a pea and the other the size of the sun.
You can divide S1 into finitely many (n) disjoint pieces A1,A2,...,An and
S2 into finitely many disjoint pieces B1,B2,...,Bn such that (after
suitable rearranging, if needed), A1 is congruent to B1, A2 is congruent
to B2, and so on until An is congruent to Bn. [Note that the word
"congruent" applies -- not merely "similar".]
Just before this statement, Newman characterizes this construction (p.
1944) as follows:
"Two distinguished Polish mathematicians, Banach and Tarski, extended the
implications of Hausdorff's paradoxical theorem to three-dimensional space,
with results so astounding and unbelievable that their like may be found
nowhere else in the whole of mathematics. And the conclusions, though
rigorous and unimpeachable, are almost as incredible for the mathematician
as for the layman."
Again, Raphael Robinson showed that for the right two different volumes, n
= 5 (perhaps 4) is enough. I do not have an exact statement of Robinson's
result handy.
Please, no more silly discussion of y = 2x or y = 4x (or 4x/3) as
"criticisms" of this result. Multiplying by 2 or 4 or 4/3 stretchs
the image. Stretched things may be similar but they are not _congruent_.
BTT is not about "mapping."
Even trying to think about BTT tends to make people dizzy. ("Visualize
whirled peas.")
I hope this helps clarify one of our stranger 20th century results.
Next time, we can dipute the poetic content of the Picard theorems for
entire functions (an entire complex function which misses two complex
values in its range is a constant). This is technical enough that people
may have more trouble arguing about it.
Thanks,
Joth Tupper
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From: Will Edgington <[EMAIL PROTECTED]>
Date: Sat, 13 Mar 1999 20:21:07 -0800
Subject: Re: Mersenne: VME claim
Henk Stokhorst writes:
M(727) is not prime. VME made the claim that they could compute the
first prime following M(727) in two seconds. Just want to know
someone who can do the same trick and what software it takes.
It took less than 2 seconds to find the next sequential prime :
7060034896770543742372772105511569658378384779628943811708504827156734575902
9962497646848024880749924272446637457099914453082421646959773690663827212173
6526607699022870679030143158018123175881930939339869708632591433883
Well, that's only 156 more than M727, so finding it is easy; the
obvious sieve will do it. Verifying it's at least pseudo-prime took
the mers package's ecmfactor program only 1.27 seconds CPU on my Linux
Pentium 200MHz just now, and proving it prime using Morain's ECPP
program took all of 50.9 seconds.
So, even if they are proving it prime, it's not a big advance.
More to the point, a better test, since they're unwilling to reveal
their method, would be to give them some large, strong pseudo-primes
mixed in with known primes of similar size. Anybody care enough to
produce such a set? The composite factors of prime exponent Mersennes
are all base-2 pseudo-primes, but that's not enough; they should
probably be Cunningham numbers (which are pseudo-prime to all bases
that aren't related to their factors).
The only stronger tests that I can think of are making the program
available via a TCP/IP server of some sort, so people like us can give
it arbitrary numbers to check in real time, and a rigorous proof of
the method, which requires making it public.
Will
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End of Mersenne Digest V1 #530
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