I set up the whole lithiverse thing as a joke, but look at it this way:
if a proton were to become a tiny supercomputer, capable of
dividing *any* two numbers, regardless of size, in one chronon
(a chronon is the time required for light to traverse the diameter
of a proton, or 1E-24 seconds, the smallest possible unit
of quantized time) and store the result within that proton,
and the entire universe were to be used to brute force
trial factor a suspected Mersenne prime, and there are
about 6E26 protons per kg and of course the universe
consists of about 90% protons by mass, but assume
every proton, neutron and electron were to become
one of these chronon dividing supercomputers, and
Sagan tells us there are billions and billions of stars in
our galaxy, so lets just round it up to a trillion (1E15)
and God knows how many galaxies there are, but again
let us say a trillion of those.
If all the above, the universe could check 1E115 (give
or take a few orders of magnitude) possible factors each second.
If Lucas Lehmer (or some such shortcut) were unknown, and
brute force factoring were used, the above process would
break down somewhere between the 14th and 15th
Mersenne primes. The 14th, having 183 base 10 digits,
could be trial factored by such a unverse in a small fraction
of a second (ignoring the speed of light limitations of course)
but the 15th Mersenne, with 386 base 10 digits, would require
on the order of 1E193 divides, which would take the super-
computer universe about 1E78 seconds, or about a
trillion*trillion*trillion*trillion years, which is longer than,
um... we have.
So. If one think one's slow dog 486 computer isnt up
to the task of Lucas Lehmer testing, check it out sometime
by doing LL testing of the 15th Mersenne, 2^1279-1. Then
ponder what that machine has just accomplished, compared
to the (trillion)^4 years it would take for a universe such as
ours where every proton, neutron and electron is a chronon
supercomputer, without that wonderful shortcut.
Thank you very much, Lucas and Lehmer. May you and all
GIMPSers live for all eternity in mathematical nerdvana. Good night.
{8^D spike
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