[extropy-chat] 44th mersenne prime verified
Hal Finney
hal at finney.org
Thu Sep 14 06:06:54 UTC 2006
Spike wrote:
> In any case, here is a list of the 44 known Mersenne primes, where the
> second column is the exponent in the form 2^n-1:
>
> 1 2 2^2 -1 = 3, prime
> 2 3 2^3 -1 = 7, prime
> 3 5 2^5 -1 = 31, prime
> 4 7 2^7 -1 = 127, prime
> 5 13 2^13 -1 = 8191, prime
> 6 17 2^17 -1 = 131071, prime
> 7 19 2^19 -1 = 524287, prime
> 8 31 2^31 -1 = 2147483647, prime etc
> 9 61 (18 digit number)
> 10 89
> 11 107
> 12 127
> 13 521
> 14 607
> 15 1279
> 16 2203
> 17 2281
> 18 3217
> 19 4253
> 20 4423
> 21 9689
> 22 9941
> 23 11213
> 24 19937
> 25 21701
> 26 23209
> 27 44497
> 28 86243
> 29 110503
> 30 132049
> 31 216091
> 32 756839
> 33 859433
> 34 1257787
> 35 1398269
> 36 2976221
> 37 3021377
> 38 6972593
> 39 13466917
> 40 20996011
> 41 24036583
> 42 25964951
> 43 30402457
> 44 32582657
This is a fascinating list, the way the numbers grow so randomly.
Just write down the first digits and group numbers with the same length:
2357, 111368, 1156, 12234499, 112248, 11278, 11236, 122233...
The weird thing is that these numbers are in a sense completely
deterministic, defined by simple rules,, while at the same time appearing
totally random. Other patterns involving the primes and many other
kinds of numbers have this property, this paradoxical combination of
determinism and randomness.
One of the oddest ideas for the nature of reality is that we may, so to
speak, live in this list somewhere. Or more generally, that we may be
creatures of mathematics, what physicist Max Tegmark calls Self-Aware
Subsystems (SASs) within mathematical structures. This mathematical
feature of randomness within deterministic systems would explain the
seeming random features of our own world.
This is usually thought of as a variant on many-worlds and parallel-
universe ideas, but I prefer another angle on it. One concept that
many-worlders use to find order amid the chaos is that of "measure",
that some systems have more of it than others, and high-measure systems
are the ones that count. Wei Dai proposed a few years ago that measure
could be defined based on Greg Chaitin's notion of algorithmic complexity.
Basically systems that can be defined by simple rules have higher measure
than those that require complex rules.
When you put all this together you get that certain conscious systems,
i.e. certain beings, have much higher measure than others. These are the
ones that happen to have relatively simple mathematical descriptions,
while at the same time being complex enough to be what we think of
as conscious. It is then very likely (within this framework of ideas)
that we are those beings; the unique ones which happen, by the raw,
abstract structure of mathematics, to have simple descriptions.
Mersenne prime patterns appear random and complex but actually are
described by simple rules. The same may be true for us, and we may
owe our existence to the combination of simplicity and complexity which
makes us mathematically "prominent" in exactly the same way that Mersenne
primes are.
We can't really expect an answer to "why" Mersenne primes have the
specific patterns that they do - it's just the nature of mathematics.
And in the same way, we should not expect to say "why" our own lives
are as they are; again, it is just the nature of mathematics that these
particular lives are the ones with the greatest measure.
Hal
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