
24, the Monster, and quantum gravity - msvan
https://plus.google.com/101584889282878921052/posts/9sKMLRJYjna
======
rkaplan
For the curious: the reason that this property with 24 holds is because 24 =
2^3 * 3. For any prime number p:

p^2 - 1 = (p+1)(p-1)

And p+1 and p-1 must both be multiples of 2 because p is odd. Furthermore, one
of p+1 or p-1 is also a multiple of 4 (because they are both multiples of 2
and only 2 apart). So, we can see where the 2^3 factor comes from in the magic
number 24. The remaining factor, 3, comes from the fact that p is prime and
not a multiple of 3, so either p+1 or p-1 must be a multiple of 3 (otherwise
p-1, p, and p+1 would be three consecutive numbers, none of which are
divisible by 3, which is impossible).

As a result, for any prime p > 3, (p+1)(p-1) is divisible by 24, so p^2 - 1 is
also divisible by 24.

~~~
woofuls
I was just curious, why is 24 x 10000+1 = 107 x 2243 not prime? I was under
the impression for any n, 24 x n + 1 is prime.

~~~
gatehouse
24x1 + 1 = 25, so we're off to a bad start.

~~~
tlammens
You forget the square root: 5 is prime.

~~~
stbarnes
huh? great-grandparent said nothing about square roots

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gajomi
So there are lots of facts here. And the facts are connected together at
various points. And I like to hear about interesting connections. But it seems
to me that unless you have looked at these things in depth (and I have not for
the most part) that you would have only a vague idea of what is being talked
about here.

But as I said, since I like connections, I am interested in moving beyond
vagaries. In particular I am wondering about this connection to quantum
gravity, and I have a few questions to this effect.

If this is about symmetries in a field theory then what is the field in this
case? If I see a representation of a permutation group or a special orthogonal
group factoring out of operations in a field theory I have some intuition
about what this is. So what about this Monster group and what, if anything
does this have to do with quantum gravity? Is it a gravity thing? Is it a
quantum thing. Both?

~~~
noobermin
Usually, requiring that the state vector in a quantum theory be invariant
under a unitary transformation require other fields (what we call gauge
fields) be added that end up being representing physical interactions.

For example, consider the state vector |y>. The norm <y|y> is invariant under
local U(1) transformations

|y> -> e^(-I* T(x,t))|y>

For

<y|y> -> <y|e^(I* T(x,t))e^(-I* T(x,t))|y> = <y|1|y> = <y|y>.

(local, as in a function of space-time, is key). Write the state as |y>=v|0>
where v is some operator, then saying <y|y> is invariant is like saying v* v
is invariant when v -> v e^(-I* T(x,t)).

In physics, a term ~v* v would represent the potential energy of a spring. To
represent the kinetic energy of a spring, we need a term like (d^{\mu}v*
)d_{\mu}v where the d_{\mu} represent derivatives w.r.t. space and time (\mu
is an index which runs from 0 to 3, 1,2,3 are x,y,z, 0 is time). Here, the
U(1) transformation does not leave this term valid, because the derivative
acts on the e^(-I T(x,t)) too...

However, if one were to make the replacement

d^{\mu} -> d^{\mu} - I e A^{\mu}(x,t)

where when

v -> v e^(-I* T(x,t))

we have that

A^{\mu} -> A^{\mu} - d^{\mu} T(x,t)

one can show (this doesn't have to be obvious!) that then the Lagrangian

~(D^{\mu}v)* D_{\mu}v - v* v

(where D is the new "covariant derivative" we described above) is now
invariant under local U(1) transformations. It turns out that this Lagrangian,
completely written out, looks a lot like classical electromagnetism, and it
is, and A^{\mu} is the "4-vector potential". In fact, A^0 represents the good
old electric potential V that I'm sure you nerds are quite familiar with. I'm
not sure how enlightening that was, but at least you've seen the start of QED
:)

Now, for this "monster group" ... from here [0] it seems that apparently,
gravity in 2+1 dimensions seems dual to this particular group, that is, take
the quantum state with that operator |y>=v|0> (v is what we call a creation
operator in QFT, it "creates" particles mathematically; in string theory, it
creates "vibrations on the string") that under monster group transforms,
apparently whatever Lagrangian Witten made out of the v's in conformal field
theory (something I should note is not my field, so I may be off here) the
gauge group that might be needed would represent gravity.

[0]
[http://en.wikipedia.org/wiki/Monstrous_moonshine#Conjectured...](http://en.wikipedia.org/wiki/Monstrous_moonshine#Conjectured_relationship_with_quantum_gravity)

~~~
gajomi
Thanks for the response. I think this is a good answer to the question of "why
would a physicist be working on the Monster group" that addresses the core
idea of required invariants, and one along the lines that I have heard before
(I suppose at this point I should confess to be an ex-physicist, although I
never studied QED at anything but the most elementary level).

But, having heard it, I feel wanting. As you have deftly illustrated, starting
with some basic requirements about invariants (from special relativity) you
can end up at the Lagrangian associated with QED. And you can then (after the
fact!) point out that the field in question is the electromagnetic one. Having
studying electromagnetism enough to know what you are talking about, I can
hear this kind of argument and it seems clever to me. There is some difference
in semantics of the states, of course (classical solution of of the confined
electromagnetic field coming in quanta, but perhaps not understood as "quantum
observables" proper). But if I had not studied these things I would probably
just be confused.

And I think this is how I feel when I hear about this advanced quantum gravity
stuff (once again, I don't mean to complain about your excellent response, I
think it just addresses a different issue then the one I am talking about).

If you are willing to entertain me, I'll try to ask a clarifying question in
terms of analogy. If someone asked me "why does the Lagrangian for QED looks
like it does", I might say to them that "the speed of light (light being made
of electromagnetic fields) is constant in all inertial reference frames (lets
hope they understand Galilean relativity) and that this fact forces QED to
look that way". This explanation makes reference to the "stuff" that the field
is made out of and the observations about invariants (stated in ordinary
language) that it must satisfy.

So, now if I ask "why does the Lagrangian for quantum gravity looks like it
does", you would say...

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seanalltogether
Interesting that if we used a base-12 numeral system then this would be
immediately obvious. I wonder what other mathematical concepts would be more
obvious if we used a different base system.

~~~
GotAnyMegadeth
Multiply and divide by N, where N is the base

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capcah
Just as a remark, proving that given a prime n, n^2 = 1(mod 24)[equivalent to
n^2-1 is multiple of 24] is pretty easy: i) prove that n^2 = 1(mod 3).
Enumerating, n = {-1,1} (mod 3) (mod 3) [since n is prime, n != 0 (mod 3)].
n^2 = 1 (mod 3) n^2 - 1 = 0 (mod 3). Exists m such that n^2-1 = 3 _m.

ii) prove that n^2 = 1(mod 8). Enumerating, n = {1,3,-3,-1} (mod 8) [ n !=
pair (mod 8) since then, it would be divisible by 2]. n^2 = {1,9,9,1} (mod 8)
=> n^2 = {1,8+1}(mod 8) => n^2 = 1 (mod 8). Exists k such that n^2 - 1 = k_8.

iii) There exists 2 integers m,k such that k _8 = m_ 3\. m must have an 8
factor and k a 3 factor. Then, there exists j such that j = m/8 = k/3 =
(n^2-1)/24\. qed.

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Aardwolf
Is there any other number than 24 with this property, or is 24 the only one?

Well, 2 also has the property: multiply any prime number other than 2 with
itself, subtract one, and it's a multiple of 2. This one is quite obvious, all
those prime numbers are odd, so of course if you subtract one of their square
(which is also odd), it's even and a multiple of 2.

But is there any other than 24 and 2? Is there one larger than 24?

~~~
mmusson
The other numbers are 3 and 4. This is what yields 24. I think the comment
above [0] explains this well.

[0]
[https://news.ycombinator.com/item?id=7839666](https://news.ycombinator.com/item?id=7839666)

~~~
AnimalMuppet
The article also mentions 6, 8, and 12 (all factors of 24).

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sekasi
I always wonder why I find things I don't understand so fascinating. It's the
same reason I read in depth articles on cryptography and play the 'wikipedia
rabbit hole' game.

Because it describes most primes, the ignorant part of me can't help but
wonder if it does have anything to do with the magic that is crypto.. but I
digress. Wikipedia might tell me more, brb losing eight hours!

Thanks for the share.

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wolfwyrd
It's also 42 backwards. Important I feel

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jimwalsh
Can someone link to this story or copy/paste it somewhere? Linking to social
media sites on Hacker News is bad news for people that work at companies
during the day that block all social media but would still like to read the
news here.

~~~
gregschlom
Here you go:

24, the Monster, and quantum gravity

By Richard Green

Think of a prime number other than 2 or 3. Multiply the number by itself and
then subtract 1. The result is a multiple of 24. This observation might appear
to be a curiosity, but it turns out to be the tip of an iceberg, with far-
reaching connections to other areas of mathematics and physics.

This result works for more than just prime numbers. It works for any number
that is relatively prime to 24. For example, 25 is relatively prime to 24,
because the only positive number that is a factor of both of them is 1. (An
easy way to check this is to notice that 25 is not a multiple of 2, or 3, or
both.) Squaring 25 gives 625, and 624=(24x26)+1.

A mathematician might state this property of the number 24 as follows: If m is
relatively prime to 24, then m^2 is congruent to 1 modulo 24. One might ask if
any numbers other than 24 have this property. The answer is “yes”, but the
only other numbers that exhibit this property are 12, 8, 6, 4, 3, 2 and 1; in
other words, the factors of 24.

The mathematicians John H. Conway and Simon P. Norton used this property of 24
in their seminal 1979 paper entitled Monstrous Moonshine. In the paper, they
refer to this property as “the defining property of 24”. The word “monstrous”
in the title is a reference to the Monster group, which can be thought of as a
collection of more than 8x10^53 symmetries; that is, 8 followed by 53 other
digits. The word “moonshine” refers to the perceived craziness of the
intricate relationship between the Monster group and the theory of modular
functions.

The existence of the Monster group, M, was not proved until shortly after
Conway and Norton wrote their paper. It turns out that the easiest way to
think of M in terms of symmetries of a vector space over the complex numbers
is to use a vector space of dimension 196883. This number is close to another
number that is related to the Leech lattice. The Leech lattice can be thought
of as a stunningly efficient way to pack unit spheres together in 24
dimensional space. In this arrangement, each sphere will touch 196560 others.
The closeness of the numbers 196560 and 196883 is not a coincidence and can be
explained using the theory of monstrous moonshine.

It is now known that lying behind monstrous moonshine is a certain conformal
field theory having the Monster group as symmetries. In 2007, the physicist
Edward Witten proposed a connection between monstrous moonshine and quantum
gravity. Witten concluded that pure gravity with maximally negative
cosmological constant is dual to the Monster conformal field theory. This
theory predicts a value for the semiclassical entropy estimate for a given
black hole mass, in the large mass limit. Witten's theory estimates the value
of this quantity as the natural logarithm of 196883, which works out at about
12.19. As a comparison, the work of Jacob Bekenstein and Stephen Hawking gives
an estimate of 4π, which is about 12.57.

Relevant links Wikipedia on the Monster group:
[http://en.wikipedia.org/wiki/Monster_group](http://en.wikipedia.org/wiki/Monster_group)
Wikipedia on the Leech lattice:
[http://en.wikipedia.org/wiki/Leech_lattice](http://en.wikipedia.org/wiki/Leech_lattice)
Wikipedia on Monstrous Moonshine:
[http://en.wikipedia.org/wiki/Monstrous_moonshine](http://en.wikipedia.org/wiki/Monstrous_moonshine)
A 2004 survey paper about Monstrous Moonshine by Terry Gannon:
[http://arxiv.org/abs/math/0402345](http://arxiv.org/abs/math/0402345)

~~~
jimwalsh
Thanks!

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adam704
Here is an accessible explanation of what the monster group is in more detail:
[http://youtu.be/jsSeoGpiWsw](http://youtu.be/jsSeoGpiWsw).

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GotAnyMegadeth
Does that not mean it is "trivial" to find all of the prime numbers because
you can just go through the 24 times table?

~~~
chopin
I'd say that it makes it at best only 24 times easier. Which cuts the problem
down approx. 1.4 order of magnitudes which doesn't matter much when the
problem scales exponentially.

~~~
GotAnyMegadeth
Ahh, I was thinking that p^2 - 1 = 24n held for any n as well as any p, but I
can see that's not necessarily the case. Thanks

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coder23
Is there any importance of the number 24.

Surely there are infinite such cases for square primes.

prime^2 = ( n * m ) + c

Where n and c are constants.

~~~
madaxe_again
All sorts - read the cited wikipedia articles for starters - the most
important being that it's a small finite number that grants exceptional
packing efficiency in the Leech lattice, which is then the basis for the CFT
tie-in.

Apart from anything else, this sort of thing is just beautiful mathematically.

