For the curious: the reason that this property with 24 holds is because 24 = 2^3 * 3. For any prime number p:<p>p^2 - 1 = (p+1)(p-1)<p>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).<p>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.
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.<p>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.<p>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?
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.
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<i>m.<p>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</i>8.<p>iii) There exists 2 integers m,k such that k<i>8 = m</i>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.
Is there any other number than 24 with this property, or is 24 the only one?<p>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.<p>But is there any other than 24 and 2? Is there one larger than 24?
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.<p>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!<p>Thanks for the share.
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.
Here is an accessible explanation of what the monster group is in more detail: <a href="http://youtu.be/jsSeoGpiWsw" rel="nofollow">http://youtu.be/jsSeoGpiWsw</a>.
Is there any importance of the number 24.<p>Surely there are infinite such cases for square primes.<p>prime^2 = ( n * m ) + c<p>Where n and c are constants.