24, the Monster, and quantum gravity

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
Wikipedia on the Leech lattice: http://en.wikipedia.org/wiki/Leech_lattice
Wikipedia on 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

#mathematics #physics #sciencesunday  
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bonne méthode pour les forts en calcul mental!
Hmm, I wonder why 24? And why not 23, or 25?
Fascinating +Richard Green, you do show us about some of the strangest math relationships and theories. 
And of course if we reverse 24 we get the meaning of life.     ☞ 42 ☜ 
Someone has a lot of time on their hands
+Satyr Icon, it's not hard to check that 24 has this property, and it's not even that hard to check that the only numbers with this property are the ones I listed. (I have set this as a final exam question in an undergraduate Number Theory course in the past.)

However, I think you are wondering if we might have been able to guess intuitively that 24 had this property. I find the result surprising, and I would not have guessed this answer without checking it.
More interesting is the properties of the number 12
No, at 10:52 PM Sunday night I am wondering why maths [number theory for example] is so precise, and that it works like the way it does. Why is it so? It's so weird that in this Universe maths should be so perfect. In short I feel perfection is an aberration of reality. An abhorrence in fact.

There was a recent documentary that talked about mathematicians seeing the universe as math functions. Not string theory or quantum foam, but math functions. And the narrator brought up the question, why does math even work? It just seems freakish.
It does seem freakish (to me, anyway) that the Monster group could have anything to do with real life, +Satyr Icon.
This was fascinating but left me feeling incredibly ignorant.
+steven edgin -- number theorists prove theorems like this for a living. You never know when they will become useful.

RSA encryption, anyone?
Isn't / wasn't RSA encryption under threat from quantum computing or something?

Google? Confirm please.
Great point, +Eric Mintz. Another good example is coding theory, without which it would not be possible to read digital media (such as CDs) accurately.
Yes I was thinking of the D-Wave Quantum computers being built. Of which Google bought shares in and also purchased a 512 bit version.

Though there is debate as to whether the Dwave is doing quantum computing really slowly, or classical computing real fast. I think it was determined that it was in fact carrying out quantum computing, but that the algorithms haven't utilised it's full potential yet.
nothing to do with gravity where we must work with continuity, this is about discrete .
That doesn't follow, +MTRE. It often happens (as in this case) that continuous objects can have interesting discrete symmetries.
Hey +Richard Green, I remember from my Uni studies there's no algorithm generating primes while this looks like one-even not 100% efficient still contradicting what we've been familiar with. What am I missing here?
It isn't an algorithm for generating primes, +Rob Kook. It is a property that holds for all primes other than 2 or 3, and the proof of the result doesn't rely on having a list of the primes.
+MTRE it is not possible to work without thinking, and thinking is discrete by nature. It is the premise of continuity that is suspect.
These are some cool facts! And yes, the connection to 42 should give us something to think about +Gary Ray R ;-)
I cannot help but notice the intimate connection between 24 and its factors to geometric primitives, esp. tetrahedron.
So, another way to say this is that the group of units of Z_24 is an elementary 2-group, and that 24 is the largest number with that property. Do I have that right?

I had no idea that informed the monster group! That's pretty sweet, actually!
Exactly, +Eric Merchant! I thought about saying that in the post, but decided that it would make it too technical.
Well, it looks like cutting the crunching time to 1/24 right away only by using the first property. I was vague in the first place by not defining what I mean by an algorithm. Of course there is one we all use since the high school. 
It's actually not even really a property of prime numbers, as I explain in the second paragraph, +Rob Kook. The reason the picture is about prime numbers is that I wanted people to be able to get a rough idea of what the post is about within 5 seconds.
Sorry for distracting readers of the thread. I get it's about the group's properties and not only about some flashy manifestation of some of them. (IMHO)Number theory needs more popularizarion than most of math. 
It is partly about the flashy manifestation, +Rob Kook, but the surprising thing is that the flashy manifestation is not something superficial.
+Richard Green I seriously doubt most people spend even 5 seconds looking at a post.

But anywho, is this why there are 24 hours in a day and 12 months in a year. 😝 I have more cheeky things to say so I'll be back later.
Cool! I had no idea there was a field theory dual to the monster group! Of course, a skeptic would point out that most conformal theories have nothing whatsoever to do with the physical world... They satisfy the axioms required to describe the physical world, but they don't match observations.

This is kind of an endemic problem in quantum gravity, actually. We have no observations to match against, so we're mostly guided by mathematical formalism. And this means there are many many incompatible ideas that we can't compare meaningfully.
From my short teaching experience I have the feeling the show you need to pull off separates the good from the average. The good don't need flashy stuff as a motivation. Ratios in my classes used to be around 1:20....which is close to 24! ;-) Anyway, I also think flashy deep stuff is unique only to math, physics and poetry.
+Richard Green -- very, very interesting, and monstrously entertaining!  So the ancients were right all along -- physics is ultimately based on numerology. :)
As professor Green pointed out, the statement about the number 24 isn't intuitive at all.  However, once you translate this statement into a statement about finite group theory, the statement becomes obvious.  Maybe one day monstrous moonshine will follow this same pattern, that is, it will become a more intuitive statement when we think about it in the right way!
poking around for 1,2,5,7,12,15 on the oeis yields http://oeis.org/A024702

but that sequence appears to be a subsequence of the general pentagonal numbers: http://oeis.org/A001318 -- which is kind of nifty from the modular functions/theta functions arena of things...
+Satyr Icon We have constructed maths from axioms gleaned inductively from nature.
So it works because it works, sort of.
That makes sense if you turn it around as p^2-1=(p+1)(p-1)=24n for p prime and n in N.
The numbers p-1 and p+1 must be consecutive even numbers, hence one is  divisible by 2 and the other is divisible by 4. Also, one of the three consecutive numbers p-1, p. and p+1 must be divisible by 3, but p is not divisible by 3 (as p is prime) so p-1 or p+1 must be.  So 2*4*3=24 must be a factor of p^2-1 ie p^2=24m + 1.
+Michael Nelson -- I call that phenomenon the "collapse into elegance".  Complex problems (which may seem random in nature) gain in complexity as we wrestle with them, until we reach a profound understanding of them, at which point they reveal the elegant simplicity that was underneath all along. 
Good find, +Owen Maresh. It's a surprise to me that it's a subsequence of the pentagonal numbers. Do you know why? Or is it a coincidence based on a few terms?

+The Scotia Academy, that's exactly right! I had originally planned to put that in the post, but I realised there was a good chance that the post would go hot, so I cut it back quite a lot. (Somehow, people tend to get angry with popular posts that contain technical details.)
+Michael Nelson, monstrous moonshine is much better understood now than it was in 1979. Richard Borcherds received the Fields Medal in 1998, largely for his work in this area.

However, it seems unlikely to me that anything involving the extremely intricate structure of the Monster group could ever reach the point of being “intuitive”.
+Richard Green Seriously? Anger at technical posts? Wouldn't 'not reading' be a faster form of revenge for them?. Also, thanks for the awesome post. This is the first I've seen Whitten outside of GR proper. Cool!
What are the primes of 24? 2 and 3, the smallest primes

Not exactly as mystical as you'd expect.
+Richard Green, I totally understand (and agree with) your decision to limit the technical language here.

After reading the post initially, I went on a long hike with my dog, and along the way, convinced myself that the group of units of Z_24 forms an elementary 2-group of size 8. So, possibly meaningless counterfactual question about the connection to the monster:

If there was a number n with the property that the group of units of Z_n was a elementary 2-group of size bigger than 8, could we use this correspondence to construct a sporadic group bigger than the monster?

Now I'm aware that anything follows from a falsehood (2+2=5 =>  I'm the king of France), so my above statement is vacuously true, but I'm hoping it has slightly more content than that...
+Richard Green The only manner in which I can imagine they'll crack RSA at the moment is by SAT solving a multiplication represented with the most minimal digital circuit. But I am not a mathematician so I am unaware of more elaborate methods.

Then again, I tried it once with naive multiplication and that blew up in my face. No idea if you can even get it to work with better representations of multiplication.

(Ah. Has been tried in "Hard instance generation for SAT", Satoshi Horie, Osamu Watanabe)
+Richard Green I too do not think that monstrous moonshine will ever become intuitive in another framework.  I think there is a limit to, what +Stephen F. Heffner calls "collapse into elegance".  When I say more intuitive, I mean something similar to how the hardy littlewood tauberian theorem becomes simplified in the framework of abstract harmonic analysis on locally compact abelian groups.  Of course, you have to learn all of the theory for it to become a better understood statement.    

+Eric Merchant The largest such group is in fact Z/24Z.  If you look at the chart on the following wiki article you'll see why:


Basically, (Z/24Z)* factors as Z/2Z x Z/2Z x Z/2Z, and it's easy to see that every non-identity element in this group is idempotent.  But any cyclic group greater than Z/24Z will always contain a term greater than Z/2Z when you factor it out.
+Michael Nelson that's why I said counter-factual.

Indeed, my statement that "the group of units of Z_24 forms an elementary 2-group of size 8" is just another way of saying "Basically, (Z/24Z)* factors as Z/2Z x Z/2Z x Z/2Z".

One could even argue that you "expanded into inelegance" my statement... ;-)
176 reshares! I can say with absolute certainty that a fraction of those people actually understand any of this. +Richard Green mixes moonshine with his posts intoxicating Plussers and controlling their minds. There's no other explanation. Or perhaps he has the One Ring...
24-1 gives you a prime. I wonder if this is related...
+Richard Green : Eyeballing oeis,  I think the prime^2 -1 one is a subsequence of the generalized pentagonal number one. This does not constitute a proof, and I am not a number theorist. oeis indicates this is the case in the crossref.

I can't help but thinking, if we had a power series with q^( (prime_{n}^2 -1)/24) it would be close to something with modular symmetries -a la the Euler function http://en.wikipedia.org/wiki/Euler_function (which is q^(-1/24) from being the Dedekind eta function, amusingly)  and measuring to what extent that function is not a modular form might give us arithmetic information about prime numbers...
+Richard Green the four fours is much more lay friendly than this one or for example the Hadamard matrices! As we established before most people don't even read it! (You can test that by embedding a question to see how many respond). So although your explanation makes sense for people who read the whole post, it doesn't work for the majority who can't be bothered to read. That's what astonishes me the most.
Most have no clue what is talking about, no insult intended, just know from experience.
Ah if only your trick worked for all of us, +Richard Green ! Many people can't use that trick since their topics of discussion don't involve numbers!

Then again, I have the advantage of talking about physics, which can be made accessible by speaking concretely. So I guess I can't complain. :)
But most of the plusses come from the skim readers, +Debashish Samaddar.

+Jonah Miller, I saw you get over 50 reshares on a long technical post recently. I'm sure you could teach the rest of us a thing or two about how to do that. I'm too scared to post anything technical that's over 10 paragraphs long, in case it goes hot and I have to deal with multiple “I hate this” comments.
I chose The irrational, Q=298,382,888.4233371779508429539887
Q is based off π thus new digits can be found
+Richard Green , I just try to talk very informally---to the extent of infantile language like "wibbly-wobbly"---and use lots of pictures and formatting to break up the tedium of text.

...That and I'm lucky in that my girlfriend,  +Alexandra Fresch , is  a professional editor and can look at each one. That's the real secret. :)

Still, it's nothing compared to the success of your posts!

I'm still very surprised people post that sort of comment! If I saw a long, technical post in the What's Hot stream and didn't care about it, I'd just skip it and get on with my life.
Me too, +Jonah Miller. Well, actually, I wouldn't, because I never read What's Hot. I only see hot posts if they come from someone I'm already following.

PS: I think you're the only person I know personally who's ever had a +200 post. That shows how few of my G+ friends are actually people I know.
+Marco Devillers in the 80s RSA was broken for encoded audio data The audio was easier to break because of its redundancy. Depends on the context.
+Satyr Icon 

Read about Gödel's Incompleteness Theorem, Turing's Halting Problem, Hilbert's 10th problem's non-solution, etc. Math is beautiful but it has its limits, there are evident flaws in number theory and logic that many people choose to ignore or disregard. Math is not perfect
+Marco Devillers my crypto professor at A&m was my reference. Let me see if U can find you a paper. They weren't decoding by factoring, jut looking at redundant symbols with the fore-knowledge it was a voice transmission.
This post is freakishly awesome +Richard Green!
Although I'm an undergrad and not so much into it so as to understand and relate all of it, but your posts fuel my in interest in mathematics even further!
Kudos to you! Keep up the good work, Sir!
i feel this world is unrealistic
Deep symmetries and connection between abstract mathematics and the real world always give me goosebumps while trying to understand.

Also +Richard Green  there's a small typo in the last line of the second paragraph, where you've written 624=(24x26)+1 instead of 62*5*=(24x26)+1
Has this been actually proven.  I think not.
That is proof that god created something with the full calculation.
Man cannot prove that prime are infinite.  If this is so then your argument falls apart.  QED
+Febri Pitra How do you feel about that?
Informally. "wibbly-wobbly"---and use lots of pictures. At least Mr Green is getting the attention of those of us curious & uneducated, maybe inspire someone to finish education or just give a S**t & try?
+Plow Mule When you want something, focus on it, until you get it!
Maybe it's the topic but i feel that's an exceptionally nice writeup, even given your standards. Nice!
In twelfty the primes end in 32 possible endings, but the squares end in just two (01, 49) and 1.49 itself is itself a square!.  But it is unusual for a base to have so few endings.  48 has 2 (1, 25) 
Take a prime number greater than 5, square it, add 35, square again and subtract 1296. The answer is always a multiple of 5760
There is no hard and fast methods of finding primes that are astronomical.  Since there is no set of rules to finding primes your argument is not inclusive.  It is a parlor trick not a proof.  If you could prove a way to extract all prime numbers you would be working at the NSA.  Obviously you are not. 
Take figures of 44 DD everytime, multiply by 7, and you have one lucky dude!
Thanks, +Cyrus Khan! It's probably not worth trying to fix the typo given that it's been reshared nearly 300 times...
excuse me i have no math, but found it interesting, Is there any repeating sequence to this?
23x23=529= (24x22)+1
                        diff   13
                         diff 5
                         diff 17
                         DIFF 13
More on Monstous Moonshine, the difference of 323. 32•3=96-96/4 =24-alphabetic index of 96 divides perfectly to XXXX.

3•23=69, forget the sexual reference, listen to Missy Elliot's "Work it", I'll know later if it's worth it btw.

Its initials are MM 1313 alph indx. 2000 in Roman Numerals. if you flip the 2nd '13' you get 1331, multiplies to 9. 1313 multiplies to 9. its also 3 to the 2nd power, thr letter "i" in the alphabet. turn MM upside down you get WW =2323 alph indx... this could go on for a long time...
+sunshaker alpha, the differences between primes are very difficult to understand, so there is probably no reasonable repeating pattern to your sequence.
+Lucas Medina not perfect yet, gaps need to be filled. Number theory is far from complete, it is just a question of work and imagination. Thus a matter of time. 
2.4,4,2,4,2,2,4,2,2.4, 4, 2... it stops at 77 though I think
but 77 is only divisible by the prime numbers 7 and 11
How come we dont use a numerical base of 12? It would work more efficiently than 10.
And 24 would be even better. Base 16 would make computing easier.
You dont think having 23 unique symbols representing the integer value with the 24th reseting to a "1" value integer followrd by a "zero" value integer would be too difficult for people to calculate effortlessly? With 12 base you just need two new one-line symbols representing 10 and 11, thatd be a slight alteration and the value in the base system lies in efficiency for its greater ability to divide itself evenly with simple integers before it. "3" numerical value would no longer be prime or "odd", and pie would be a ration number.
+Richatd Green Yes it does, dividing 22 by 7 is'nt the same because it is no longer "22" by the base 10 standard the fractional value of it, and consequently pi would both be represented differently.
But 22/7 is not actually pi. It's just an approximation, and not even a very accurate one.
There is no rational representation of PI. 
Alright I could care less really. Interesting to think on but things change.
For those that actually do care, here is a good explanation of why even in using a base system of pi, it is still an irrational number. 

I was sure it was irrational but could not come up with a good explanation, so I found this one.

Part of the explanation:
The distinction you are missing is that there are numbers, which have certain properties, and then there are numerals, which are sequences of symbols that we use to represent numbers.

The base is a choice about how to represent a number as a numeral. For example, in base 10, the number 100 is represented with the symbol 100. In base 7, it is represented as 202; in base 13 it is represented as 79. In all cases, it is a perfect square; it is an even number; it is a multiple of 5, and so forth, because it is the same number. It is still equal to $36+64$, regardless of whether we write that equation as 36 + 64 = 100 or as 51 + 121 = 202 or as 2A + 4C = 79.

It is possible here's how: I should have mentioned this I talked about once but forgot. You can make pi (0.4) in base 12 but then almost every integer that does exist becomes irrational. Which is fine I think because how rational ARE numbers in base 10 anyways? Thats philisophical and sounds foolinsh but could a numeric system using both bases collaboratively with the ability to be easily intervhangeable when efficient be possible? You can eat steak and and broccoli on one plate we can use base 10 and 12 when our taste for efficiency calls for it, doesn't seem that farfetched IMO.

Rationality has nothing to do with numerical base. 
This is a rational comversation 
+Richard Green -- has anyone ever observed that every possible sentence in every Roman alphabet language is a legal integer base 36, if that base's digits proceed beyond hexadecimal's A-F through Z?  In fact, using ASCII-128 and base 128, we could make every written communication a legal integer.
+Richard Green -- oh, and BTW, in base "pi", the result of every numeric calculation is necessarily an approximation. :(  That makes "pi" what I would call a base base... :)
Pi in base 22 is quite interesting.  Recall that pi is 3:17 is near 3:1717.1717...  This is in base 22 3:3.3, and 3:3,3 is 1.17 square.
So why dont these concepts that could improve god knows what never become applied? I also hace this theiry that binary would be inferior to trinary in IT
shorter lines of code, and in 12 base system 3 isnt an odd number you could replace 8bit with 6 and from there the possibility of 3 dinentional codeing vs 2 because a hexagon is the most complex poly that will fit into itself evenly, infinately, and has the ability to be constructed into efficient clusters of spherical shaped parts
And some say binary is rhythmic of the simplicity of matter in the universe being there is positive and negative(dark) energy, I disagree because whike those do exist, we dont use a symbol to represent neutrality that occurs sonewhere in between. Trinary would more accurately mirror the polarity of physics 
and if you believe the 3 parts of an atom to be true it mimics that aswell. The power of 3 if applied and developed has technological potential that is much greater than just 2
Balanced trinary is the largest set where the digits are closed o multiplication.  It does not have a siign bit, but you can compare numbers in log2(digit) passes.  In essence, you take a number, eg 14 = 1mmm, and then write down the sign of each pair of digits until you get 1, eg 1mmm -> 1m -> 1 = positive.  The comparison of two numbers does not involve subtraction.  I designed a three-input adder-subtractor (with ripple carry) based on a single CMP gate.
Are you thinking about computers when you say this... lol
I think that you can build all other discrete functional maths with bits. Including modeling triangles and hexagons. I don't see the point of complicating.
These comments are turning into a dramatized parody of the sort of comments I tend to get on my posts. Pinging +Debashish Samaddar, who may also appreciate the show.
Yep.  Thought that myself. Math brings out the very best in people. ヅ
In highschool, at lunch, at parties, in class, wherever. We'd shamelessly talk with confidence without softening our voices about our 'private' and what things we were gonna try to acomolish on friday. Come to find out, math nerds hit the books just as hard.. Ill be damned lol.
Huh? The property that n² mod 24 = 1 is true for any odd number that's not a multiple of 3 (of course, the primes after 3 are a proper subset of this set). n²-1 = (n1)(n+1). For any odd n, n-1 and n+1 are even numbers, and one of them is divisible by 4. If n is not divisible by 3, then either n-1 or n+1 is. So the product of n-1 and n+1 definitely contains factors 2, 3 and 4 (=24).
Great post.

I am looking forward to doing some Wikipedia Archaeology starting with the terms I'm not familiar with in this post. 
+Satyr Icon All purely math-based encryption schemes are at risk from quantum computing (when we achieve it).  The reason is that a qubit can exist as both 0 and 1 simultaneously.  This property means that when you "compute" any given math equation using qubits the 'bits' don't need to 'flip' in order to reach the final result.

A visual example would be to imagine a three identical flash cards with a 1 on one side and a 0 on the other.  You could place these on a table as, 010 to represent the number 2.  Now let's place three more identical flash cards (010) above that to represent binary addition (010 + 010).  "Doing the math" here would result in 100, or 4.  This is because the bits 'flip' which is apt because traditional computers are really just jillions of tiny switches (transistors) that can be flipped from an "off" (0) state to an "on" (1) state.  The rate at which a computer can flip these bits is limited to the speed of it's CPU.

Now let's do the same thing using quantum flash cards...  Instead of three flat flash cards imagine a transparent (see-through) greeting card with two folds (accordion-like), \/\ where each "page" represents a qubit.  Each page will have both a 1 and a 0 printed on it, "\10/10\10".  To add 010+010 we'll need two of these flash cards and use some tape to cover up the first 1, the second 0, and the third 1 on both cards so they look like, "\0/1\0".

Now if we line them up and shine a flashlight through we'll see "100" projected on to the wall!  "Spooky!"

The key difference is that no "flipping" of bits needed to occur.  You could stack as many of these cards as you wanted (making an algorithm) and the "answer" would be come just as fast; instantaneous results every time.

Here's why that would render factorization-based cryptography obsolete:  If you can construct the right algorithm the quantum computer can produce the end result instantly without having to go through all the intermediary steps (factorization).  While a regular computer could take thousands of years to factor a set of really large primes a quantum computer would give you the answer right away.
Zach O
+Satyr Icon Because 1 less than a prime, written as p^2-1,  factors out to (p-1)*(p+1). p-1, p, and p+1 are three consecutive integral numbers. If p is not even then p-1 and p+1 must be consecutive evens, since one more or one less than an odd is an even. In two consecutive evens, one is always a multiple of 4. Thus the product of (p-1) and (p+1) is a multiple of 8. Also, if p is not a multiple of 3 then either p-1 or p+1 is a multiple of 3, since p-1, p, and p+1 are 3 consecutive numbers. If p>3 then it cannot be a multiple of 2 or 3 since it is prime and can have only a factor of 1 and itself. So p^2-1 is a multiple of 24 whenever p is greater than 3.
Wow.  It also leaves me thinking that perhaps we shouldn't switch to the metric system, since the standard system is partially based on numbers that have many factors...  And is it a coincidence that our day has 24 hours?
Thanks, +Harold Hausman and +Giovanni Totaro! I was wondering why this post was still getting plusses despite being two days old.

(In other news, it looks like Alexander Shulgin died yesterday. He did well to live so long without being mysteriously killed by a governmental agency.)
+The Scotia Academy Nice explanation of why the 24 phenomenon is not surprising when considered from a certain point of view.  The number 24 is not so arbitrary.  It's the product of 2, 3, and 4 which are significant in the way you explained.  I'm not just being a hater, though!  I really enjoyed +Richard Green 's post! :)
+Nick Roosevelt no, it's not a coincidence, 24 is just so nicely divisible that I can say "I spend a third of my day working" and you know that's 8 hours.

In metric time, I would say "my workday is 3.33 deci-days" and that just doesn't trip off the tongue so nicely...
+Andreas Geisler well, there's 10 deci-days, which each work out to be 2 hrs and 24 mins in your primitive earthling time…
These, of course contain 10 centi-days which you would consider to be 14 minutes and 24 seconds...
+Eric Merchant that's not part of the SI.
Metric doesn't mean "units of ten", it means the set of measurement unit scales (like the meter) that make up the SI.
+Andreas Geisler  well, you've got me then.
So if hours are part of the SI (I don't know what that is, but that's never stopped me from using an acronym yet) then we should refer to a day as 2.4 deca-hours?

I'm totally cool with that too. I'm not really expecting either of these to catch on...
+Eric Merchant The second is the basic SI (System International, or "metric") unit of time.
So there are milliseconds, microseconds, nanoseconds, etc.
But 60 seconds is a minute and 60 minutes is an hour, and 24 hours is a day.
So a coffee break in European countries is what, about 10 kiloseconds?
+Richard Green , check this out:

+Anthony Ire et al. -- when IBM was designing a new mainframe in "Project Stretch", they considered using ternary (not "trinary", BTW) instead of binary, and concluded it would be more efficient.  But they were stymied by the unavailability of a reliable tri-stable device to use.
+Stephen F. Heffner Was this recently or years ago? I wonder if they would ever try it again, I imagine the funding for R&D would be extemely expensive.
+Hamilton Carter I remember the Israeli scientists attack, it seems to depend on feeding a computer the right messages and observe non trivial noise generated by the algorithm. But I don't really understand RSA apart from that it is factoring plus reasoning modulo something which someone once told me is trivially explainable with high-school math too. (And I forgot the explanation.)

If someone could crack RSA by feeding it certain data without observing the algorithm that would surprise me. (Maybe a SAT solver, again? It's an unlikely approach, though.) But don't bother finding the link.

You could try various things. To decrypt a SAT solver would need to solve m^c0 mod c1 = c2. No idea about the digital circuit representation of that but it would be one of the things which would be nice to try. (Forget about factoring, hope that the computation exposes an overconstrained solution (unlikely, I wouldn't know why), and the SAT solver races through the bits.)
+Anthony Ire -- yes, it was back in the mists of time -- in the early '60s, just before the 360 family appeared on the scene; in fact, many of the 360's concepts were pioneered in Stretch.  The project's result was the 7030, IBM's first fully transistorized computer, which was a commercial failure.  (I was teaching myself 7090 assembler from a book shortly after that.)
Yes! 400 reshares on a post with (a) no flashy visuals, (b) a definition, and (c) a statement of a theorem.
Something about 24   ☞42  is fascinating I guess.
Thanks for the post. 
This is a really interesting subject. I've come across the term "monstrous moonshine" before, but was unprepared for the knowledge and absorbed very little of it.

So many many many comments, lots to say:

Although in  base 3 and higher, 24 is 3*2^3 , a beautifully symmetric representation. There is arbitrariness to that beauty though, 5*3^5 is probably not important. 

Knowing that the square of every prime more than 24 is one more than some UNKNOWN multiple of 24 ( or = 1 mod 24 if you prefer) doesn't tell you a lot more about the primes. You still don't know which multiples to sift through, and it doesn't really decrease the computational time of finding large primes that much.

It's odd that 24 seems to be such a big player.  I would figure that all primes are created equal, just not distributed equally.

A talk I saw from a mathematician said that there will never be a pattern to primes in the sense of a writable function of natural numbers, f(n), such that f(n) and f(n+1) are always adjacent primes. He likens the shape of the prime distribution to a musical chord. A note, or chord in music is the superposition of an infinite number of waves whose frequencies are related by a pattern, a 'harmonic series'. The pattern is in the waves that make the notes, not in the notes themselves. Or some thing like that, I suggest  further reading here :


I can't remember the lecturer's name, I keep thinking Hardy, but that wasn't it. I'll keep my eye out for the original video, it was one of the best general-audience talks I've ever seen on mathematics. 

Harmonic series:

Intuition supposes what? I know when something is intuitive to me, but I couldn't define it. I find many things intuitive to me that aren't to others, but that street isn't one way. The capacity of a theory or a subject to be intuitive depends also on the maturity of the research. If you look at how Newton did calculus it is brutally complex, the knowledge is more palatable over years of distillation, reinterpretation. Somethings though are probably just too far from our scale of experience to relate to knowledge we obtained from our senses alone. speaking about or non-relativistic, macroscopic life. But look at what modern highschool chemistry teaches about orbitals, and compare that to solving Schrodinger's equation for a multi-body system. The distillation process takes longer with richer complexity but long isn't never.
 especially if you consider that as the complexity of our knowledge grows, so too does our potential to enhance our thinking ability through invention, a.k.a. Transhumanism. Surely bio-electro-mechanically enhanced abilities to think will lead to more ideas being intuitive..

+Anthony Ire 
Last thought I have time for : 22/7 is a dumb approximation of pi. A great one is to take the first 3 odd numbers, write them each twice 11 33 55 and divide the second half by the first: 355/ 113 gives you pi to 8 decimal places!


As usual, you've sparked my interest +Richard Green , I'll be studying monstrous mathematical moonshine now.
+Steven Sarasin When you mention Transhumanism are you refering to the theoretic concecept of human and computer merging into one seemless conciousness? I watched an episode of Through the Wormhole that predicted we will have that by the year 2050. Organic evolution and technological evolution mimic each other's growth patterns suggesting that fundamentally they are one and the same.

Makes me wonder that if we created technology we have today and that if the same tech were to become sophisticated enough to wonder how it became so, could something similar have happened that designed the making of us?
Am I a machine built for a purpose and does realizing it mean I am broken?
+Richard Green I enjoy your mathematics posts, they encourage me to branch out and learn new things. I never tire of that! I appreciate the time and effort you put into making the information clear and easy to understand. I, for one, read all the comments on your posts, since they are often entertaining, add information to your post, and give new information, as well. Thank you for sharing!
And the multiple of the integers preceding the first square is 1x2x3x4 = 24.
Anyway, I can't see myself in a space of 196883 dimensions.
Even more, if physicists haven't a clear idea of the nature of time, the fourth dimension after three centuries of deliberations, how many years will they need to explain the following 196.879?
+Mauricio Luque 
How many numbers do you need to describe exactly where you are in a city. Street, Avenue, Building floor#? Perhaps the time you're there as well. 4 numbers needed ok, that's reasonable. Now what if I wanted to describe where two people are, I'd need 3 + 3, for their respective street,avenue, floor, and maybe 1 or 2, times. Now we need 7 or 8 numbers to describe the situation. We have then a 7 or 8 dimensional system. How can we have 196,883 SPATIAL dimensions, now that I can't explain, feel, or understand myself.
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