Comments on: The miracle of 163 http://www.neverendingbooks.org/index.php/the-modular-miracle-of-163.html lieven le bruyn's blog Fri, 20 Jan 2012 16:50:41 +0100 hourly 1 http://wordpress.org/?v=3.3.1 By: John McKay http://www.neverendingbooks.org/index.php/the-modular-miracle-of-163.html/comment-page-1#comment-8788 John McKay Sat, 10 Apr 2010 18:05:22 +0000 http://localhost/?p=10#comment-8788 <p>Gukov-Vafa: RCFT & complex multiplication arxiv:hep-th/0203213 Comm Math Phys 246 (2004) 181-210. in which (page 21 line 9) is the remark that the class number h(sqrt(-163)) = 1 implies the uniqueness of a certain rational conformal field theory (RCFT) with chiral ring of dim 163. Now for the monster group, M, and its moonshine functions MM, we have (see Conway-Norton, Monstrous Moonshine Bull Lond.Math Society 1979, p 310 (1) and p319) the calculations giving the (column) rank as 163 for the MM fns (conceived as vectors of their q-coefficients, which are all rational integers). This 163 is obtained from the 194 conjugacy classes of M from which we derive 172 classes of cyclic subgroups. There are 9 linear relations between the corresponding 172 rational MM fns. Tables of these are in McKay p 117-118 of CRM/AMS Proc. vol 30 ed. McKay & Sebbar 2001. We also have a paper in Les Houches (2003) Proceedings II pp 373-386. in which we note the possible significance in physics of the number of classes in a finite group. Irrational character table entries for M are the 22 2x2 cells of complex quadratic entries. There are two classes, 27A & 27B that have he same MM fn. but have centralizers of order 486 and 243. This curiosity remains to be explained.</p> Gukov-Vafa: RCFT & complex multiplication arxiv:hep-th/0203213
Comm Math Phys 246 (2004) 181-210.
in which (page 21 line 9) is the remark that the class
number h(sqrt(-163)) = 1 implies the uniqueness of a certain
rational conformal field theory (RCFT) with chiral ring of dim 163.
Now for the monster group, M, and its moonshine functions MM, we
have (see Conway-Norton, Monstrous Moonshine Bull Lond.Math Society
1979, p 310 (1) and p319) the calculations giving the (column) rank
as 163 for the MM fns (conceived as vectors of their q-coefficients,
which are all rational integers). This 163 is obtained
from the 194 conjugacy classes of M from which we derive
172 classes of cyclic subgroups. There are 9 linear relations
between the corresponding 172 rational MM fns. Tables of these are
in McKay p 117-118 of CRM/AMS Proc. vol 30 ed. McKay & Sebbar 2001.
We also have a paper in Les Houches (2003) Proceedings II pp 373-386.
in which we note the possible significance in physics of the number
of classes in a finite group.
Irrational character table entries for M are the 22 2×2 cells of
complex quadratic entries. There are two classes, 27A & 27B that have
he same MM fn. but have centralizers of order 486 and 243. This
curiosity remains to be explained.

]]> By: John McKay http://www.neverendingbooks.org/index.php/the-modular-miracle-of-163.html/comment-page-1#comment-6198 John McKay Fri, 28 Nov 2008 01:21:46 +0000 http://localhost/?p=10#comment-6198 <p>It is also the rank of the monstrous moonshine functions. This is not a coincidence, it is significant.</p> It is also the rank of the monstrous moonshine functions. This is not a coincidence, it is significant.

]]> By: tpc http://www.neverendingbooks.org/index.php/the-modular-miracle-of-163.html/comment-page-1#comment-261 tpc Sun, 24 Jun 2007 04:03:13 +0000 http://localhost/?p=10#comment-261 I remember martin gardner wrote an april fool's joke in 1975 that announced a proof that e^{\pi \sqrt{163}} is an integer! I remember martin gardner wrote an april fool’s joke in 1975 that announced a proof that e^{\pi \sqrt{163}} is an integer!

]]> By: Jonathan Vos Post http://www.neverendingbooks.org/index.php/the-modular-miracle-of-163.html/comment-page-1#comment-213 Jonathan Vos Post Fri, 22 Jun 2007 04:01:59 +0000 http://localhost/?p=10#comment-213 <a href="http://www.research.att.com/~njas/sequences/A108764" rel="nofollow">A108764 Products of exactly two supersingular primes (A002267).</a> 4, 6, 9, 10, 14, 15, 21, 22, 25, 26, 33, 34, 35, 38, 39, 46, 49, 51, 55, 57, 58, 62, 65, 69, 77, 82, 85, 87, 91, 93, 94, 95, 115, 118, 119, 121, 123, 133, 141, 142, 143, 145, 155, 161, 169, 177, 187, 203, 205, 209, 213, 217, 221, 235, 247, 253, 287, 289, 295, 299 There are exactly 15 supersingular primes: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 41, 47, 59, and 71 (A002267). The supersingular primes are exactly the set of primes that divide the group order of the Monster group.... A108764 Products of exactly two supersingular primes (A002267).

4, 6, 9, 10, 14, 15, 21, 22, 25, 26, 33, 34, 35, 38, 39, 46, 49, 51, 55, 57, 58, 62, 65, 69, 77, 82, 85, 87, 91, 93, 94, 95, 115, 118, 119, 121, 123, 133, 141, 142, 143, 145, 155, 161, 169, 177, 187, 203, 205, 209, 213, 217, 221, 235, 247, 253, 287, 289, 295, 299

There are exactly 15 supersingular primes: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 41, 47, 59, and 71 (A002267). The supersingular primes are exactly the set of primes that divide the group order of the Monster group….

]]> By: javier http://www.neverendingbooks.org/index.php/the-modular-miracle-of-163.html/comment-page-1#comment-155 javier Wed, 20 Jun 2007 10:23:53 +0000 http://localhost/?p=10#comment-155 Hi! Been away for a while, so I didn't see the redesign till today. Love the new design! Just a small accesibility issue: in ffox 2.x for linux (cannot say for windows) compulsory fields in any form appear with light yellow background. So for the name, email and website (don't know exactly why also this one) I am forced to reading white text over light yellow background, which is not a very pleasant experience. More on-topic: Very nice property of 163! I must have a really twisted mind, though, since first thing I thought about it was "Ohhh, I could use it for the arbitrary-precision computing practice for my CS students, and tell them that it *is* a whole number unless they are introducting carrying mistakes" and then thinking of the feeling of walking around the clasroom, smiling for myself, while they desperately try to find errors in their codes... Maybe I'll do it, at least it will show them not to believe something true just because the teacher told it :D Hi! Been away for a while, so I didn’t see the redesign till today.
Love the new design! Just a small accesibility issue: in ffox 2.x for linux (cannot say for windows) compulsory fields in any form appear with light yellow background. So for the name, email and website (don’t know exactly why also this one) I am forced to reading white text over light yellow background, which is not a very pleasant experience.

More on-topic: Very nice property of 163! I must have a really twisted mind, though, since first thing I thought about it was “Ohhh, I could use it for the arbitrary-precision computing practice for my CS students, and tell them that it *is* a whole number unless they are introducting carrying mistakes” and then thinking of the feeling of walking around the clasroom, smiling for myself, while they desperately try to find errors in their codes… Maybe I’ll do it, at least it will show them not to believe something true just because the teacher told it :D

]]> By: Doug http://www.neverendingbooks.org/index.php/the-modular-miracle-of-163.html/comment-page-1#comment-92 Doug Mon, 18 Jun 2007 14:29:09 +0000 http://localhost/?p=10#comment-92 The mathematical near miss is conceptually reminiscent of the 1999 Physics Nobel Award, from the Presentation Speech: “... 't Hooft and Veltman tell us instead: calculate as if the number of dimensions were slightly less than four, four minus epsilon, i.e., 3.99999. This approach proved to be highly effective. The nasty infinities became less frightening. They could be collected, harnessed and interpreted ...” http://nobelprize.org/nobel_prizes/physics/laureates/1999/presentation-speech.html Notice these curiosities: 3 + 1 = 4 = 2^2 43 + 1 = 44 = (2^2) * 11 67 + 1 = 68 = (2^2) * 17 163 + 1 = 164 = (2^2) * 41 These are similar to but different from: 196884 - 1 = 196883 = 47 * 59 * 71 Yet it appears as though all of these numbers can be related to some form of supersingular primes. I do not know if this is significant or merely happenstance. The mathematical near miss is conceptually reminiscent of the 1999 Physics Nobel Award, from the Presentation Speech:
“… ‘t Hooft and Veltman tell us instead: calculate as if the number of dimensions were slightly less than four, four minus epsilon, i.e., 3.99999. This approach proved to be highly effective. The nasty infinities became less frightening. They could be collected, harnessed and interpreted …”
http://nobelprize.org/nobel_prizes/physics/laureates/1999/presentation-speech.html

Notice these curiosities:

3 + 1 = 4 = 2^2

43 + 1 = 44 = (2^2) * 11

67 + 1 = 68 = (2^2) * 17

163 + 1 = 164 = (2^2) * 41

These are similar to but different from:

196884 – 1 = 196883 = 47 * 59 * 71

Yet it appears as though all of these numbers can be related to some form of supersingular primes.
I do not know if this is significant or merely happenstance.

]]> By: lieven http://www.neverendingbooks.org/index.php/the-modular-miracle-of-163.html/comment-page-1#comment-28 lieven Sat, 16 Jun 2007 18:49:01 +0000 http://localhost/?p=10#comment-28 Kea & Christine : thanks! Though I love to restyle my blog from time to time it's a time consuming exercise so it is always nice if al least some people like the result. Doug : as to 1. no, there are infinitely non-supersingular primes and just 9 p's such that the ring of integers in [tex]Q(\sqrt{-p})[/tex] is a principal ideal domain (which is the reason for the near miss). as to 2. the pdf-files are not perfect but will have to do for now. probably i will recycle/update some old posts when they are on-topic here (such as the Mathieu post and some on modular groups). To all of you I apologize for having set my options so that anyone can post a comment provided they have one comment approved. So, only your first comment will not appear automatic. I learned the hard way it saves me time removing spam-comments. Kea & Christine : thanks! Though I love to restyle my blog from time to time it’s a time consuming exercise so it is always nice if al least some people like the result.
Doug : as to 1. no, there are infinitely non-supersingular primes and just 9 p’s such that the ring of integers in [tex]Q(\sqrt{-p})[/tex] is a principal ideal domain (which is the reason for the near miss). as to 2. the pdf-files are not perfect but will have to do for now. probably i will recycle/update some old posts when they are on-topic here (such as the Mathieu post and some on modular groups).
To all of you I apologize for having set my options so that anyone can post a comment provided they have one comment approved. So, only your first comment will not appear automatic. I learned the hard way it saves me time removing spam-comments.

]]> By: Doug http://www.neverendingbooks.org/index.php/the-modular-miracle-of-163.html/comment-page-1#comment-27 Doug Sat, 16 Jun 2007 16:51:38 +0000 http://localhost/?p=10#comment-27 1 - Could this near miss be due to 43, 67, 163 not being supersingular primes? 2 - Thanks for the pdf registry. I thought I had a computer dump of your blog, but it was only a print dump of 20 pages inclusive from 'bloomsday-end' to the fifth part of your NCG geometry course 'the NC manifold of a Riemann surface' without any diagrams or figures. 1 – Could this near miss be due to 43, 67, 163 not being supersingular primes?

2 – Thanks for the pdf registry. I thought I had a computer dump of your blog, but it was only a print dump of 20 pages inclusive from ‘bloomsday-end’ to the fifth part of your NCG geometry course ‘the NC manifold of a Riemann surface’ without any diagrams or figures.

]]> By: Christine http://www.neverendingbooks.org/index.php/the-modular-miracle-of-163.html/comment-page-1#comment-21 Christine Sat, 16 Jun 2007 11:55:01 +0000 http://localhost/?p=10#comment-21 What a wonderful surprise! Congratulations on your beautiful and carefullly done new blog! Christine What a wonderful surprise! Congratulations on your beautiful and carefullly done new blog!

Christine

]]> By: Kea http://www.neverendingbooks.org/index.php/the-modular-miracle-of-163.html/comment-page-1#comment-10 Kea Fri, 15 Jun 2007 22:07:15 +0000 http://localhost/?p=10#comment-10 Beautiful! Thanks for creating such a wondeful new blog. Beautiful! Thanks for creating such a wondeful new blog.

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