Tuesday, November 22, 2011

Theory Update 127

Dolan's triality studies conformal field theories at $c = 24 = 8 \times 3$, where an $8$ is associated to the qubit edge of a tetractys. It should not surprise us then that massive theories contain trialities using the centre of the tetractys. As it happens, the four largest Heegner numbers all equal $1$ mod $6$.

These are the numbers $d$ that generate near integers using the j-invariant value

$j( \frac{1 + \sqrt{-d}}{2} ) = k^3$

for $k$ an integer. The number $k^3$ always has a factor of $27 \times 8$. The Euler primes that generate the Heegner numbers are supersingular. Observe that the largest Heegner number uses the dimension of the bioctonion Jordan algebra.
Strangely, if we restrict to the massless dimensions, such as $24$ in $27$, we obtain the following.


  1. Here is an interesting comment that discusses a moonshine link between Heegner numbers are supersingular ones.

  2. Further, strangely, we have

    $67 = 4 \cdot 6 + 43$
    $163 = 20 \cdot 6 + 43$
    $163 = 16 \cdot 6 + 67$

  3. Thus the secret dimensions linking $163$ to the other three large Heegner numbers are $16$, $20$ and $24$, which are just $4 \times (4,5,6)$.

  4. Another way of saying that is

    $163 = 6 \cdot 24 + 19$
    $67 = 2 \cdot 24 + 19$
    $43 = 1 \cdot 24 + 19$

    so I guess $19$ really is a very nice prime. Maybe that's why Pythagoras said that the last row of the tetractys ($19$ + $8$) completes ...

  5. The reduction in the Monster group character table goes from 194 to 171 to 163 independent 'moonshine' mini-j functions (a span of 31, 23 and 8). I wonder if this figures in the Planck scale QG and the hierarchy mass problem. In addition could this be related to a graded series of the j-invariant coefficient 196884 to the kissing number 196560 by CM (complex multiplication) a span of 324?

  6. Louise Dolan's paper was from a talk at Alabama in 1989
    which I attended and got from my copy of the proceedings.
    It is on my web site at
    Although that was a long time ago, my recollection is
    that she is a very good person who had some interesting
    experiences as a female in physics.

    As to the number 19, Rashad Khalifa said:
    "... "... Why 19 ? Because 19 means "God Is One" ...
    ... this is the message of Quran. ...
    ... The Quran consists of 114 chapters, and this number is a multiple of 19 ( 19x6 ). The last chapter is No. 114, and consists of 6 verses.
    ... The first Quranic revelation consisted of 19 words ... found in the first 5 verses of Chapter 96
    ... The first Quranic revelation consisted of 76 letters, and the number 76 is a multiple of 19 ( 76 = 19x4 ) ...
    ... At the time of revelation of Quran, the numerals as we know them today were not in existence. Instead, the alphabet letters were used as numerals ... In Arabic, the word "One" means "Waahid" ... the numerical value ... is ... [ 6+1+8+4 = 19 ] ...".

    However, his mathematical studies of the Quran led him to question the two suras 9:128-129
    for that he was killed (in 1990 in Arizona by the radical Islamic sect Al Fuqra).


  7. I should have added that Luise Dolan's Alabama talk is a bit earlier than the Phys. Lett. B paper and that there is also a later paper at


  8. Thanks, Mark and Tony. Apparently, in regards to Tony's remarks, Pythagoras was considered divine, presumably because there was a clear continuity of ideas back to his time.

  9. Kea,

    I took from your links the discussion on super-singular and other low prime numbers put them into my grid and got some rather interesting patterns.

    The adding of one as in the 56 or 57 dimensions does refer to more than even and odd and how we see as did the Greeks of zero or one as the first dimension. (without zero they developed a more geometric style- more rhetorical than syncopated view). We may add a dimension or see some negative absolute value as less than zero- but more can be observed here than something presumed non-linear. (I use Pythagorean concept of gender as an old view of number-not to be mystical in the sense such ideas are now viewed.)

    I also presented a rather recreational model of the 15 supersingularity primes in the colour cubes- for now of interest for its own sake and to show perhaps that things may be a little wider in view than absolutely symmetrical.

    What these things would be if they are particles I am not sure- but I do agree with you things like mentioned in your next post may be a little vague and fanciful- mirror neutrinos in and hard to see- but no gravitons, gravitinos or fairies. (of which we may also have a historical disconnect as well that between physics and mathematics.

    The Pe Sla


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