# 4 11 26 dimensions?

5 x 5 = 1 x 24 + 1
7 x 7 = 2 x 24 + 1
11x11 = 5 x 24 + 1
13x13 = 7 x 24 + 1
17x17 =12 x 24 + 1
19x19 =15 x 24 + 1
23x23 =22 x 24 + 1

For those who think this is totally irrelevant, I've just realised the link with the article by John Baez, refered to earlier.

The other basic magical formula involving 24 is:

1x1 + 2x2+ 3x3 + ....... + 24x24 = 70x70.

Proof:
1x1 + 2x2 + ...nxn = n/6. (n+1).(2n+1).

24/6 = 4 = 2x2 [as Baez says it all boils down to 4x6 = 24]
24+1 = 5x5
2.24+1 = 7x7

2x5x7 = 70

arivero
Gold Member
$$\epsilon_{0} = \frac{d-2}{2} \sum_{m=1}^{\infty} m$$

A similar computation can be done in superstring theories, but this gives d=10.

Dimitri, do you have a good reference for the "similar computation" in superstring theories? Naively, I'd expect it to use $$\sum_{m=1}^{\infty} m (-1)^m$$

arivero
Gold Member
Opps, it was a 2006 post...

arivero
Gold Member
Dimitri, do you have a good reference for the "similar computation" in superstring theories? Naively, I'd expect it to use $$\sum_{m=1}^{\infty} m (-1)^m$$

I think I found it, or a part of it for a concrete superstring example. For the vacuum in the NS sector of the open superstring, the sum
$$\sum_{m=1}^{\infty} m = 1 + 2 + 3 + 4 + 5 + 6 + ... = -1/12$$
is replaced by a difference between a bosonic and a fermionic piece
$$2 (\sum_{m=1}^{\infty} m - \sum_{r=1/2}^{\infty} r ) = (2 + 4 + 6 + 8 + ...) - ( 1 + 3 + 5 + 7 + 9 + ...) = (-1 + 2 - 3 + 4 -5 +6 - ...) = -1/4$$

But:

-the GSO truncation removes the tachion, so it seems less meaningful that in the pure bosonic case.

-One would prefer a more straightforward way, linking the $(-1)^m$ term to superspace.

EDIT: A lot of related topic flourish here, also Dirichlet series and Dedekind Eta Functions. For the latter, perhaps Conway et al have some better explanations about the links between 24 and 8.

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