### Mersenne MUBs

Consider the MUB matrices in prime dimensions $p$. If we demand that the entries form a finite field of $p + 1$ elements (that is, including $0$) it follows that $p$ must be a Mersenne prime.

It is not known if there are an infinite number of such primes or not, but Euler showed that any perfect number $n$ may be written in the form

$n = \frac{1}{2} M_i (M_i + 1)$

for some Mersenne prime $M_i$. In particular, the perfect number $6$ comes from the Mersenne prime $3$.

It is not known if there are an infinite number of such primes or not, but Euler showed that any perfect number $n$ may be written in the form

$n = \frac{1}{2} M_i (M_i + 1)$

for some Mersenne prime $M_i$. In particular, the perfect number $6$ comes from the Mersenne prime $3$.

## 3 Comments:

Interesting, and of course you mean that any

evenperfect number takes that form. The case for odd perfect numbers is still unsettled.So do the MUBs form a finite field for Mersenne primes?

Really interesting. Mersenne primes define in p-adic mass calculations physically preferred p-adic mass scales L_p, p =about 2^k. M_k, k=89, 107, 127 correspond to intermediate gauge bosons, hadronic space-time sheet, and electron. k =127 is the largest p-adic Mersenne scale which is not completely super-astrophysical.

I would be happy to find mathematical and physical justifications for why the primes very near to powers of 2 are winners in fight for survival at elementary particle level.

Also Gaussian Mersennes are important. The one near 2^k, k=113 corresponds to muon mass scale. There is entire bundle of them- those near to 2^k, k=151,157,163,167 - in the biologically interesting range from 10 nm to 2.5 micrometers.

I was just thinking aboiut this a tiny bit more. If you look at the more general case where the dimension of the MUB is a prime power p^n, then you might also have examples where p is two and 2^n+1 is a Fermat prime. I think there is only one other case where prime powers differ by one which is 3^2 = 2^3+1.

But can any of these sizes of MUB give cases where the entries actaully form a finite field? There might be some interesting exceptional structures if they do.

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