_{n}=2

^{n}-1 obviously satisfy this condition optimally. The proposal generalizes to GAussian Mersenne primes M

_{G,n}=(1+i)

^{n}-1. It is now possible to understand preferred p-adic primes as so called ramified primes of an algebraic extension of rationals to which the parameters characterizing string world sheets and partonic 2-surfaces belong. Strong form of holography is crucial: space-time surfaces are consctructible from these 2-surfaces: for p-adic variants the construction should be easy by the presence of pseudo-constants. In real sector very probably continuation is possible only in special cases. In the framework of consciousness theory the interpretation is that in this case imaginations (p-adic space-time surfaces) are realizable. Also p-adic length scale hypothesis can be understood and generalizes: primes near powers of any prime are preferred.

The definition of p-adic length scale a convention to some degree.

- One possible definition for L
_{p}is as Compton length for the smallest mass possible in p-adic thermodynamics for a given prime if the first order contribution is non-vanishing.

- Second definition is the Compton length L
_{p,e}for electron if it would correspond to the prime in question: in good approximation one has L_{p}= 5^{1/2}× L_{p,e}from p-adic mass calculations. If p-adic length scale hypothesis is assumed (p≈ 2^{k}) one has L_{p,e}== L(k,e)=2^{(k-127)/2}L_{e}, where L_{e}is electron Compton length (electron mass is .5 MeV). If one is interested in Compton time T(k,e), one obtains it easily from electrons Compton time .1 seconds (defining fundamental biorhythm) as T(k,e)= 2^{(k-2×127)/2}× .1 seconds. In the following I will mean with p-adic length scale T(k,e)≈5^{-1/2}× T(k).

Mersenne primes M_{n}=2^{n}-1 are as near as possible to power of two and are therefore of special interest.

- Mersenne primes corresponding to n∈{2, 3, 5, 7, 13, 17, 19, 31, 61} are out of reach of recent accelerators.

- n=89 characterizes weak bosons and suggests a scaled up version of hadron physics which should be seen at LHC. There are already several indications for its existence.

- n=107 corresponds to hadron physics and tau lepton.

- n=127 corresponds to electron. Mersenne primes are clearly very rare and characterize many elementary particle physics as well as hadrons and weak bosons. The largest Mersenne prime which does not define completely super-astrophysical p-adic length scale is M
_{127}associated with electron.

- n∈{2, 3, 5, 7, 11, 19, 29, 47, 73} correspond to energies not accessible at LHC. n= 79 might define new copy of hadron physics above TeV range -something which I have not considered seriously before. The scaled variants of pion and proton masses (M
_{107}hadron physics) are about 2.2 TeV and 16 TeV. Is it visible at LHC is a question mark to me.

- n=113 corresponds to nuclear physics. Gaussian Mersenne property and the fact that Gaussian Mersennes

seem to be highly relevant for life at cell nucleus length scales inspires the question whether n=113 could give rise to something analogous to life and genetic code. I have indeed proposed realization of genetic code and analogs of DNA, RNA, amino-acids and tRNA in terms of dark nucleon states.

- n= 151, 157, 163, 167 define 4 biologically important scales between cell membrane thickness and cell nucleus size of 2.5 μ m. This range contains the length scales relevant for DNA and its coiling.

- n=239, 241 define two scales L(e,239)=1.96× 10
^{3}km and L(e,241)=3.93× 10^{3}km differing by factor 2. Earth radius is 6.3 × 10^{3}km, outer core has radius 3494 km rather near to L(2,241) and inner core radius 1220 km, which is smaller than 1960 km but has same order of magnitude. What is important that Earth reveals the two-core structure suggested by Gaussian Mersennes.

- n=283: L(283)= .8× 10
^{10}km defines the size scale of a typical star system. The diameter of the solar system is about d=.9 × 10^{10}km.

- n=353: L(353,e)= 2.1 Mly, which is the size scale of galaxies. Milky Way has diameter about .9 Mly.

- n=367 defines size scale L(267,e)= 2.8× 10
^{8}ly, which is the scale of big voids.

- n=379: The time scale T(379,e)=1.79× 10
^{10}years is slightly longer than the recently accepted age of the Universe about T=1.38× 10^{10}years and the nominal value of Hubble time 1/H=1.4× 10^{10}years. The age of the Universe measured using cosmological scale parameter a(t) is equal to the light-cone proper time for the light-cone assignable to the causal diamond is shorter than t.

For a summary of earlier postings see Links to the latest progress in TGD.