p-Adic length scales hypothesis for twin primes

A theorem of Yitang Zhang about distribution of distances of twin primes (see this) has caught the attention of media. The theorem states that there exists number of order 10⁷ such that the number of twin prime pairs with mutual distance smaller than this number is infinite. The naive expectation is that minimum distance between twins increases and becomes infinite at the limit as also the average distance does.

In TGD framework the result is not so counter-intuitive.

1. p-Adic length scale hypothesis implies that elementary particles correspond to p-adic primes characterizing the effective p-adic topology assignable to them. If cognition is accepted to be part of world order then one must generalize physics by gluing together real and various p-adic physics and p-adic space-time sheets are correlates for cognition. The basic hypothesis emerging from the comparison of the results of p-adic mass calculations with experimental numbers is that physically preferred p-adic primes correspond to primes p≈ 2^k, k positive integer. has as preferred values primes and Mersenne primes. Also Gaussian Mersennes assignable to Gaussian (complex) integers appear an define important p-adic length relevant to living matter (scales between cell membrane thickness and cell nucleus size. One can also consider twin pairs for $k$ and these will be indeed discussed below.
   
   
2. If one accepts the idea that elementary particles correspond to primes, one can turn the wheel around and ask whether physics might help in understanding the distribution of primes. Elementary particles form bound states. These bound states would correspond to twin pairs or larger clusters of primes near to each other. If this is the case then one can make number theoretical conjectures based on approximate scale invariance. Twin prime pairs are analogous to bound stats of particles and have some minimal size and the number of pairs with this size is infinite just like the number of twin primes. Same applies to bound states of arbitrary many twin primes. Besides twin primes there are, triple primes, etc. and also they form bound states. Same conjecture applies to the number of them having minimal size.
   

As such the idea that twin primes might be important in physics does not look interesting. One can however consider p-adic length scale hypothesis restricted so that one considers only primes p≈ 2^k, k prime (another such restriction considered earlier is that the second scale of twin scale pair corresponds to Gaussian Mersenne). What does one obtain if one considers only pairs of twin primes (k,k+2)? The twins form pairs of p-adic length scales L(k) and L(k+2) differ by scaling of 2 and already during the first years of p-adic physics I noticed that this kind of pairs might be relevant in biomatter and also in particle physics. Twins appear as basic scale pairs also in solar system as following considerations show. A reasonable conjecture is that twin pairs define an important set of fundamental length scales allowing to get a glimpse about entire physics from CP₂ scale up to longest cosmological scales.

First some background about twins and p-adic length scales hypothesis.

1. From Wolfram Mathworld ) one finds the following list of twin pairs. The first few twin primes are of form n± 1 for n=4, 6, 12, 18, 30, 42, 60, 72, 102, 108, 138, 150, 180, 192, 198, 228, 240, 270, 282, .... Explicitly, these are (3, 5), (5, 7), (11, 13), (17, 19), (29, 31), (41, 43), ... . All twin primes except (3, 5) are of the form $6n± 1$.
   
   
2. From this one can calculate p-adic length scales L(k), k=n± 1 and mass scales assuming p≈ 2^k. It is convenient to use electrons Compton length in the scale involved rather than p-adic length scales proper (as a matter fact, I confused for a long time these two scales). By L_p=L(k)∝ p^1/2≈ 2^k/2 one has L_e(k)= 2^(k-151)/2L_e(151), where scaled up electron Compton length L_e(151)= 2^{(151-127)/2=12)}L_e(127) can be also written as L_e(151) ≈ L(151)/(5+x)^1/2, where x<1 (most naturally very near to zero) is a parameter left free by uncertainties of the model behind p-adic mass calculations.
   

   The reference scale can be taken as L_e(k=151)≈ 10 nm, which is cell membrane thickness. Note that this suggests that Cooper pairs of electrons and perhaps even their light and dark variants are important. Macroscopic quantum phases of dark electrons are highly suggestive. The Gaussian integer (1+i)^k_G=151-1 defines Gaussian Mersenne as also k_G= 157,163,167. Such a large number Gaussian Mersennes in so small an integer interval can be regarded as a number theoretical miracle and must relate to the very special physics of living matter systems. The natural guess is that both p-adically scaled up and dark variants of strong interaction physics, weak physics, and electromagnetism appear in these scales varying up to the size of cell nucleus.
   
   

3. p-Adic mass scales are obtained from electron mass m_e= .5 MeV by the scaling m(k)= 2^(127-k)/2 × m_e/(5+x)^1/2. It seems physically more natural to use as the mass scale also the mass m_e(k) of electron if it were be characterized by k.
   

Consider first the twin length scales assignable to elementary particle and atomic physics.

1. n=102 defines pair (101,103). I have earlier assigned these p-adic prime to to b and c quark. One has m_e(101)= 4 GeV and m_e(103)= 2 GeV.
   
   
2. n=108 defines pair (107,109). k=107 is assigned with hadronic space-time sheets and with proton. The mass scales are m_k(107)=512 MeV and m_k(109)=256 MeV. k=109 could be assigned with deuteron. An interesting question is twin pair is relevant for the understanding of also QCD. The current masses of u and d quark (with the mass assignable to the magnetic body not included) are rather light - in the range 5-20 MeV. This means that their Compton lengths are much longer than those of hadron itself! The interpretation is that they are associated with the magnetic body of the hadron, say proton. This could explain the anomalous finding that the charge radius of proton seems to be slightly larger than it should be.
   
   
3. n=138 gives rise to pair (137,139). The pair defines atomic length scales possibly highly relevant for the condensed matter and molecular physics. The length scale pair is (L_e(137)=.79 Angstroms and L_e(139)=1.57 Angstroms. A new physics in these length scales is strongly suggestive if p-adic length scale hypothesis is accepted.
   
   

The next group of twin length scales can be assigned to living matter.

1. n=150 gives rise to the pair (149,151) corresponding to the thicknesses of lipid layer of cell membrane and cell membrane itself. The scales are L_e(149)=5 nm and L_e(151)=10 nm. Especially the scale of 10 nm appears very often in bio-systems as a basic scale: for instance, the thickness of DNA coil is of this order of magnitude.
   
   
2. n=180 defines pair (179,181) differing by a scaling by 2¹⁵≈ .512× 10⁶ from the pair defined by cell membrane. The scales are L_e(179)=2.5 mm and L_e(179)=5 mm and might relate to binary columnar structures appearing in the cortex (orientation and ocular dominance columns). The thickness of cortex is of order 2-3 mm.
   
   
3. n=192 defines pair (191,193) related by scaling with 2×10⁶ to lipid-cell membrane pair. The scales are L_e(191)=1 cm and L_e(179).
   
   

The last two scales in the length scale range considered could relate to the physics of solar system.

1. n=270 defines pair (269,271) related by scaling by scaling 10¹⁸ to cell membrane system. One has L_e(269)=5 Mkm and L_e(271)=10 Mkm. These scales are larger than the size scale .7 Mkm defined by solar radius but smaller than astronomical unit AU defined size scales for planetary orbital radii. In the model of solar system based on the notion of gravitational Planck constant Earth corresponds to n=5 Bohr orbit with radii proportional to n². The radius of n=1 orbit would correspond to the length scale AU/5²≈ 5.98 Mkm rather near to L_e(169) so that the size scale of gravitational atom would be in question. The pair of atomic length scales (137,139) would be replaced with its gravitational variant (269,271). The interpretation proposed earlier is that the invisible gravitational orbitals correspond to dark matter around which no visible matter has condensed (yet).
   
   
2. n=282 defines pair (281,283) related by scaling 32 to the previous pair. The scales are 160 Mkm and 320 Mk. The radius of the orbit of Earth defines astronomical unit AU=149.60 Mkm and again the order of magnitude is same.
   
   

It is remarkable that electron Compton scales in various p-adic length scales are in question which suggests that dark electrons are involved in an essential manner. To my opinion these findings are so remarkable that p-adic length scale hypothesis deserves serious consideration since it also leads to excellent predictions for elementary particle masses.

There is also an exceptional twin pair not considered above. It is formed by primes 2 and 3. Recently I have been developing a model of music harmony based among other things on the interplay of these primes. Octaves in music correspond to powers of 2 and under octave equivalence the notes of 12-note scale constructed by using quint rotation correspond to powers of 3/2 of equivalently of 3. One has both 2-padic and p-adic aspects. The natural distance for notes corresponds to the number of quints related the notes in quint rotation. This hypothesis allows to understand basic facts about harmony. Some time ago I commented about an article that I received from Jose Diez Faixat (see this) claiming what in TGD framework translates to the statement that 3-adicity is realized at the level of biological time scales. This conforms with the TGD based icosahedral model for biosystems suggesting a close relationship between twelve note scale and aminoacids and also leads to a proposal for icosahedral realization of genetic code (see this).