Remarkable Properties of the Eddington Number 137 and Electric Parameter 137.036 excluding the Multiverse Hypothesis

Considering that the Large Eddington Number has correctly predicted the number of atoms in the Universe, the properties of the Eddington electric number 137 are studied. This number shows abnormal arithmetic properties, in liaison with the 5th harmonic number 137/60. It seems that Egyptians was aware of this, as the architecture of the Hypostyle Karnak room reveals, as well as the Ptolemaic approximation π ≈ 2 + 137/120, together with a specific mention in the Bible, and overhelming connexions with musical canonic numbers. The SO(32) characteristic superstring number 496, the third perfect number, connects directly with the three main interaction parameters, and is very close to the square root of the Higgs boson-electron mass ratio, so the Higgs boson discovery excludes the Multiverse, favoring rather a unique Cosmos being a finite computer using 137 and its extension 137.036 as calculation basis. The direct liaison between the mean value of the two main cosmic radiuses and the Bohr radius through the simplest harmonic series excludes any role of chance. Precise symmetric relations involving the Lyuty-Kotov non-Doppler period permit to propose precise (100 ppb) values for the weak and strong interaction constants, as well as G ≈ 6.6754635 × 10-11 kg-1 m3 s-2, at 2σ the tabulated value. 1. The Large Eddington Number: the most astonishing prediction in Physics of all times Eddington believed he had identified an algebraic basis for fundamental physics, which he termed "E-numbers" (representing a certain group– a Clifford algebra). These in effect incorporated spacetime into a higher-dimensional structure. While his theory has long been neglected by the general physics community, similar algebraic notions underlie many modern attempts at a grand unified theory. Moreover, Eddington's emphasis on the values of the fundamental constants, and specifically upon dimensionless numbers derived from them, is nowadays a central concern of physics. In particular, he predicted a number of hydrogen atoms in the Universe [1] 136 x 2256, or equivalently the half of the total number of particles protons + electrons. When equalized with the non-dark energy equivalent number of hydrogen atoms (the factor 10/3, as well as the critical condition, are trivial in the simplest cosmology [2]) NH = (3/10) Rc /GmH => R = 13.8 Glyr (1) this corresponds to a Universe radius R = 13.8 Giga light year, a value predicted for years from universal constants using an atomic-cosmic symmetry [3], and compatible with c-times the socalled Universe age 13.80(4) Gyr, as determined by the recent mission Planck (March 2003). This formula traduces the double large number correlation in the manner Eddingon presented it, with lH = ħ/mH c: R/2lH ≈ √(M/me) ≈ ħc/Gmemp => R = 13.8 Glyr (2) which implies directly the gravitational force in the Hydrogen atom [2][3]. The Eddington original form is R/2σ = √N, but Eddington was not able to deduces the above formula exhibiting the electron-proton symmetry, which was one of his essential hypothesis, because of the error of an order of magnitude of the first estimation of Lemaître for the redshift constant (an erroneous value strangely confirmed by Hubble et Humason). 2. Eddington and 137 Eddington has demonstrated [1] that, in reduced units, the square of electric charge must be 137. When it was precisely measured, it turned to be 137.036, and Eddington approach was rejected. This rejection is not conform with the traditional 'approach' method of physics. Indeed , there is a dircet relation bvetween these numbers: 1372 + π2 ≈ 137.0362 (3) So, it is worth asking mathematician 'does 137 appear special in Number Theory?' The general answer is '137 is unknown in Number Theory, it has no remarkable property' This appear as a contradiction 'How can the Nature can be driven by mathematics, when appears a number unknown by mathematician ?' Two possible answers Multiverse Solution : Such an electrical constant is a random number, characterising a Universe among a multitude : it is the Multivers hypothesis. So the number is a completely 'free' number, and there is no need to look for any special mathematical properties. Universe Solution : 137 has special properties belonging to a part of mathematics undechifred by present mathematicians. We show here that the second way seems the right way 3. The largest primes in Harmonic Numbers Indeed, 137 is the 33ième prime number. Now the distribution of prime numbers is tied to the zéta Riemann function, itself a generalisation of the harmonic series Σ(1/n). Now, let us decline the prime numbers emerging in numerators of these harmonic numbers: 3, 11, 5, 137, 7, 11, 761, 7129, 61, 863, 509, 919, 1117, 41233, 8431, 1138979, 39541, 7440427, 11167027, 18858053, 227, 583859, 467183, 312408463, 34395742267, 215087, 375035183, 4990290163, 17783, 2667653736673, 535919, 199539368321... (4) Believe it or not, 137 appears as an arithmetic monster, not detected by our brillant mathematicians during a century ! One reason for this is that at the epoch of the mathematical fundators, the number 137 was not revealed by physical measurements. And after 137 was finally revealed by physics, modern mathematicians generally do not care with physics. In fact, the above series is described in the 'on-line encyclopedia of integer sequences' under the following complicated definition: 'largest prime factor of Stirling numbers of first kind s(n,2)', unstead of the simple one 'largest prime factor in the numerator of harmonic series'. But this identification was not published. So the 5th harmonic number exhibits 137: 1+1/2+1/3+1/4+1/5 = 137/60 (5) Now 137 is also a 'central polygonal number of the type x(x+1)/2+1, with x = 16 for 137. This means that cutting a cake in 16 strikes give a maximal 137 parts. In turn, 16 is the number of parts for 5 strikes. Let us write this: 137 = C(16) = C(C(5) (6) while C(60) = 1831 (7) and one observes: √(C2(60) + C2(16)) = √(18312+1372) ≈ 6π5 (8) which is, to 43 ppb, the famous Lenz approximation of the proton-electron mass ratio. 4. The number 11 of superstring dimensions The number 11 appears two times in the above harmonic series, so it appears also as a monster, and, moreover, it is both the number of dimensions in the superstring theory and 11 = C(4) = C(C(C(1)). So the question: is there a relation tying 11 and 137 ? Indeed : 112 + 42 = 137 (9) So 137 have the triple property, tied to x = 4 and y = x+1 = 5: 137 = 1+ x2(x2+1)/2 = x2 + (x(x+1)/2 +1)2 = 1+ (1+ y(y+1)/2)( 2+ y(y+1)/2)/2 (10,11,12) the two first relations reduce in (x-4) (x+1)2 = 0 (13) showing x = 4 is the only positive solution. So 11 (superstring dimension number) and 4 (usual dimension number) are tied together, through 137, which can be considered a number of dimensions in Eddington's Theory. Moreover their ratio 11/4 is, in the standard cosmic statistical theory, the ratio of the temperatures of the background fields (photons / neutrinos). It seems that Nature uses it as an approximation of the optimal base 'e'. Indeed, with d ≈ 1.001159652, the abnormal electron magnetic moment, and aF ≈ 573007.4 the Fermi-electron mass ratio : 11/4 ≈ e d10 ≈ √(6aF)/alna (14) Now C(11) = 67, which is related to 137 and the other monster '61' in the above 'harmonic series': 2 × 67 + 3 = 137 (15) 67 = 61 + 6 (16) Where 6 is the first perfect number. Note also that the mass of the scalar boson, by respect to the electron one, is closed to, with aF the Fermi ratio and as the inverse of the strong interaction constant as ≈ 1/0.1184(7) √s ≈ √(134 p) ≈ 496 ≈ aF/asa (17) where 496 = 24 (25 − 1) is the third perfect number, tied to the Mersenne prime 31 = 25 – 1. Now, the number 496 is a very important number in superstring theory. In 1984, Michael Green and John Schwarz realized that one of the necessary conditions for a superstring theory to make sense is that the dimension of the gauge group of type 1 string theory must be 496. The group is therefore SO(32). Their discovery started the first superstring revolution. This would confirm the central importance of the scalar boson (or Brout-Englert-Higgs). But the present work shows it is a strong argument against the Multiverse. 5. The Harmonic series and Egyptians Now Egyptians used only entire fractions of unity, so they probably was aware of the above singular property of 137, tied to the harmonic series. Indeed, the Hypostyle Room in Karnak shows 134 huge columns placed between the second and the third pillars of the Amon Temple. On each side of the main axis there are 61 columns + 6 huge ones, which is precisely the above relation (16). The 61 colums are separated by a 'royal axis' into 28 and 33 ones 61 = 33 + 28 (18) Now 137 is the 33th prime number, while 28 is, after 6, the second perfect number (equal to the sum of its divisors, including 1). Note that on each side, in the row of 6 huge columns, the extremal one is partly inserted in the wall, as if the architech has tried to represent the root of 137, which is very close to 11+1/√2. In fact the approximation is better for √a: √a ≈ 11+1/√2 (19) Note that the 61 columns show a square of seven ones, with the separation 61 = 72 + 2 × 6. Now 2 × 67 = 134 = 7 + 127 (20) where 127 is the Mersenne number or order 7. And 7 itself is the one of order 3. So the sum of the Series (kown as the Combitational Hierarchy CH) gives 137, from 3 = M(2): 137 = 3 + M(3) + M(M(3)) (21) The following term is M(M(M(3))) = 2^127 1, which is known to be also a prime, is about half the Hubble radius, by using the electron wavelength le = ħ/mec as unit, to 0.6%.: (2127 – 1) le ≈ 13.9 Glyr / 2 (22) One may consider that Egyptians have devined that this term (which is the ending one in the CH) was of cosmic significance, because the story tells that the pharaon was acustomed to meditate at the center of this Hypostyle room [4]. Note that 3 and 7 = M(3) which are considered as magic numbers in all times, appear in the above prime number series, and moreover, 7 is the numbers of parts in a 3 strikes maximal cut. So they form a very particular duality, and their sum is histotically known as the 'tetractis': 3 + 7 = 1 + 2 + 3 + 4 = 10 (23) By symmetry one must consider the sum completd by the 5 and 11, 3+5+7+11 = 26 = 10 + 16 (24) to get 26, the dimension number of the bosonic string theory. One cannot escape the conclusion that the ancian egyptians have a predilection for perfect numbers 6 and 28, and they managed to make a correspondance with 137 and its ordinal number 33, containing the 11, with also the liaison to the Catalan Sequence. This is also called the Combinatorial Hierarchy, but for the later, the following term 2127 – 1 is the ending one. Note that the Ptolemaic approximation for π contains also the fith harmonic number 137/60: π ≈ 377/120 = 2 + 137/120 (25)  Let us recall that, among non-resolved mathematical problems is that of the fractional development of π : 3,7,15,1,292.6346, this last 'monstruous' term being n/2π within 4 ppm imprecision, where n ≈ 1838.6836 is the neutron-electron mass ratio. 6. The Bible and 137 From the above observations, it was proposed long ago that the number 137 would be known by ancient civilisations. Indeed, according to the Bible, while Jesus lived 33 years (137 is the 33 prime), the two sons of Abraham lived for 137 and 180 years, so 137 is related to 60. Now 180 is very close to (n/a)2, and: 180 = 5 × 62 (26) corresponding to one quark u = 5 and two quarks v = 6. In this numerical hypothesis [5], the proton would corresponds to 6 × 52 = 150. Indeed (6 × 52)3/2 ≈ 6π  5 + 1 (27) where p ≈ 65 is the above famous Lenz approximation for the proton-electron mass ratio. Eliminating it with n gives: