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Remarkable Properties of the Eddington
Number 137 and Electric Parameter 137.
excluding the Multiverse Hypothesis
Francis M. Sanchez
February-May 2015
Abstract. 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 Kotov non-Doppler period permit to
propose precise (100 ppb) values for the weak and strong interaction constants, as well as G ≈
6.675464346 × 10
kg
m
s
, at 2 σ higher than the tabulated value.. This value is confirmed
by a direct connection with the Superspeed ratio C/c ≈ 6.945480 × 10
1. Eddington and 137
Eddington has established [1] that, in reduced units, the square of the inverse electric charge must
be 137. But, when it was precisely measured, it turned out to be a ≈ 137.036, and Eddington
approach was rejected. This rejection is not conform with the traditional 'approach' method of
physics. Indeed, there is a direct relation between these numbers:
137² + π² ≈ 137.036² (1.1)
a/137 ≈ a
1/a² (1.2)
precise respectively to 0.1 and 0.7 ppm. So, it is worth asking mathematicians 'does 137 appears as
a special whole 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 arise:
Multiverse Solution : Such an electrical constant is a random number, characterising a Universe
among a multitude, each with a different set of numerical parameters: it is the Multiverse
hypothesis. So the number a would be a completely 'free' number, and there is no need to look for
any special mathematical properties.
Universe Solution : 137 and a have special properties, belonging to a part of mathematics not
revealed by present mathematicians.
We show here that the second way seems the right one.
Note that Armand Wyler [2] has established a formula, from the consideration of geometrical
coefficients in hyper-spheres of higher dimensions, in relation with
p ≈ 6 π
the famous empiric Lenz formula, approaching the proton-electron mass ratio p to 18.824 ppm.
Note that the Wyler formula can be written in function of V 5 (1/π) = (8π²/15)(1/π)
, the volume of
the 5-sphere of radius 1/π, and the volume of the 3D sphere of radius √√2 :
√a ≈ V 3 {√√2} /V 5
{1/π} (1.4)
giving a to 0.61 ppm. Wyler was invited a year at Princeton by Tyson, but the american staff did not
take this work seriously, in spite of its characteristic holographic character (the Holographic
Principle was not expressed at the epoch), and its supplementary work was not even published by
Princeton!
2. 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). Indeed, this 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 2
, 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, as recalled below [3])
N
H
= (3/10) Rc
/Gm H
=> R = 13.8 Glyr (2.1)
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 [4], and compatible with c -times the so-
called 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 (2.2)
which implies directly the gravitational force in the Hydrogen atom [3][4]. The Eddington original
form is R/2 σ = √N, but Eddington was not able to deduce the above formula exhibiting the electron-
proton symmetry, which was precisely 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 later by Hubble et Humason, on the basis of an unique very far galaxy).
The elimination of ħ in (2) leads to M = ( m H
/ m p
) Rc²/2G , where the main term Rc²/2G is the
classical formula of the horizon radius of a black hole, and corresponds to the 'cosmic critical
condition', justified below. Note that the corrective factor ( m H
/ m p
) is eliminated, by replacing in
(2), M/me by M/m'e , with m' e = me / (1+1/ p ) the corrected electron mass in the Hydrogen atom. This
means that the relation bretwee M and R/2 is given by the elementary 3-fold (Length-Mass-Time)
dimensional analysis excluding ħ.
Note that the first correct estimation of the atomic dimension was established three years before
Bohr by Arthur Haas, simply by equalizing theree expressions of energy, including the Planck
energy form E = hf , and identifying the frequency f = v/2 π r with that of a circulating electron in
the simplest model: this gives half the bare atomic radius : =
E = mv²/2 = e²/r = hf (2.12)
The elimination of the electron speed v leads to the length 2 ħ²/e²me i.e. twice the above value.
It follows that, while it is not a-priori clear why the speed c does not enter the Bohr radius, this
is mandatory for a cosmic radius. So the cosmic calculation is simpler that the atomic one, and this
gives exctly half the redshift radius. So the above Haas equalisation was called the 'Coherence
Principle', generalizing the Virial Theorem, and this leads directly to the calculation of the above
factor 10/3 appearing between the total Universe energy and the matter energy. So the so-called
dark energy problem is not a problem at all.
This would mean that dark matter is a special form of ordinary matter; the simplest explanation
being it is vibrating in quadrature with ordinary matter-antimatter vibration.
Note that another elimination of c gives a length close to the above redshifty radius. Indeed
eliminating c betwwen the Planck length l P
= (G ħ c
and the classical electron radius r e
ħ/mNc , where m N
= ame is the Nambu mass, of central importance in particle physics (see below)
this defines the half length R'/2:
R '/2 = r e
/ l P
= ħ² /G m N
This value is greater than the above length R /2 by the factor:
R'/R = m e
em p
p m H
H / m N
revealing a quasi-holographic relation relating the lengths associated with the Universe Bekeinstein-
Hawking entropy of a R-radius black hole:
π (R / l P
)² = π (R / lr e
( R'/2R ) ≈ (2 π /3) (R / r e
This corresponds to half the redshift sphere volume/. By introducing the length l such that
l
= l P
² R
/3 r e
≈ R² r e
This introduses a double full-sphere holographic system:
π( r / l P
)² = (4 π /3) (R / lr e
π( R / l )² ≈ 2 π R / l 2.18)
This length is l corresqponds to a ligth time of 9684.17 s, too close to the Kotov period of the non-
Doppler oscillations (called the 'coherent period)' t c
≈ 9600.61 s, observed in several quasars. Such
a phenomena is such a ruputure to traditional physics that it is not taken seriously by ordinary
physicists. But in the real of a coherent cosmology, as said above, speeds far greater than c are
needed. So, such a phénomena was anticipated by the author, and, since the ratio to the electron
time l e
/ c is of the order P, the mass ratio Planck-electron, it took only seconds to find the first part
of the relation:
t c
/t e
≈ PF/√(pH) ≈ μ
/a (2.19)
with F the Fermi mass 573007.33 m e
. defined to 0.25 ppm. The second part involves the muon-
electron mass ratio μ ≈ 206.76829 (30 ppb). Thus a cosmic precise phenomena is directlu associated
with precise parameters of particle physics. At the epoch of this discovery, F was defined only in
the 10 ppm range, so the relation permitted to anticipate correctly about a factor 40 in the
determination of F. Now this parameter is precisely deduced from the muon lifetime T μ
, by its
relation with the muon period t μ
= ħ/m μ
c
T
μ
/t μ
= (8π
( F/ μ)
/π (2.20)
Now (19) implies
F/ μ = μ √(pH) /a (2.21)
so that (20) is independant of μ under the form:
T
μ
/t μ
= (8π
F
pH/πa
3. The Black Atom model
In the Black Atom model [4], [5], [9], the quantification of kinetic momentum leads to the
following formula for the mean distance between proton and electron, With le = ħ/mec as unit length,
the sums running between 2 and N = √(RR')/ le , one gets, with the Euler constant γ ≈ 0.
Σ(1/n)/Σ(1/n²) = (lnΝ + γ −1)/(π²/6−1) = 137.1145 ≈ (1+1/ p ) × 137.040(8) (3.1)
This is precisely, in the 1% precision of R , the Bohr radius, which is rB = a(mH/mp)ħ/mec , taking
into account the classical correction 1+1/p ≈ H/p. This formula interprets the rough estimation a ≈
ln(aG) , which is central in the famous article of Carr and Rees [8], but destroys completely their
'anthropic' interpretation favoring the Multiverse.
4. The largest primes in Harmonic Numbers
Coming back to fundamental, 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 maximal prime numbers emerging in numerators of these harmonic
numbers:
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 (5.2;3;4)
the two first relations reduce in
(x-4) (x+1)² = 0 (5.5)
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 coefficient, and aF ≈ 573007.4 the Fermi-electron mass ratio :
11/4 ≈ e d
≈ √(6 aF )/ alna (5.6)
Now C(11) = 67, which is related to 137 and the other monster '61' in the above 'harmonic series':
2 × 67 + 3 = 137 (5.7)
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,
√s ≈ √(134 p ) ≈ 496 ≈ aF / asa (5.9)
with aF the above Fermi ratio and a s
the inverse of the strong interaction constant as ≈ 1/0.1184(7)
: where 496 = 2
− 1) is the third perfect number, tied to the Mersenne prime 31 = 2
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.
6. The Harmonic series and Egyptians
Note that 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 the Amon
Temple of Karnak shows 134 huge columns placed between the second and the third pillars of the
Amon Temple. On each side of the main axis (called 'divine') 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
Now 137 is the 33
th 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 (6.2)
Note that the 61 columns show a square of seven ones, with the separation 61 = 7² + 2 × 6, showing
that 7 is a privilegied number. Indeed:
2 × 67 = 134 = 7 + 127 6.3)
where 127 is the Mersenne number or order 7. And 7 itself is the one of order 3. This is known as
the Catalan sequence. So the sum of this Series (kown as the Combitational Hierarchy [7]) gives
137, indeed, from 3 = M(2):
137 = 3 + M(3) + M(M(3)) (6.4)
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%:
One may consider that Egyptians have devined that this term was of cosmic significance, because
the story tells that the pharaon was acustomed to meditate at the center (the common point of the
divine and royal axis) of this Hypostyle room [5].
Note that 3 and 7 = M(3) which are considered as magic numbers in all epochs, 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':
By symmetry one must consider the sum completd by the 5 and 11,
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 2
- 1 is the ending one. Note that
the Ptolemaic approximation for π contains also the fifth harmonic number 137/60:
π ≈ 377/120 = 2 + 137/120 (6.8)
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.
7. The Bible and 137
From the above observations, the author 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
th prime), the two sons of Abraham lived for 137 and 180 years, relating 137 with 60. Now
180 is very close to ( n/a )², and:
these two large numbers approches R'/le (to 0.03%) and R/le (to 0.6%), involving the two principal
cosmical radiuses: R being the normal redshift radius, while R' is the radius of the holographic
sphere representing the Grandcosmos behind [8].
The scale n° 4 is with 306 ≈ π
≈ p /6 notes. This writes 2
, or 2
a
a /
. So
p / ≈ a
a (8.3)
Replacing 3 by the optimal base e , with d ≈ 1.001159652 the abnormal electron magnetic moment:
e
1836/ed√d ≈ a
a (8.4)
Note that the definition of the optimal base e is the value for which x
1/x is maximal. So the above
formula have clearly an informative signification. Now the number 1836/ d√d shows a double
remarkable singularity :
1836/ d√d ≈ e² (2 π )
The two last expressions show a deviation close to p / 6 π
, so:
p ≈ e² π
precise to 0.8 ppm, and the following remarkable relation shows symmetry between and its biblic
value 3. :
( π
/e)
The large number associated to the above 'extraordinary n° 5 musical scale' shows an overwhelming
relation with μ ≈ 206.7683 the muon mass (with electron mass as unity), f the Fermi mass, and Z,
W the weak boson masses :
≈ μ
a ≈ a
exp(5) ≈ p
Z/p ≈ W
2f/137² (8.8)
Curiously, the triangular number 666 is mentioned in the Bible as a 'demoniac number'. Note the
relations, with H the Hydrogene-electron mass ratio:
p/√ ( H6 π
) ≈ 137/ a (8.9)
( R'/R )
= ( pH / a
≈ ( 3H/2p )( 6 π
/p )
precise both in the ppb range. This means p, 6 π
, H, 137 and a are tied in the ppb range (
Now, H is eliminated in (8.10) leading to:
p
( p /(6π
≈ A (3×137²/2) (8.11)
with
A ≈ 3 a
Coming back to the very origin of quantum mechanics, it was assumed that the electron in the
Hydrogen atom do not radiate power. The latter would be precisely
PH = ( m e c ²)²/( Aħ ) (8.13)
So, it seems significative that this canonic number A enters Eq. (40) in the ppb range (
).The
relation (8.11) will be used in Section 14 to confirm the G value proposed in Section 13.
9. The Golden Number ϕ and 137
137 and 60 are also related by:
1+1/2+1/3+1/4+1/5 = 137/60 ≈ ϕ
It follows that the generalised Fibonacci series, begining by 1, 4, is very particular, containing both
60, 127 and 137 :
1, 4, 5, 9, 14, 23, 37, 60, 97, 157, 2× 127, 3×137 , 665 ... (9.2)
which contains also the remarquable above number
665 = 666 -1 = 36×37/2 -1 (9.3)
All this cannot be due to chance. This milits for the unicity of the Universe, a strong argument
against this scientific deviation, called the Multiverse.
Also
a ≈ ϕ^(a²/(1836 + 1/ϕ) (9.4)
precise to 2 ppm, while 1836+1/ϕ ≈ 6 π
+1/2, to 41 ppb.
Now, there is a relation involving the 360° circonference : the 'golden angle' 360/ ϕ
is about
137.5°. This could explain the liaison between 137 and 180 in the Bible. See section 16 for another,
more direct liaison.
10. The Ratio a /
The Ratio a /137 shows overhelming properties. First of all, it is a caracteristics part of the indian
comma 2
, in the following ppb formula:
a /137 ≈ 2
Secondly, the number of dimension N = 196883 of the 'monster' in group theory seems to appear in
the expression of P , the ratio between the Planck and electron masses:
The Lyuty-Kotov period of Coherent Oscillations tcc ≈ 9600.6 s , observed in several quasars
without any Doppler effect, apart stable dephasages, must be an essential cosmic period. Now the
solar Wolf period (tWf ≈ 11 years) shows overwhelming connections with tcc, T = R/c , and the Bohr
time tB= rB/c:
tWf ≈ ( tcc²T )
≈ ( tB T
Eliminating t Wf, the study of deviation leads to the remarkable relation:
( T/tcc )
≈ π
1/2π ( tcc /tB )
≈ π
a+1/^ π^ ≈ (1/2)(3/√2)
2 ×3×5×
Now there is a dramatic connexion implying 136, the original Eddingon's value, and the minor third
π
1/2^ π^ ≈ (a -136)
≈ e
5²/a ≈ 6/5 ≈ 3
leading to the discovery:
showing a new musical property of a , clearly distinct from 137.
13. Predictions for G, aF and as
The coherent period t cc was identified [4] in a formula eliminating c between the gravitational and
the weak interaction constants. With the Planck time tP = √(G ħ /c
) and te = ħ /mc², the above
electron time, a further identification was made, with the well measured (30 ppb) muon ratio μ ≈
X = tcc tP/te
≈ F/√(pH) ≈ μ
/a (13.1)
So, the value
F ≈ 573007.3857 (60 ppb) (13.2)
was predicted years before its recent precise measurement from 9 ppm to to 250 ppb: 573007.33.
This permitted also to propose a value for G :
G ≈ 6.6754294 × 10
kg
m
s
at 2σ from the tabulated value, which is subject to serious objections. This value for G is
consolidated by the following observation, implying the radius R = cT, with T = 2teP²/pH ≈ 13.
Gyr of the Hubble (or redshift sphere) [3] [4]:
(4π /3) ( R/cte )
≈ 2 π
H
/e
p ≈ 2 π (π/ e )
p (13.4)
where the proton-electron mass ratio p appears to 0.5 ppm (with the official G value, this is about
3.5 ppm). This seems to imply that both the optimal mathematical basis e and the geometrical one π
are used in a computing Cosmos. Now H is related to X by an holographic relation H ≈ X
/9. With
d ≈ 1.001159652, the electron abnormal magnetic coefficient, study of deviation leads to the hightly
symmetric relation:
X
≈ 9pnH/6 π
d ≈ 2 π fF (13.5)
in the 100 ppb range. This permits to predict the value for f, only known presently to 2%
f ≈ 8.43450353 ≈ 1/0.118560624 (13.6)
with a direct relation with the muon-electron mass ratio:
as ≈ μ
at 0.03%. These predictions, if confirmed, would be a supplementary argument for the existence of
a Great Theory unifying gravitation and standard particle model, refuting the Multiverse
Hypothesis.
14. Dramatic confirmation of the prediction for G.
The above value for G is confirmed by the relation, in the ppb range, involving 2π
a
, the surface
of the 4-sphere of radius a :
R/le = 2 P
/ pH ≈ (2π
a
H/6 π
By itself, this relation leads to a slightly different value (2.3 ppm) :
G ≈ 6.67546446 × 10
kg
m
s
By supressing an excessive π
factor, the above relation relates with the topologic function [4]
exp(
D/
), for the boson string dimension D = 26, :
6R/le = 12 P
/ pH ≈ (2π a
H ≈ exp(
Now, coming back to the ppb relation (8.11) :
p
( p /(6π
a
The dramatic third root of 6π
calls for an elevation to high power of a , involving larger numbers,
which is consistent with the hypothesis that a is a calculation basis. Adding the gravitationnal large
number P = mPl / me (with the above selected value for G), in the search for correlation, between 3/2,
a, 137, p, P and (6π
, the computer shows, in les than the ppm domain :
a
p
( p /(6π
≈ P (3/2)
permitting an elimination of the numerical factor 3/2 in :
e^((2^(26/4))/6 ≈ u
(2 π
a
≈ u
(3^3)^(3^3) ≈ (4/3) u
R'/l e ' = P
p/a
H ≈ (4/3) u
with the corrected electron wavelength (4/3): l e ' = l e '/(1+1/p), the last one gives a within 44 ppb. It is
remarkable that the best precise relation involves the cosmic radius R'. Considering the simplest
economic large number of order 4, it shows singular properties:
(e^(e^(e^e)))
≈ (Pn/H)
e^(e^(e^e)) ≈ a^(√Z (p-1)) ≈ a^√(We^(e^e ) ≈ a^a²e
e e (15.10)
clearly showing that a is calculation basis, from which one deduces, with pG ≈ P/
Z/W ≈ e^(e^e)/pG 6 π
pG appears in the dramatic double relation:
pG
F ≈ aZ √(pH) ≈ P/Wn
permitting to predict the following values Z ≈ 178451.209 and W ≈ 157334.3376.
16. Large numbers associated to the electron confirms the prediction for G
Jean Maruani has specially studied the large numbers associated with the electron. He
considered firstly the electric-gravitational ratio of force between two electrons : this is
Fel /Fgr = ħc / aGme
= P
/a (16.1)
A second large number is the ratio between the Hubble radius and the classical electron radius re
This last radius re = ħ/came = le / a is of the order the nucleon radius: for this reason, it was chosen
by Dirac in his definoition of canobical large number. But, at this epoch, the Hubble radius was
erroneous by a factor 8. This large number is
R /re = 2a P
/pH (16.2)
Thus, the ratio between the above two large numbers is
x = (Fel /Fgr )/(R /re ) = pH/2a
This 'electron special ratio' exhibits a dramatic property
x = pH/2a
≈ ln(aG) = ln(R/2le) ≈ 127ln2 (16.4)
relates rather to aG. Now these two numbers aG and ln(aG) are related above in the Dark Atom
model. Moreover, 2 x ≈180 = 5×
≈ (n/a)
which was associated (Sanchez , Holic Principle, 1995)
to neutron with quark combination udd, while the proton combinaison 5
×6 = 150 is close to H
to 13 ppm, and ( 6× π
, to 0.3 ppm. This overwhelming singularity produces the following
0.15% approximation for a
a
≈ 6× 5
We recall also the strange mention of the number 180 in the Bible, in connection with 137 (Section
7). The simplest explanation for this could be the liaison with the golden number (Section 8).
Now, the study of deviation from (16.2) shows the overwhelming correlation :
x = pH/2a
≈ ln(3R/le) (16.6)
with a direct liaison with the Topologic Axis function for the bosonic string dimension D = 26 :
R/le ≈ (1/3) exp(pH/2a
) ≈ (1/6) exp(
Moreover, the deviations are connected, leading to the formula :
ln(3(H-p)P
/pH) ≈ pH/a
corresponding to :
G ≈ 6.6754643453 × 10
kg
m
s
Note the use of the correcting factor (H-p), which makes this value consistent in a few ppb range to
the one defined by the overwhelming formula :
R/le ≈ (2 π
a
(H/6× π
corresponding to:
G ≈ 6.6754643460 × 10
kg
m
s
The ppb proximity between the two last formulas for G let very small doubt about their pertinence.
17. A Synthetic Formula for a
A first estimation was given about 2000, by recognizing that a is related to the factor appearing
in the black body formula e
g with g = 5(1-e
-g ) ≈ 4.965114232, the reduced Wien coefficient :
a ≈ e
g
- 2 π (17.1)
So, it was looked for a trigonometric formula, leading to the discovery of the 65 ppb formula :
a ≈ 44 π – Arccos(1/e) (17.2)
References
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