THE SUM OF RECIPROCAL FIBONACCI PRIME NUMBERS

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THE SUM OF RECIPROCAL FIBONACCI PRIME
NUMBERS CONVERGES TO A NEW CONSTANT
Ing. Pier Franz Roggero, Dott. Michele Nardelli, P.A. Francesco Di Noto
Abstract:
In this paper there is a sum of the reciprocal Fibonacci primes that converges to a
new constant. Furthermore, we have described some new possible mathematical
connections with the universal gravitational constant G , the Einstein field equations
and some equations of string theory linked to Φ and π
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Index:
1.
ARE THERE AN INFINITE NUMBER OF FIBONACCI PRIMES ?……………………………................................. 3
2.
SUM OF RECIPROCAL FIBONACCI PRIME NUMBERS AND OTHER MATHEMATICAL CONNECTIONS…..5
2.1 APPROXIMATION OF THE VALUE OF NEW CONSTANT FP…………………………………................................23
3
REFERENCES…………………………………………………………………………………………………………………..25
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1.
ARE THERE AN INFINITE NUMBER OF FIBONACCI
PRIMES?
In the case of the Fibonacci prime numbers always increases the distance D(x) between
the Fibonacci prime numbers:
12
23
35
4 13
5 89
6 233
7 1597
8 28657
9 514229
10 433494437
11 2971215073
The first 10 terms are the following:
D(x) = 1, 2, 8, 76, 144, 1364, 27060, 485572, 432980208, 2537720636, …
Now since we know that the Fibonacci numbers are infinite in number this succession
of D(x) it is equivalent to an infinite sum of "1", because each gap D (x) is exactly 1
because there is only and always a D( x), one for one, one for 2, one for 8, one for 76
and so on:
1 + 1 + 1 + 1 + 1 + 1 + ... = ∞
Even if the sum tends to infinity this very slowly, however, shows that the Fibonacci
primes are infinite in number.
Since the distance D(x) between the Fibonacci numbers grows indefinitely it could be
that exists always a Fibonacci prime number because Fibonacci numbers are infinite.
In fact we know that the number of Fibonacci N(x) numbers is less or equal to x:
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N(x) ≈
1
log(x 5 ) = 2,0780869212350275376013226061178 … * log(x 5 )
log ϕ
with φ golden ratio = 1,6180339887498948482045868343656 …
From this formula we can derive that the number of Fibonacci primes F(x) less than or
equal to a certain threshold x is approximately:
F(x) ≈ 2φ * log [log(x 5 )] = 3,236 * log [log(x 5 )]
For x → ∞ we have F(x) → ∞ and the number of Fibonacci primes grows to infinity as
the natural logarithm of the natural logarithm even if in a very slow but always
endlessly.
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2. SUM OF RECIPROCAL FIBONACCI PRIME NUMBERS
A Fibonacci prime is a Fibonacci number that is prime, a type of integer sequence
prime.
It is not known whether there are infinitely many Fibonacci primes. The first 23 values
are the following
12
23
35
4 13
5 89
6 233
7 1597
8 28657
9 514229
10 433494437
11 2971215073
12 99194853094755497
13 1066340417491710595814572169
14 19134702400093278081449423917
15 475420437734698220747368027166749382927701417016557193662268716376935476241
16 529892711006095621792039556787784670197112759029534506620905162834769955134424689676262369
17 1387277127804783827114186103186246392258450358171783690079918032136025225954602593712568353
18 3061719992484545030554313848083717208111285432353738497131674799321571238149015933442805665949
19 10597999265301490732599643671505003412515860435409421932560009680142974347195483140293254396195769876129909
20 36684474316080978061473613646275630451100586901195229815270242868417768061193560857904335017879540515228143777781065869
21 96041200618922553823942883360924865026104917411877067816822264789029014378308478864192589084185254331637646183008074629
22
3571035606419098607209077741390634544455699265828433067940419974763010711027675704833435635185100078003041954440805185626309
0002738649893394461921019285676835268346883175442323421797852576592104074729131668157655686149077313521486178287771656087968
6368266117365351884926393775431925116896322341130075880287169244980698837941931247516010101631704349963583400361910809925847
7213008027417055194123065229412024294379288260338854166569679715599027431502632522294562989922630081267195892034304073852282
3036162849486017212970227117292646950080234260872200642074558629726792905250905915434096834850958055230714864200143847031622
9
23
5001956361269572929050245125969728066958033451362433489705652881794353613138049565055817826376346124779796798932751033961473
4865076200759493751080454114500230430286734100629849340431965738212320115800718825260655080669453532923225685105665637237964
9097735304781630173812454531781511107460619516018844320335033801984806819067802561370394036732654089838823551603083295670024
4534775890931199183865663976776102742138373919545911476030544426503268279807811402759414252171724284486981617108417406880425
8720416125608491416676254900701271392217274825969056661458006268219660646649810257162768372671848322957804434364673769443640
6261444368327649097401550241341102704783841619376027737767077127010039900586625841991295111482539736725172169379740443890332
2343411043104709074498984155224148052103411380633509997307499509201472506832277987802648112156477065425116810278253908827707
62662185410080310045261286851842669934849330548237271838345164232560544964315090365421726004108704302854387700053591957
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If the sum of reciprocal Fibonacci prime numbers converges to an irrational number, so
we prove that exist infinitely many Fibonacci prime numbers.
Let’s the sum as Fp, with p to denote prime number, we have:
1
1
1
1
1
1
1
1
1
1
1
+ + +
+
+
+
+
+
+
+
  =
2
3
5
13
89
233
1597
28657 514229
433494437
p: p∈Fp  p 
1
1
1
+
+
+
2971215073
99194853094755497
1066340417491710595814572169
1
+ … = 1,1264472276728533386016660044138 + …
19134702400093278081449423917
Fp =
∑
The sum for the first 14 values gives as result:
Fp ≈ 1,1264472276728533386016660044138 …
The Fp value is a new constant but we don’t know if it’s irrational number.
If we were sure that the Fibonacci prime numbers are infinite then the constant
would be irrational because each reciprocal add one decimal place to the constant
and this could no longer be rational.
The reciprocal of the 15th prime Fibonacci is in fact 2,1 * 10-75 a very small value so
that gives almost no contribution to the sum and so it goes for all other reciprocals.
The value Fp, as calculated, is approximate and we have to wait then a slightly bigger
but very little than previously estimated because we must consider the sum of the
reciprocals of ALL Fibonacci prime numbers.
The value Fp is a new constant irrational because we have proven in paragraph 1 that
the Fibonacci primes are infinite in number.
It is interesting to note that this new irrational constant is much smaller than the Brun’s
constant - the sum of the reciprocals of twin primes - a value roughly equal to
1.902160583104 ...> 1.1264472276728533386016660044138, because the number of
the Fibonacci primes it is significantly less than the number of the twin primes below a
certain threshold x.
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It has in fact, below a certain threshold x a number less than or equal to the twin primes
numbers equal to

x 

2 
 (log x ) 
π2(x) ≈ 2C 
where:
C = 0,66016181584686957392781211001455 … constant of the twin primes
For the Fibonacci prime numbers below a certain threshold x a number less than or
equal to
F (x) ≈ 2φ * log [log (x)] = 3.236 * log [log (x)]
Then
π2(x) >> F(x)
The value Fp is of course less than ψ - sum of the reciprocals of all the Fibonacci
numbers - that is irrational because Fibonacci numbers are infinite, known to be
approximately:
Ψ = 3,35988566624317755317201130291892717968890513373…
The value that would be expected to be definitely lower than:
Fp < log Ψ = 1,2119069454923344087062057571675…
and this happen successfully.
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The sum S(x) of the reciprocal of Fibonacci P(x) prime numbers is given by:
− 2ϕ
+ 1,1264472276728533386016660044138 =
x log x
− 3,236
− 3,236
=
+ 1,1264472276728533386016660044138 =
+ Fp
x log x
x log x
S(x) ≈
Furthermore, we know that:
n
n
1  1 + 5   1 − 5  
 −
 

Fn =
5  2   2  


With the following formula, from any number n (imput) correspond a Fibonacci’s
number (output).
This formula concerning the aurea ratio. Indeed:
n
1+ 5 
n


 2  =Φ


while
1− 5
2
is the negative solution of the equation from which is obtained the golden number,
equivalent to:
−
1
Φ
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NOTE 1
Examples of integrals equations containing the golden ratio Φ =
5 +1
and
2
mathematical connections with the universal gravitational constant G , the Einstein
field equations and some equations of string theory linked to Φ and π
1) We have the following expressions:
Φ = 2 ⋅ cos
π
5
∞
;
π4
x3
dx
=
∫0 e x − 1 15 .
Thence, we obtain the following relationship:
 3 ∞ x3

 3 π4 
π
2 ⋅ cos 3 ∫ x dx  = 2 ⋅ cos 3 ⋅  = 2 ⋅ cos = 2 ⋅ cos 0,6283185... = 2 ⋅ 0,809017 ≅ 1,618034...
5
 π 15 
 π 0 e −1 
2) There exist a link between Φ and π to the factor of the universal gravitational
constant G (6.67 x 10^-11) by the relationship: π2 – 2Φ = G , with Φ =
5 +1
is the
2
golden ratio.
The common link of Newtonian constant and that of the fine structure (α = 2πe2 / (hc),
that as we see, is linked to π) to π thus implies that between the two above mentioned
constants α and G. Furthermore, by 432 / π = 137 = 1 / α follows that α = π / 432 =
1/137 (where 432 is the frequency of the natural La) and similarly, by G = π2 – 2Φ is
obtained G = 0,0155 x 432. (M. Nardelli)
From
http://dev.physicslab.org/document.aspx?doctype=3&filename=universalgravitation_un
iversalgravitationforces.xml
The Law of Universal Gravitation states that every object in the universe attracts every
other object in the universe with a force that has a magnitude which is directly
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proportional to the product of their masses and inversely proportional to the distance
between their centers squared.
F =G
M 1M 2
R2
where
•
•
•
•
G is the gravitational constant, 6.67 x 10-11 Nm2/kg2
M1 is the mass of the first body in kg
M2 is the mass of the second body in kg
R is the distance from the center of M1 to the center of M2
Now we take an example of an integral containing the gravitational constant G that is
connected with φ by the relationship: π2 – 2Φ = G above mentioned. It involves
calculating the gravitational force between a point mass M and an extended rod of mass
m, length L, and mass per unit length, λ.
To begin, divide the rod into a finite (countable) number of segments of mass
located at a distance x from M.
Each of these segments will contribute a gravitational force of attraction.
n
F = ∑ GM
i =1
∆mi
xi2
If we take the limit as ∆m approaches zero, then our expression for F becomes
each
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n
F = lim ∑ GM
∆m → 0
i =1
∆mi
;
xi2
F =∫
L+a
a
GM
1
dm
x2
Before we can integrate we must express ∆m in terms of x .
λ=
dm
;
dx
λdx = dm
Substituting and integrating gives us
F =∫
L+a
a
GMλ
1
dx ;
x2
F = GMλ ∫
L+ a
a
 a − (L + a ) 
 ;
F = −GMλ 
 a (L + a ) 
1
dx ;
x2
1
 1
F = −GMλ 
− ;
L+a a
GMλL
GMm
F=
; F=
a (L + a )
a (L + a )
F = GMλ
 −L 
 ;
F = −GMλ 
 a (L + a ) 
− 1 L+a
;
x a
where
m = λL
Thence we have that, for π2 – 2Φ = G:
F=∫
L+ a
a
GM
(
)
1
GMλL
π 2 − 2Φ MλL
dm
=
=
a(L + a )
a(l + a )
x2
From Wikipedia:
3) The differential form of Gauss's law for gravity states
∇ ⋅ g = −4πGρ ,
where ∇ denotes divergence, G is the universal gravitational constant, and ρ is the mass
density at each point.
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Relation to the integral form
The two forms of Gauss's law for gravity are mathematically equivalent. The divergence
theorem states:
∫
∂V
g ⋅ dA = ∫ ∇ ⋅ gdV
V
where V is a closed region bounded by a simple closed oriented surface ∂V and dV is an
infinitesimal piece of the volume V (see volume integral for more details). The
gravitational field g must be a continuously differentiable vector field defined on a
neighborhood of V.
Given also that
M = ∫ ρdV
V
we can apply the divergence theorem to the integral form of Gauss's law for gravity,
which becomes:
∫ ∇ ⋅ gdV = −4πG ∫ ρdV
V
V
which can be rewritten:
∫ (∇ ⋅ g )dV = ∫ (− 4πGρ )dV .
V
V
This has to hold simultaneously for every possible volume V; the only way this can
happen is if the integrands are equal. Hence we arrive at
∇ ⋅ g = −4πGρ ,
which is the differential form of Gauss's law for gravity.
The two above expressions, for the relation π2 – 2Φ = G, can be rewritten also as
follows:
∫ (∇ ⋅ g )dV = ∫ [− 4π (π
V
V
2
)]
− 2Φ ρ dV
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(
)
∇ ⋅ g = −4π π 2 − 2Φ ρ
4) Also in the Einstein field equations, we have the term G and thence is possible to
obtain the mathematical connection with π and Φ.
The Einstein field equations are the 16 coupled hyperbolic-elliptic nonlinear partial
differential equations that describe the gravitational effects produced by a given mass in
general relativity. As result of the symmetry of Gµν and Tµν , the actual number of
equations reduces to 10, although there are an additional four differential identities (the
Bianchi identities) satisfied by Gµν , one for each coordinate.
The Einstein field equations state that
Gµν = 8πTµν ,
where Tµν is the stress-energy tensor, and
Gµν = Rµν −
1
g µν R
2
is the Einstein tensor, with Rµν the Ricci curvature tensor and R the scalar curvature.
The Einstein field equations can be written also as follows:
1
8πG
Rµν − Rg µν + Λg µν = 4 Tµν
2
c
that include a cosmological constant term Λ . A positive value of Λ is needed to
explain the accelerating universe . The Ricci curvature tensor, also simply known as the
Ricci tensor (Parker and Christensen 1994), is defined by
Rµκ ≡ R λ µλκ
Where R λ µλκ is the Riemann tensor.
The covariant derivative of the Riemann tensor is given by
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Rλµνκ ;η
1 ∂
=
2 ∂xη
2
 ∂ 2 g λν
∂2g
∂2g
 κ µ − κ µνλ − ∂ µg λκν + ν µκλ
 ∂x ∂x
∂x ∂x
∂x ∂x
∂x ∂x





Permuting ν , κ , and η (Weinberg 1972, pp. 146-147) gives the Bianchi identities
Rλµνκ ;η + Rλµην ;κ + Rλµκη ;ν = 0
which can be written concisely as
R α β [λµ ;ν ] = 0
(Misner et al. 1973, p. 221).
The Riemann tensor (Schutz 1985) R α βγδ , also known the Riemann-Christoffel
curvature tensor (Weinberg 1972, p. 133; Arfken 1985, p. 123) or Riemann curvature
tensor (Misner et al. 1973, p. 218), is a four-index tensor that is useful in general
relativity. Other important general relativistic tensors such that the Ricci curvature
tensor and scalar curvature can be defined in terms of R α βγδ .
The Riemann tensor is in some sense the only tensor that can be constructed from the
metric tensor and its first and second derivatives,
α
α
µ
α
µ α
Rα βγδ = Γβδ
,γ − Γβγ ,δ + Γβδ Γµγ − Γβγ Γµδ ,
where Γαβγ are Christoffel symbols of the first kind and Aκ is a comma derivative
(Schmutzer 1968, p. 108; Weinberg 1972). In one dimension, R1111 = 0 . In four
dimensions, there are 256 components. Making use of the symmetry relations,
Riklm = − Rikml = − Rkilm ,
the number of independent components is reduced to 36. Using the condition
Riklm = Rlmik ,
the number of coordinates reduces to 21. Finally, using
Riklm + Rilmk + Rimkl = 0 ,
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20 independent components are left (Misner et al. 1973, pp. 220-221; Arfken 1985,
pp. 123-124).
Thence, we have that :
1
8πG
Rµν − Rg µν + Λg µν = 4 Tµν
2
c
for π2 – 2Φ = G can be rewritten also as follows:
Rµν −
or
(
)
1
8π π 2 − 2Φ
Rg µν + Λg µν =
Tµν ;
2
c4
1
8π 3 − 16πΦ
Rµν − Rg µν + Λg µν =
Tµν ;
2
c4
We note that with regard the 256 components of the Riemann tensor in four dimensions
256 = 144 + 55 + 34 + 21 + 2; 36 = 2 + 13 + 21; and 21 are all Fibonacci’s numbers.
Furthermore, 256 = 32 × 8 = 22 × 82 , where the number 8, and thence the numbers 64 = 82
and 32 = 22 × 8 , are connected with the “modes” that correspond to the physical
vibrations of a superstring by the following Ramanujan function:
∞ cos πtxw'


− πx 2 w '
e
dx 
∫0 cosh πx

142
4 anti log
⋅ 2
πt 2
t w'
−
w'

e 4 φw' (itw') 
1 
.
8=
3
  10 + 11 2 
 10 + 7 2  
+ 

log  



4
4
 




5) Also in string theory the term G and thence is possible to obtain the mathematical
connection with π and Φ.
With regard the string theory, we know that the asymptotic value of the dilaton is:
Φ 0 = lim Φ( X ) . (1)
X →∞
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The string coupling is then given by
g s = e Φ 0 . (2)
With regard the action for a D = 26 spacetime, this is the low-energy effective action of
the bosonic string and is given from the following relationship:
S=
1
2κ
2
0
∫d
26
1


X − G e − 2Φ  R − H µνλ H µνλ + 4∂ µ Φ∂ µ Φ 
12


(3)
where we have taken the liberty of Wick rotating back to Minkowski space for this
expression.
We defined the constant part in (1); it is related to the string coupling constant. The
varying part is simply given by
~
Φ = Φ − Φ0
(4)
~
In D dimensions, we define a new metric Gµν as a combination of the old metric and
the dilaton,
~
~
Gµν ( X ) = e −4 Φ / ( D − 2 )Gµν ( X ) (5)
One can check that two metrics related by a general conformal transformation
~
Gµν = e 2ω Gµν , have Ricci scalars related by
~
R = e −2ω R − 2(D − 1)∇ 2ω − (D − 2 )(D − 1)∂ µ ω∂ µ ω
(
)
~
With the choice ω = −2Φ / (D − 2) in (5), and restricting back to D = 26 , the action (3)
becomes
S=
1
2κ
2
∫d
26
~ ~ 1 ~
1 ~ ~
X − G  R − e −Φ / 3 H µνλ H µνλ − ∂ µ Φ∂ µ Φ  (6)
12
6


The gravitational part of the action takes the standard Einstein-Hilbert form. The
gravitational coupling is given by
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κ 2 = κ 02 e 2Φ ≈ l s24 g s2 (7)
0
The coefficient in front of Einstein-Hilbert term is usually identified with Newton’s
constant
8πG N = κ 2
From Newton’s constant, we define the D = 26 Planck length 8πG N = l p24 and Planck mass
M p = l p−1 .
For the relationship π2 – 2Φ = G eq. (3) can be rewritten also in the following
equivalent form:
S=
 2  5 + 1   −2Φ 
1
26
 π − 2
 e  R − H µνλ H µνλ + 4∂ µ Φ∂ µ Φ  (8)
−
d
X
∫




2κ
12


 2 

1
2
0
We know that the value of the Planck constant is:
h = 6,626070040 ×10 −34 J s = 4,135667662 ×10 −15 eV s
and that the value of the reduced Planck constant is:
h=
h
= 1,054571800 ×10 −34 J s = 6,582119514 × 10 −16 eV s
2π
In physics, the Planck length, denoted ℓP, is a unit of length, equal to
1.616199(97)×10−35 metres. It is a base unit in the system of Planck units, developed by
physicist Max Planck. The Planck length can be defined from three fundamental
physical constants: the speed of light in a vacuum, the Planck constant, and the
gravitational constant.
The Planck length ℓP is defined as
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lP =
hG
≈ 1,61619 ×10 −35 m
3
c
where c is the speed of light in a vacuum, G is the gravitational constant, and ħ is the
reduced Planck constant.
We note immediately, as the number 1,61619 is very near, about equal, to the value
of the golden ratio 1,61803398… = (√5 + 1) / 2
The Planck length is about 10−20 times the diameter of a proton.
In physics, the Planck mass, denoted by mP, is the unit of mass in the system of natural
units known as Planck units. It is defined so that
mP =
hc
G
≈ 1,2209×1019 GeV/c2 = 2,17651×10−8 kg = 21,7651 µg (microgram) =
1,3107×1019 amu.
where c is the speed of light in a vacuum, G is the gravitational constant, and ħ is the
reduced Planck constant.
We note that the value 21,7651 is very near to 21 that is a Fibonaccì’s number.
The Planck time is defined as:
tP =
hG
≈ 5,39106 ×10 −44 s
5
c
We have that Fp ≈ 1,1264472276728533386016660044138 … Now 1,126447 × 5 =
5,632235 value very near to the value of tP that is also very near to the number 5,
that is a Fibonacci’s number. Furthermore, we know that the constant of the twin
primes is C = 0,66016181584686957392781211001455. Now: C × 8 ≈ 5,28129 value
very near to the value of the Planck’s time. Note that 8 is a Fibonacci’s number.
In physics, Planck energy, denoted by EP, is the unit of energy in the system of natural
units known as Planck units.
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hc 5
EP =
≈ 1,956 × 10 9 J ≈ 1,22 × 10 28 eV
G
We have that Ψ = 3,35988566624317755317201130291892717968890513373…
Furthermore, we have log Ψ = 1,2119069454923344087062057571675…value that
is very near to the value in eV of EP and of mP in GeV/c2. Furthermore, 1,956 is a
value very near to the Brun’s constant B2 ≈ 1,902160583104. (The sum of the
reciprocals of the twin prime numbers tends to a constant B2). Also here the twin
prime constant C × 3 = 1,98048 value very near to the Planck’s energy. Note that 3
is a Fibonacci’s number.
An equivalent definition is:
EP =
h
,
tP
where tP is the Planck time.
We have Ψ = 3,35988566624317755317201130291892717968890513373…
We note that Ψ / 5 = 0,671977, value very near to the final spin of the black hole
that is produced calculated by the observations of the gravitational waves, i.e. 0,67.
Interesting also that the divisor, i.e. 5 , is a Fibonacci’s number .
Also
(1,61803398)3
π
= 0,6551 is very near to 0,67 and in this formula there is also π .
Furthermore, log Ψ = 1,2119069454923344087062057571675… is very near to the
36 / 29 = 1,2413793…, where 36 and 29 are the values of the solar masses of the two
initial black holes.
The Hilbert-Einstein action of the gravitational field is:
Sg = −
c3
d 4 x − g ( x )R ( x ) , (9)
∫
16πG
where g is the determinant of g µν and R is the scalar curvature,
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µν
R = g Rµν
ρ
ρ
 ∂Γµν

∂Γµρ
σ
ρ
ρ
σ 

. (10)
=g
−
+
Γ
Γ
−
Γ
Γ
µν
σρ
σν
µρ
 ∂x ρ

∂xν


µν
The variation to volume fixed from the integral (9) is
δ ∫ d 4 x − g R = ∫ d 4 x − g Rµν δg µν + ∫ d 4 xRδ − g + ∫ d 4 x − g g µν δRµν . (11)
We have that
δ −g =−
1
1
∂g
δg = −
δg µν . (12)
µν
2 −g
2 − g ∂g
now
g µν =
1 ∂g
, (13)
g ∂g µν
we have
∂g
= gg µν , (14)
∂g µν
thence, the eq. (12) become
δ −g =−
1
− g g µν δg µν . (15)
2
Substituting this expression in the (11), we obtain




1
2
δ ∫ d 4 x − g R = ∫ d 4 x − g  Rµν − g µν R δg µν + ∫ d 4 x − g g µν δRµν . (16)
The second term of the eq.(16) is a surface term which does not contribute to the
variation of the action. Then, we have


δ ∫ d 4 x − g R = ∫ d 4 x − g  Rµν −
1

g µν R δg µν . (17)
2

The principle of stationary action
δS g = −
c3
δ ∫ d 4 x − g R = 0 , (18)
16πG
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implies, for the (17),
Rµν −
1
g µν R = 0 . (19)
2
that is the Einstein’s equation of the gravitational field in the absence of the sources.
Consider now a source of energy represented from a generic field ϕ . The action of ϕ
has the form
Sϕ =
1 4
d x − g L , (20)
c∫
where L is the density of Lagrangian. Putting equal to zero the variation of Sϕ with
respect to ϕ , we obtain the equation of motion for the field ϕ . We are interested only
the variation of Sϕ with respect to g µν , that is
δSϕ =
=
(
)
(
)
(
)

∂ − gL
1 4 ∂ − gL
d x
δg µν +
δ∂ ρ g µν  =
µν
µν
∫
c
∂∂ ρ g
 ∂g

(
∂ − gL
1 4 ∂ − gL
d x
− ∂ρ
µν
∫
c
∂∂ ρ g µν
 ∂g
)δg

µν
. (21)
Introducing the tensor Tµν defined by
(
)
(
)
∂ gL
∂ − gL
1
− g Tµν =
− ∂ρ
, (22)
µν
2
∂g
∂∂ ρ g µν
the eq. (21) can be rewritten as follows
δSϕ =
1
d 4 x − g Tµν δg µν . (23)
2c ∫
The full action of the system constituted by the field ϕ and by the gravitational field g µν
is
S = S g + Sϕ = −
c3
1
d 4 x − g R + ∫ d 4 x − g L . (24)
∫
16πG
c
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The interaction between the two fields is incorporated in the second term. Using the eq.
(17), we find that the stationarity condition for S is
 c3 
 µν
1
 1
δS = ∫ d x − g −
 Rµν − g µν R  + Tµν δg = 0 , (25)
2
 2c
 16πG 

4
and from this are obtained the Einstein’s equations in the presence of sources,
1
8πG
Rµν − g µν R = 4 Tµν . (26)
2
c
For the relationship π2 – 2Φ = G, we have that

c3

1

1

µν
δS = ∫ d 4 x − g −
 Rµν − g µν R  + Tµν δg = 0 , (27)
2
16
(
−
2
Φ
)
2
2
c
π
π




with Φ =
5 +1
that is the golden ratio.
2
Furthermore, we have the following mathematical connection between the eq. (27) and
the eq. (6):
~  ~ 1 −Φ~ / 3
1 ~ ~
26
d
X
−
G
H µνλ H µνλ − ∂ µ Φ∂ µ Φ  ⇒
R − e
∫
2κ
12
6






3

 µν
c
1
1


⇒ ∫ d 4 x − g −
 Rµν − g µν R  + Tµν δg = 0 . (28)
2
 2c
 16π  π 2 − 2 5 + 1   

 2 








S=
1
2
For the relationship π2 – 2Φ = G, we have finally that
(
)
1
8π π 2 − 2Φ
Rµν − g µν R =
Tµν ;
2
c4
(29)
From this new expression we can to obtain a new way of obtaining Φ . Indeed, we have
Torino, 15/02/2016
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that:
1

 4
 Rµν − g µν R c
2
 + 1π 2
Φ = −
16πTµν
2
or:
1

 4
 Rµν − g µν R c
5 +1
2
 + 1 π 2 . (30)
=−
2
16πTµν
2
Thence, the golden ratio Φ is directly proportional to the gravitational field g µν , the
Rµν , that is the Ricci curvature tensor and R , that is the scalar curvature and c that is the
speed of the light; and inversely proportional to the Tµν , that is the stress-energy tensor.
}
2.1
APPROXIMATION OF THE VALUE OF NEW CONSTANT Fp
A good approximation of the value of the new constant Fp is given by:
√√Ф = 4 ϕ 1,127838485561682260264835483177 , with Ф = (√5 + 1) / 2
Fp ≈ 1,1264472276728533386016660044138 …
with a relative error of 0.1235% and an absolute error of about only 0.00139, about a
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thousandth and a half.
WARNING: this approximate value does not tend to the new constant Fp that is
independent of the golden ratio and is a new irrational number probably even
transcendental.
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3. REFERENCES
1) PROOF THAT THE PRIMES OF FIBONACCI ARE INFINITE IN NUMBER
Ing. Pier Francesco Roggero, Dott. Michele Nardelli, Francesco Di Noto
2) Wikipedia
3) Mathworld
4) Vincenzo Barone – “Relatività” – Bollati Boringhieri – Nov. 2004
5) www.damtp.cam.ac.uk/user/tong/string/seven.pdf
6) Properties of the binary black hole merger GW150914 - The LIGO Scientific
Collaboration and The Virgo Collaboration (compiled 11 February 2016)