Fermat's Library | On some equations concerning the general "unitary" formula, which derives from DN Constant. New possible mathematical connections with the DN Constant, Ramanujan Recurring Numbers and some parameters of Number Theory, Cosmology and String Theory II annotated/explained version.

1

On some equations concerning the general "unitary" formula, which derives

from DN Constant. New possible mathematical connections with the DN

Constant, Ramanujan Recurring Numbers and some parameters of Number

Theory, Cosmology and String Theory II

Michele Nardelli

1

, Antonio Nardelli

Abstract

In this paper (part II), we analyze some equations concerning the general "unitary"

formula, which derives from DN Constant. We obtain new possible mathematical

connections with the DN Constant, Ramanujan Recurring Numbers and some

parameters of Number Theory, Cosmology and String Theory

1

M.Nardelli studied at Dipartimento di Scienze della Terra Università degli Studi di Napoli

Federico II, Largo S. Marcellino, 10 - 80138 Napoli, Dipartimento di Matematica ed Applicazioni

“R. Caccioppoli” - Università degli Studi di Napoli “Federico II” – Polo delle Scienze e delle

Tecnologie Monte S. Angelo, Via Cintia (Fuorigrotta), 80126 Napoli, Italy

A. Nardelli studied at the Università degli Studi di Napoli Federico II - Dipartimento di Studi

Umanistici – Sezione Filosofia - scholar of Theoretical Philosophy

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Srinivasa Ramanujan (1887-1920)

https://www.moduscc.it/ramanujan-il-grande-matematico-indiano-13453-131115/

3

Introduction

In this paper, an octahedron could serve as a mathematical or conceptual model of the

universe in the quantic phase, while the spherical surface could be used to describe

the geometry of the bubble-universe.

The values (2√2)/π, the golden ratio φ, ζ(2) and π, can be connected to the proposed

cosmological model. Here's how they might be connected:

Ratio (2√2)/π the so called DN Constant:

This relationship may have a connection with the geometric properties of the

octahedron and the sphere, which have been considered as mathematical models of

the early universe and bubbles universe in eternal inflation.

Golden Ratio φ:

The golden ratio is a mathematical constant that appears in many natural and artistic

contexts and is often associated with harmonious proportions and aesthetic beauty. Its

emergence in this context could suggest a kind of intrinsic symmetry or harmony in

the structure of the early universe and bubbles universe.

Value of π:

The value of π is a fundamental mathematical constant that appears in many

geometric formulas and relationships, including the geometry of the sphere. Its

appearance could indicate a direct connection between the geometry of bubbles

universe and the mathematical properties of spherical surfaces.

Ultimately, the results obtained can be interpreted as manifestations of the geometric

and mathematical properties of the models proposed for the early universe and

universe bubbles. This suggests that there is a profound connection between

geometry, mathematics and cosmological physics, and that through the analysis of

4

these relationships we can deepen our understanding of the universe and its

fundamental phenomena.

The above values (2√2)/π, the golden ratio φ and π, can be connected to the proposed

cosmological model. This hypothesis is certainly plausible.

The various mathematical solutions and relationships can be seen as representations

of the principles and laws that govern the formation and evolution of the universe.

Regarding the fundamental mathematical values, they could emerge as a consequence

of the geometric and physical laws that govern the structure and evolution of the

quantum universe and bubbles universe.

The multidisciplinary approach involving complex mathematical solutions and

cosmological concepts can offer deeper insight into the fundamental nature of the

universe and its processes. Exploring these connections could lead to new discoveries

and insights into our understanding of the early universe and its complexity.

Proposal:

The initial octahedron: Let's imagine a regular octahedron, with perfectly

symmetrical faces. Each face represents an ideal symmetry.

The emerging sphere: Inside the octahedron, there is an inscribed sphere. This

sphere represents the bubble of the universe that emerges from the perturbations of

the quantum vacuum during eternal inflation.

Expansion and transitions: As time passes, the universe expands. The faces of the

octahedron begin to break, symbolizing "symmetry breaks." The sphere continues to

grow, representing the expanding universe.

Constants and numbers: We integrate the mathematical results you obtained. For

example, the golden ratio (φ) could be represented by a proportion between the

dimensions of the octahedron and the sphere.

Entropy and complexity: Entropy increases as the universe evolves. We can

represent this with a disordered growth of structures within the emerging sphere.

Imagine this scene as an abstract work of art, where geometric shapes and

cosmological concepts merge

5

In Fig.1 and Fig.2 let's imagine a regular octahedron representing the universe in its

phase of high symmetry and very low entropy. Inside the octahedron we have an

inscribed sphere that emerges from perturbations of the quantum vacuum during

eternal inflation. As time passes, the universe expands, the faces of the octahedron

break (symmetry breaks), and entropy increases. Spheres emerge from the octahedra,

symbolizing the transition phases from a regime of very low entropy to a universe in

which, with the passage of time, entropy increases, increasing the complexity of the

universe itself.

Fig. 1

6

Fig. 2

7

Now, we have that:

Octahedron Sphere

From the octahedron volume V = 1/3*√2 l

3

and, from the sphere volume,

V = (4/3*π*r

3

) , we consider the following relationship, for r = x:

4/3*π*x^3 = 1/3*√2*l^3

Input

Exact result

Alternate forms

8

Real solution

Solutions

Integer solution

Implicit derivatives

9

From the alternate form

for l = 8, we have that:

8/(sqrt(2) π^(1/3)) = 8/(2sqrt2 * Pi)^1/3

Input

Result

Logarithmic form

Thence:

l/(sqrt(2) π^(1/3)) = l/(2sqrt2 * Pi)^1/3

Input

10

Logarithmic form

Now, we have that:

l/(2 sqrt(2) π)^(1/3) = (2sqrt2)/Pi

Input

Exact result

Plot

11

Solution













12

From:

A NEW TYPE OF ISOTROPIC COSMOLOGICAL MODELS WITHOUT

SINGULARITY - A.A. STAROBINSKY - Volume 91B, number 1 PHYSICS

LETTERS 24 March 1980

We analyze the following equation:

We consider:

(d^2*f)/(d*ξ^2)+M^2*ξ^(-2/3)*f^(-1/3)-(M^2*f)/(12*H^2*ξ^2)+K(√12 M^2*f^(-

5/3)-3^(-1/2)*ξ^(-4/3)*f^(-1/3)*(M^2/H^2)+1)+K^2*ξ^(-2/3)*f^(-5/3)*(1-

(M^2/H^2))

Input

Exact result

Alternate forms

13

Derivative

Indefinite integral

14

From the indefinite integral result

we obtain:

(d^2 f)/(2 ξ^2) - (d f M^2)/(12 H^2 ξ^2) + (d (-K^2 M^2 + H^2 (K^2 + f^(4/3)

M^2)))/(f^(5/3) H^2 ξ^(2/3)) + d EllipticK(1 + (M^2 (6 - f^(4/3)/(H^2

ξ^(4/3))))/(sqrt(3) f^(5/3)))

Input

Exact result

Expanded form

15

Alternate forms

Derivative

Indefinite integral

16

Again, from the indefinite integral result

we obtain:

(d^3 f)/(6 ξ^2) - (d^2 f M^2)/(24 H^2 ξ^2) + (d^2 (-K^2 M^2 + H^2 (K^2 + f^(4/3)

M^2)))/(2 f^(5/3) H^2 ξ^(2/3)) + 1/2 d^2 EllipticK(1 + (M^2 (6 - f^(4/3)/(H^2

ξ^(4/3))))/(sqrt(3) f^(5/3)))

Input

Exact result

17

Alternate forms

Expanded form

Derivative

18

Indefinite integral

From the exact result

(d^3 f)/(6 ξ^2) - (d^2 f M^2)/(24 H^2 ξ^2) + (d^2 (-K^2 M^2 + H^2 (K^2 + f^(4/3)

M^2)))/(2 f^(5/3) H^2 ξ^(2/3)) + 1/2 d^2 EllipticK(1 + (M^2 (6 - f^(4/3)/(H^2

ξ^(4/3))))/(sqrt(3) f^(5/3)))

for d = 1, f = 2, ξ = 4, H = 8, K = 16:

(2)/(6 4^2) - (2 M^2)/(24 8^2 4^2) + ((-16^2 M^2 + 8^2 (16^2 + 2^(4/3) M^2)))/(2

2^(5/3) 8^2 4^(2/3)) + 1/2 EllipticK(1 + (M^2 (6 - 2^(4/3)/(8^2 4^(4/3))))/(sqrt(3)

2^(5/3)))

Input

19

Exact result

Plots (figures that can be related to the open strings)

Alternate forms

20

Expanded form

Numerical roots

21

Series expansion at M=0

Series expansion at M=∞

22

Derivative

Indefinite integral

23

From the exact result

1/48 - M^2/12288 + (-256 M^2 + 64 (256 + 2 2^(1/3) M^2))/1024 + 1/2 EllipticK(1 +

((6 - 1/(128 2^(1/3))) M^2)/(2 2^(2/3) sqrt(3)))

for M = 6:

1/48 - 6^2/12288 + (-256 6^2 + 64 (256 + 2 2^(1/3) 6^2))/1024 + 1/2 EllipticK(1 +

((6 - 1/(128 2^(1/3))) 6^2)/(2 2^(2/3) sqrt(3)))

Input

Exact result

Decimal approximation

Alternate complex forms

24

Alternate forms

Expanded forms

25

Alternative representations

Series representations

26

The general "unitary" formula, which derives from DN Constant, is the following:



 

 















 



     









  



for C = 1.616255*10

-35

(Planck Length), R = 1.265120782997423× 10

48

that is equal

to the radius of the Multiverse and 2.33*10

-13

is the mean temperature of the CMB is

equivalent around to 2.73 Kelvin , we obtain:

27



 

 















 



 



   



  







Multiplying the previous result 12.815 (cos(-0.019978) + i sin(-0.019978)) from the

above expression, we obtain:

(√(2×(2∙(((2√2)/π)^(1/16)))/(1/(π∙0.9991104684)(1.616255∙10^(-

35)×1.265120782997423*10^48×2.33∙10^(-13)))))(12.815(cos(-0.019978) + i sin(-

0.019978)))

Input interpretation

Result

Alternate complex forms

Polar coordinates

20.7351

28

From which, we obtain:

(76+7)*(√(2×(2∙(((2√2)/π)^(1/16)))/(1/(π∙0.9991104684)(1.616255∙10^(-

35)×1.265120782997423*10^48×2.33∙10^(-13)))))(12.815(cos(-0.019978) + i sin(-

0.019978)))+8

Input interpretation

Result

Alternate complex forms

Polar coordinates

1729.01

This result is very near to the mass of candidate glueball 



 scalar meson.

Furthermore,  occurs in the algebraic formula for the j-invariant of an elliptic

curve 



 



. The number  is one less than the Hardy–Ramanujan number

(taxicab number, as it can be expressed as the sum of two cubes in two different

ways 



 







 



 and Ramanujan's recurring number).

29

Since bosons are made of gauge bosons and scalar bosons (meson), then this number

theoretic analysis perhaps confirms that the number , confirm the fact that both

the gauge and scalar bosons are actually different states of a single bosonic string,

and that these states are isomorphic or that the states vibrations are synchronised with

the state of the bosonic string. This also imply that each state lives inside a cubic or

octahedron as a spherical cloud, and that the total sum of these two states is the state

of the bosonic string. Taking the cross section of the bosonic string, we realise that it

must be a rectangular, or a two shaped octahedron. As the string vibrates in

difference frequencies, so is the two spherical cloud states inside the string. That is,

the string vibrations simply excites the gauge bosons i.e Photon, gluon, W and Z

inside one cube/octahedron, and the scalar boson i.e. Higgs inside the other

cube/octahedron.

Furthermore, if we bring the picture of loop quantum gravity (LQG) with the

property of a discontinues quantum geometry, we can therefore, think of the graviton

living on the vertices of the rectangles or the octahedrons. This graviton then acts a

glue binding the bosonic strings lattice together forming a complete cross section of

alternating states of between the gauge bosons and scalar bosons. This arrangement

of states then gives a precise supersymmetric quantum picture of the vacuum

geometry at low entropy.

But the geometry further reveals very important fact, that since the vacuum geometry

is discontinues, then we observe that there is no relation whatsoever between the

quantum vibrational frequencies of the strings, and that of the vertices of the vacuum

geometry where the graviton lives. Ashtekar et al., (2021) asserted that gravity is

simply a manifestation of spacetime geometry. Thus, the graviton cannot be a string

boson, however, there is a duality between gravity and strings. Also, gauge bosons

have spin-1, while the graviton has spin-2. Then lastly, because of the

thermodynamic constraints we were able to arrive at the results we have, now this

bring us to this fundamental question; that string theory and LQG theory are two

intrinsic aspects of a complete quantum gravity theory we are after? That is, without

the other no complete and compelling quantum geometry can be attained, as it is done

here? This need to be investigated further.

30

(1/27(((76+7)*(√(2×(2∙(((2√2)/π)^(1/16)))/(1/(π∙0.9991104684)(1.616255∙10^(-

35)×1.265120782997423*10^48×2.33∙10^(-13)))))(12.815(cos(-0.019978) + i sin(-

0.019978)))+8)-1))^2

Input interpretation

Result

Alternate complex forms

Polar coordinates

4096.06 ≈ 4096

31

The number 4096 = 64

2

, is the Ramanujan Recurring Number, that when multiplied

by 2 give 8192. The total amplitude vanishes for gauge group SO(8192) for bosonic

string SO(8192), while the vacuum energy is negative and independent of the gauge

group. The vacuum energy and dilaton tadpole to lowest non-trivial order for the

open bosonic string. While the vacuum energy is non-zero and independent of the

gauge group, the dilaton tadpole is zero for a unique choice of gauge group, SO(2

13

)

i.e. SO(8192). This could be the implications for a pre-big bang scenario where only

self-perturbative bosonic strings lived when the enthalpy was extremely low as

discussed above. This regime contains all the intrinsic properties of superstrings

inherent in the bosonic strings, would at the big bang give effect to the properties of

matter (fermions) as Higgs Boson. This number theoretic connection to the gauge

group SO(8192), gives a much more compelling relevance of the bosonic string

theory SO(8192), to quantum gravity and places this string theory where it should

appropriately be in the evolution of the universe from a quantum gravity perspective

rather than it be neglected because it doesn’t include fermionic strings to confirm to

post big-bang reality. The vanishing of the bosonic string’s amplitude could be

explained by the effect of extreme low entropy on the quantum vacuum geometry.

Thus, as the entropy increases infinitesimally as a result of the vacuum self-

perturbation then also is the amplitude of the vibrating bosonic strings from zero.

Thus, was right to indicate that the “vanishing of the amplitude of the bosonic string

could be the results of string theory itself”, but here, we give a much more elaborate

explanation of what could be happening.

32

((76+7)*(√(2×(2∙(((2√2)/π)^(1/16)))/(1/(π∙0.9991104684)(1.616255∙10^(-

35)×1.265120782997423*10^48×2.33∙10^(-13)))))(12.815(cos(-0.019978) + i sin(-

0.019978)))+8)^1/15+(MRB const)^(1-1/(4π)+π)

Input interpretation

Result

Alternate complex forms

Polar coordinates

1.64494 ≈ ζ(2) = π

2

/6 = 1.644934 (trace of the instanton shape and Ramanujan

Recurring Number)

33

We analyze the following equation:

(k_2/(2880*π^2))(R^l*R-2/3*R*R_1-

1/2*g*R_2*R^(l*m)+1/4*g*R^2)+(k_3/(2880*π^2))*1/6*(2*R-2*g*R^l-

2*R*R_1+1/2*g*R^2)

Input

Exact result

Alternate forms

34

Expanded form

Periodicity

Root for the variable k

3

Series expansion at m=0

35

Derivative

Indefinite integral

From the indefinite integral result

we obtain:

(-(6 g k_2 R_2 R^(l m))/(l log(R)) + k_2 m R (3 g R + 12 R^l - 8 R_1) + k_3 m (R (g

R - 4 R_1 + 4) - 4 g R^l))/(34560 π^2)

Input

36

Expanded form

Alternate forms

Alternate forms assuming g, l, m, R, R

2

, R

1

, k

2

, and k

3

are positive

37

Reduced logarithmic form

Series expansion at m=0

Derivative

Indefinite integral

38

Again from the indefinite integral result

we obtain:

(-(12 g k_2 R_2 R^(l m))/(l^2 log^2(R)) + k_2 m^2 R (3 g R + 12 R^l - 8 R_1) + k_3

m^2 (R (g R - 4 R_1 + 4) - 4 g R^l))/(69120 π^2)

Input

Expanded form

Alternate forms

39

Alternate form assuming g, l, m, R, R

2

, R

1

, k

2

, and k

3

are positive

Reduced logarithmic form

Series expansion at m=0

Derivative

40

Indefinite integral

Again, from the indefinite integral result

(-(36 g k_2 R_2 R^(l m))/(l^3 log^3(R)) + k_2 m^3 R (3 g R + 12 R^l - 8 R_1) + k_3

m^3 (R (g R - 4 R_1 + 4) - 4 g R^l))/(207360 π^2)

for g = 2, k = 4, l = 8, m = 16 and R = 32:

(-(36 2 4 32 32^(128))/(8^3 log^3(32)) + 4 16^3 32 (3 2 32 + 12 32^8 - 8 32) + 4

16^3 (32 (2 32 - 4 32 + 4) - 4 2 32^8))/(207360 π^2)

Input

41

Decimal approximation

-9.6395909288989614…*10

185

Alternate forms

Expanded forms

42

Alternative representations

43

Series representations

44

Integral representations

45

From which, after some calculations:

4ln((-(36 2 4 32 32^(128))/(8^3 log^3(32)) + 4 16^3 32 (3 2 32 + 12 32^8 - 8 32) + 4

16^3 (32 (2 32 - 4 32 + 4) - 4 2 32^8))/(207360 π^2))+16

Input

Exact result

Decimal approximation

46

Alternate complex forms

47

Polar coordinates

1729

This result is very near to the mass of candidate glueball 



 scalar meson.

Furthermore,  occurs in the algebraic formula for the j-invariant of an elliptic

curve 



 



. The number  is one less than the Hardy–Ramanujan number

(taxicab number, as it can be expressed as the sum of two cubes in two different

ways 



 







 



 and Ramanujan's recurring number). Since bosons are

made of gauge bosons and scalar bosons (meson), then this number theoretic analysis

perhaps confirms that the number , confirm the fact that both the gauge and

scalar bosons are actually different states of a single bosonic string, and that these

states are isomorphic or that the states vibrations are synchronised with the state of

the bosonic string. This also imply that each state lives inside a cubic or octahedron

as a spherical cloud, and that the total sum of these two states is the state of the

bosonic string. Taking the cross section of the bosonic string, we realise that it must

be a rectangular, or a two shaped octahedron. As the string vibrates in difference

frequencies, so is the two spherical cloud states inside the string.

48

That is, the string vibrations simply excites the gauge bosons i.e Photon, gluon, W

and Z inside one cube/octahedron, and the scalar boson i.e. Higgs inside the other