Stealth Elliptic Curves and The Quantum Fields

Thierno M. SOW∗

2013

February.13 ABC

June.8 GoldbachAugust.6 Riemann

December.13 Erdos− Straus

Abstract

The complete abstract was accepted by the organizing committee of

the International Congress of Mathematicians as a part of the short

communications at Coex in Seoul 2014.

The present paper is a series of preprints called SO PRIME. The goal

of which is to build and to applied a new and strong approach in number

theory according to the ABC Conjecture, the Beal Conjecture, the Erdos-

Straus Conjecture, the Goldbach Conjecture, the Riemann Hypothesis

and the Twin Primes Innity. So far, as a major advance result, we

can observe how to break through the RSA cryptography protocol.

Finally, as the consequence of the Riemann Hypothesis we will give, in the

next article, a complete statement on the Prime Number Theorem π(N).

Mathematics Subject Classication 2010 codes: Primary: 11MXX; Secondary: 11P32, 11D68

∗We dedicate the present article to the vivid memory of Nelson Mandela.

1

Contents

1 INTRODUCTION 3

2 THE ABC CONJECTURE 42.1 DEFINITION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.2 THE FORMULATION . . . . . . . . . . . . . . . . . . . . . . . . 42.3 THE NEW APPROACH . . . . . . . . . . . . . . . . . . . . . . 4

2.3.1 THE DIAMOND . . . . . . . . . . . . . . . . . . . . . . . 5

3 THE RIEMANN HYPOTHESIS 63.1 FORMS AND SHAPES OF THE RIEMANN ZETA . . . . . . . 73.2 Miscellaneous . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

4 THE GOLDBACH CONJECTURE 94.1 DEFINITION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94.2 THE STATEMENT . . . . . . . . . . . . . . . . . . . . . . . . . 94.3 THE GOLDBACH STAIRS . . . . . . . . . . . . . . . . . . . . . 104.4 THE GOLDBACH CORRELATIONS . . . . . . . . . . . . . . . 11

4.4.1 ATOMS, ELEMENTS AND MOLECULES . . . . . . . . 114.4.2 Miscellanous . . . . . . . . . . . . . . . . . . . . . . . . . 11

5 THE ERDOS-STRAUS PROBLEM 125.1 NOTATION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125.2 THE STATEMENT . . . . . . . . . . . . . . . . . . . . . . . . . 12

5.2.1 FIXING THE TRIPLES X,Y,Z . . . . . . . . . . . . . . . 135.3 THE ERDOS-STRAUS PLOT . . . . . . . . . . . . . . . . . . . 145.4 BEYOND THE ERDOS-STRAUS PROBLEM . . . . . . . . . . 16

5.4.1 GOLDBACH AND THE ESP . . . . . . . . . . . . . . . . 175.4.2 THE BEAL CONJECTURE . . . . . . . . . . . . . . . . 175.4.3 THE BIRCH AND SWINNERTON-DYER . . . . . . . . 185.4.4 A VULNERABILITY IN THE RSA CRYPTOGRAPHY 185.4.5 THE RSA AND THE PRIME NUMBER THEOREM . . 185.4.6 THE ESP OVER THE QUANTUM FIELDS . . . . . . . 18

6 CONCLUSION 19

2

1 INTRODUCTION

The original manuscript is over 727 pages long1, using very tough demonstra-tions and complicated gures. Nevertheless, we do believe that Mathematicsresult on a careful analysis. In particular, the complexity is not mathematicalsince Mathematics are about the truth. One way to see this is to consider theFermat last theorem. So, let us give an explicit proof to the little theorem.We assume: There are no coprime solutions to the equation xn + yn = zn since

the sum of the exponents equals to n2 with n > 1.Along this line, In 1657, Fermat wrote back to Cureau that he was in agree-

ment that nature always acts by the shortest and simplest path. It turns outthat Fermat had founded, with a brilliant sense of humours, that the margin istoo large.... We do believe that Fermat, Gauss and Euler knew this theorem.

As another interesting example, let's consider the character of the famousirrational number π, which is used in the present article. Under suitable loga-rithms conditions, we assume: For any function f(ω)and constant ∆ > 1

ˆ 3

−∞

[ω +

log( 1∆ )

log( 1

∆√

Ω)

]f(ω)dω = π

ˆ 3

−∞f(ω)dω, (1)

where

Ω = (π − ω)−2.

Now, if we build a new set of primes on the form 2p + 3, according to theAfrican fractions and the Twin Primes conjecture, we assume: for every pair of

twin primes x, y there exists α, β positive integers such that

ζ = β

(1

x+

1

y+

1

α

)(2)

where β denotes the product of x, y, α and ζ is a twin prime.With a usefull smart algorithm we can generate the following partition(

x αy ζ

)(

2 13 11

) (3 25 31

) (5 37 71

) (11 413 239

)(

17 519 503

) (29 731 1319

) (41 1143 2687

) (59 1461 5279

)(

71 1673 7487

) (101 7103 11831

) (101 9103 12239

) (101 36103 17747

)At last, the complete proof will be very signicant to testify the deep con-

nections between twin primes and they innitude.

1thierno.sow@sciences-po.org - 10@one-zero.eu - www.one-zero.eu - - Typeset LATEX

3

2 THE ABC CONJECTURE

2.1 DEFINITION

We shall denote the new stealth elliptic curves of the (a,b,c) triples by ♦...abc

. We

shall write Q...abc

for the quality triples of (a,b,c).

2.2 THE FORMULATION

The formulation of the abc-conjecture is : For every ε > 0, there are only nitelymany triples of coprime positive integers a+ b = c such that c > d1+ε, where ddenotes the product of the distinct prime factors of abc.

Theorem 4. There are no triples of coprime positive integers a + b = csuch that c > d1+ε.

Proof.

d1+ε =log(πc)

log(√π2abc)

=1

ac+

1

bc< 1 < c. (3)

2.3 THE NEW APPROACH

Theorem 5. The Mode Smart I. Let 1 < ♦...abc

< 2

♦...abc

=| bc|n + | a

b|n= 1 + ε. (4)

Theorem 6. The Mode Smart II. The highest quality triples Q...abc

→ 1.

Q...abc

= limq→1| cb|= 1 + ε. (5)

Proof.

Q...abc

=

log(πc)

log(√πab)

a > 1

1 +

[log(πc)

log(√πbc)

= ε

]a > 1

(6)

If a = 1, then replace πab by π2ab and πbc by π2bc.

Example 1 & 2 With n = 1

♦...abc

Q...abc

a b c

1, 00 ≈ 1, 00 ≈ 3 13.712 216

1, 00 ≈ 1, 00 ≈ 1 23.112.61 310

4

2.3.1 THE DIAMOND

We observe that ♦...abc

has the properties and the polyhedric forms of a diamond.

For the physical relevance, we observe a polygon form and amplitude whichhas the properties of the triangle and the square with a strong triangulation indual space-time. May it be the perfect path and skills of the next generation ofstealth aircrafts. Hopefully, we assume that this new structure of stealth elliptic

curves has the same singularity of the Amplituhedron well re-introduced inPhysics by Nima Arkani-Hamed and his research group. Until we learn muchmore about this IFS Fractal or this DOGU gure: What is thisMatter-Atom-Hedric-Diamond? W. MAHDI shall be it's name.

3D Graph of ♦...abc

5

3 THE RIEMANN HYPOTHESIS

In 1859, Bernhard Riemann founded that the distribution of prime num-bers follows a relatively straightfoward residue characteristic of the qualitativeproperties of an function called the Riemann Zeta function. The Riemann Hy-pothesis asserts that the real part of any nonreal zero of the Riemann Zetafunction is 1/2 such that

ζ(s) =

∞∑n=1

=1

ns. (7)

Theorem 7. ˆ s is log(n−1)

log(n−2s)ds = ζ(s)

ˆ s

f(s)ds (8)

hence

ζ(s) =

1

2+is log(n−1)

log(n2s)n > 1

1

2−is log(n−1)

log(n−2s)i = 1

(9)

Proof. The abc and the Riemann hypothesis are related to one another bymeans of the formula

d1+ε =log(π−1)

log(π−ab) ζ(abc) =

b log(π−1)

log(π−ab)= 1

2︸ ︷︷ ︸ . (10)

Where a > 1.Nevertheless, we can oberve in the abc conjecture the special cases where

a = 1 log(π−1

)log (π−3ac)

+log(π−1

)log(π−

32ac)+

log(π−1

)log (π−3bc)

+log(π−1

)log(π−

32 bc) =

1

ac+

1

bc(11)

hence with the Riemann Hypothesis we have

d1+ε =log(π−1)

log(π−ab) ζ(abc) =

b/2 log(π−1)

log(π−ab)= 1

2 .︸ ︷︷ ︸ (12)

The second proof is given by the means of the new RSA factorization tech-nique well introduced in the next sections.

6

3.1 FORMS AND SHAPES OF THE RIEMANN ZETA

Polar & 3D Graph of Riemann Zeta

ˆ s is log(n−1)

log(n−2s)ds = ζ(s)

ˆ s

f(s)ds (13)

7

3.2 Miscellaneous

That is to say, the Riemann Zeta fonction has the similar topology and thehomeomorphic surface of a Mobius strip. It can be modeled in 3-dimensionalspace. It will result at the nonreal zero spectrum both separate strips alongthe center line with two surfaces and two boundaries. Finally, the edge of theRiemann strip has several smart properties, highlighting the perfect innitygure Z like ζ.

The Spectrum

8

4 THE GOLDBACH CONJECTURE

4.1 DEFINITION

The expression G︸︷︷︸ may be understood as referring to the Goldbach's even

integer greater than 2. We shall write G2 for the square part of c. We shalldenote the pairs of primes by γn

a,b. We shall write G︸︷︷︸

γ

to represent the Goldbach

partition. The underbrace ︸︷︷︸ in the notation may be thought of as denoting

the residue of a Goldbach's even integer. Then, we shall write Gc︸︷︷︸γ2

a,b−a′,b′

for the

even integer c, for the set of 2 pairs of primes satisfying the relationa+b=ca′+b′=c

.

4.2 THE STATEMENT

Christian Goldbach wrote a letter to Leonard Euler, as of June 7, 1742, but,we do know that it's Leonard Euler himself who formulated the terms of one ofthe most popular conjecture in number theory. As the consequence of the articlethe Prime Square, we propose two strong proof of the Goldbach conjecture.The second proof is given by the means of the Erdos-Straus theorem which willbe clearly introduced in the next section.

Theorem 8. For every even integer greater than two, there exists a real

ϕ ≥ 1 and ψ prime such that

G︸︷︷︸ = limϕ→∞

(2 +

log(ψϕ)

log(√ψ)

)(14)

hence ˆ1

[2 +

log(ψϕ)

log(√ψ)

]f(ϕ)dϕ = G︸︷︷︸ ˆ

1

f(ϕ)dϕ. (15)

9

4.3 THE GOLDBACH STAIRS

Polar of the Goldbach function

ˆ1

[2 +

log(ψϕ)

log(√ψ)

]f(ϕ)dϕ = G︸︷︷︸ ˆ

1

f(ϕ)dϕ. (16)

3D Graph of the Goldbach function

The Goldbach function is the perfect illustration of some natural elementslike the nautilus, the snail, the staircase of a green lighthouse or a modernescalator and above all the logarithmic spiral.

10

4.4 THE GOLDBACH CORRELATIONS

4.4.1 ATOMS, ELEMENTS AND MOLECULES

The pair of primes γn around the even numbers are similar, in many dierentways, to the molecules and the atoms elds. So, the Goldbach Theorem remainsof interest for the deep connections between primes and the atoms it wouldprove. The following are examples of the Goldbach connections.

Figure 4.1:

Figure 4.2:

Figure 4.3:

4.4.2 Miscellanous

The importance of the Goldbach Theorem cannot be overstated. Many othertechniques may be used to demonstrate such structures correlations with theGoldbach Partition more eectively in Physics and Mathematics including theDesign and the High-Tech Industry. Let us observe for example the square partof the G4︸︷︷︸

γ12,2

as illustrated below:

11

5 THE ERDOS-STRAUS PROBLEM

5.1 NOTATION

The expression G︸︷︷︸ may be understood as referring to the Goldbach's even

integer greater than 2. We shall write Qxyz

for the quality of the Erdos-Straus

triples. The expressionNeHmay be thought of as denoting the Erdos-Straus el-

evator. Then, we shall write4

n

(NeHQ

)or

4 exn eyQ ez

for the matrices of the

Erdos-Straus triples.

5.2 THE STATEMENT

The formulation of the Erdos-Straus conjecture is: for all integers n ≥ 2, there

exists x, y, z positive integers such that4

n=

1

x+

1

y+

1

z.

Theorem 9. For all integers n ≥ 2 there exists x, y, z positive integers suchthat 4/n = 1/x + 1/y + 1/z if and only if n couldn't divide the sum of x, y, z and

(xyz)2

(n?)2 = 4. (17)

12

More precisely, there exists two major set for the Erdos-Straus triples x, y, zsuch that

2x = z? (18)

andx = z? (19)

We note the importance of n and the clue number 2 in the both sides of theequation. For example

x y z → 2x z?2 4 20 → 2.2.20 = 4.202 5 10 → 2.2.5 = 2.10︸ ︷︷ ︸

4/5

1 4 12 → n.y = x.z1 6 6 → 6 = 6.12 2 3 → 2.3 = 2.3︸ ︷︷ ︸

4/3

For the λ case, which means that4

n=

1

x+

1

y+

1

z+

1

λ, one integer x or y or

z = 1 =1

2+

1

2such that λ = 2.

5.2.1 FIXING THE TRIPLES X,Y,Z

As we learn it from the Topology, counting the Erdos-Straus triples does notmake sense as well as the physical dimensions have no eect on the mathematicalproblem. Then, we begin by xing the set of the Erdos-Straus triples such thatx, y and z are constants and

4

n=

1

QNeH

, (20)

where1

Qdenotes the quarter part of the Erdos-Straus triples.

Theorem 10. WithNeH6= −3

4

n=

1

1

4+

1

2+

1

4

NeH

. (21)

Proof.4

n=

log(π−1)

log(√π−2Q)

. (22)

Hence(−10, 5

) 4 12 1

0, 5 −1

︸ ︷︷ ︸

4/2

,

(0

0, 75

) 4 13 1

0, 75 0

︸ ︷︷ ︸

4/3

,

(2

1, 25

) 4 15 1

1, 25 2

.

︸ ︷︷ ︸4/5

13

5.3 THE ERDOS-STRAUS PLOT

Polar of the Erdos-Straus Problem (ESP)

14

Diving into the polar of4

nproves that the ESP function will follow a straight-

forward residue around zero which is very similar to the supersonic wave withthe edge of the cone forms of the diraction of a shock wave moving at super-sonic velocity (faster than the speed of sound). The NASA is studying and

testing devices that could be used on aircraft to lessen the noise and window-

rattling eects of supersonic ight . We assume that the Erdos-Straus theoremmay be used to determine the Mach angle and the edge at the vertex of theshock cone.

15

The sinusoidal waves correspond to the harmonic phenomena between theset of all Erdos-Straus triples and, nally, may be consider as a new way to putquantum gravity predictions to experimental tests.

Time and Space by the Erdos-Straus Problem (ESP)

5.4 BEYOND THE ERDOS-STRAUS PROBLEM

The Erdos-Straus statement is one of the most powerful master key in numbertheory. What's more, let us observe how it can be applied to some related

16

conjectures like the Goldbach conjecture, the Beal conjecture and the Birch andSwinnerton-Dyer conjecture as well.

5.4.1 GOLDBACH AND THE ESP

Theorem 11. Let n = a + b where a and b primes then, for every integer

n > 3 there exists an even integer G︸︷︷︸ such that

a+ b︸ ︷︷ ︸n

= 2 +log(π

G︸︷︷︸)log(π2)

. (23)

Proof.

log(πG︸︷︷︸+23

)

log(

√π

2( G︸︷︷︸+23))

=log(π

G︸︷︷︸)2 log(

√π2n)

+log(π23

)

2 log(π2n)= 1. (24)

5.4.2 THE BEAL CONJECTURE

Andrew Beal has formulated a strong conjecture so close to several similarproblems in number theory. This inequality can be stated in very simple termsand it can be applied to some related conjectures. So far, the Beal Conjecturecan be formulated as follows:

If Ax + By = Cz , where A,B,C, x, y and z are positive integers withx, y, z > 2, then A, B and C have a common factor.

For example, the connections between both diophantine equations 1+23 = 32

and 33 + 63 = 35 are true, we know that. It can be expressed as below:

Theorem 12.4

4=

log(π(Ax+By+Cz))

log(√πCz2

). (25)

Proof. ∆ proves that A, B and C have a common factor since ∆ = Cz2

.

We have the expected result with the primes triples and the pythagoreantriples.

17

5.4.3 THE BIRCH AND SWINNERTON-DYER

Theorem 13. (y2 + n2x+ x3

)2x32 = 4. (26)

5.4.4 A VULNERABILITY IN THE RSA CRYPTOGRAPHY

Theorem 14. For every odd number on the form N = pq, where p and q are

primes, there exists ζ +NeHsuch that

ζ +NeH

=

N

(1

p+

1

p+

1

2p

)

N

(1

q+

1

q+

1

2q

) (27)

whereNeHis any positive integer and ζ may takes only the both values

ζ =

2 +

1

2

7 +1

2

(28)

This is an explicit proof which testify the very serious vulnerability ofthe RSA cryptographic technique. In the next article, we will release an inno-vative security system to reinforce the RSA software encryption technique.

5.4.5 THE RSA AND THE PRIME NUMBER THEOREM

The previous theorem has a huge consequence in number theory since the rela-tion fails only if N is Prime.

Although, it is the perfect ilustration of the veracity of the Riemann Hy-pothesis since the set is on the form

ζ =1

2+ it. (29)

5.4.6 THE ESP OVER THE QUANTUM FIELDS

The major keynote of the Erdos-Straus theorem may be understood as a newparticle physics hypothesis since every particle n may exists simultaneously at

dierent points in the space represented by the three cordinates1

x+

1

y+

1

z. In the

present article we give an explicit illustration of the deep connections betweenthe Erdos-Straus theorem and the synodic period and phases of the Moon. If it

18

becomes true by the light of the physical experiments, it will conrm that anypoint around an elliptic surface like the planets can be reached by means of theFour Seasons Theorem. Finally, it may change our way to navigate and tocommunicate beyond the globe.

6 CONCLUSION

Since we know, by the light of the African fractions, how to rewind and tosurround the innity between 0 and 1, then, a new age in number theory andcryptography begins.

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