The Diophantine Equation (E-book)Fermat’s Proposal Verification, A Thorough InvestigationΣυγγραφέας: Αλιπράντης Σπύρος

The Diophantine Equation (E-book)

The Diophantine Equation (E-book)Fermat’s Proposal Verification, A Thorough InvestigationΣυγγραφέας: Αλιπράντης Σπύρος




PREFACE
This text represents my work on the solutions of the Diophantine Equation: αν = βν + γν, for ν = 2, 4, 3, 5, 7, 11… (odd prime).
Admitting that the present essay is subject to further development, I will appreciate it, if, in time, readers send me their comments and remarks through my publisher’s mail (info@lexitipon.gr) in order to improve the next edition.
The proof of the non-existence of solutions of the above equation, for: ν = 3, 4, 5, 7, 11…, is also a proof of the corresponding FERMAT’S proposal known as: “FERMAT’S LAST THEOREM” (FLT).
S. D. Aliprantis


ISBN: 978-960-597-190-8
Έτος έκδοσης: Αθήνα 2018
Διαστάσεις: 17x24
Σελίδες: 142

PART 1

“The used factorizations at the Diophantine Equation

αv = βν + γν, and their applications (v = 2, 3, 5, 7,…: prime)”


PART 2

“The necessary forms of the: (α, β, γ), for the Solution

of the Diophantine Equation: αν = βν+ γν,

for v = 2, 3, 5, 7, 11, 13…: prime”


PART 3

“The Solutions of the Diophantine Equations:

3.1 αν = βν + γνfor v = 2

3.2 A2 = B2 + 2Γ2(use at v = 4)

3.3 A2 + B2 = 2Γ2(use at v = 4)

3.4 A2 + B2 = Γ2+ Δ2(use at v = 3, 5, 7, 11…)”


PART 4

“The non-existence of Solutions of the Diophantine Equation

αν = βν + γν, for v = 4”


PART 5

“The non-existence of Solutions of the Diophantine Equation

αν = βν + γν, for v = 3”


PART 6

“The non-existence of Solutions of the Diophantine Equation

αν = βν + γν, for v = 5, 7, 11, 13,…: prime ≥ 5”


“The used factorizations in the Diophantine
Equation: αν = βν + γν and their applications
(v = 2, 3, 5, 7, … prime)”

General Relations and Definitions
(X, Ψ) = Ζ: means that the maximum common divisor of the
X and Ψ, is the Z.

In an equation of the form: A·Nσ = Β·Ντ + Γ·Δ, if (Ν, Γ) = 1,
it follows that Ν divides Δ.

It must be: (α, β) = (β, γ) = (γ, α) = 1 (1.01)
Because, if two of the (α, β, γ) have one common prime divisor
p, then p must divide and the other, and we wish to have
finished this procedure.

Because the expression: αν = βν + γν, is symmetrical at the
letters β and γ, and because v must not divide both β and γ
(: 1.01), we shall consider from the view of divisions by the v,
only the two cases:
(β, ν) = (γ, ν) = 1 (1.02)
or (β, ν) ≠ 1, (γ, ν) = 1 (1.03)

Due to (1.01), only one, or none of the (α, β, γ) can be
divided by the V. Taking into account (1.02) and (1.03), we
shall distinguish the following 3 general cases:

Case I (α, ν) = 1 (β, ν) = 1 (γ, ν) = 1
Case II (α, ν) ≠ 1 (β, ν) = 1 (γ, ν) = 1 (1.04)
Case III (α, ν) = 1 (β, ν) ≠ 1 (γ, ν) = 1


          

•        NAME: Spiros (of Dionissios) Aliprantis

 

•        YEAR – PLACE OF BIRTH: 1944 – Cefalonia

 

•        LOWEST  - LOWER EDUCATIONS: At Cefalonia

 

•        FINAL YEAR AND MARK OF THE LOWER EDUCATION: 1962 – 18,3

 

(11 lessons, MAXIMUM MARK: 20)

 

•        MASTER EDUCATION: Mechanical and Electrical Engineer at the Athens National (“Metsovion”) Polytechnic School

 

•        FINAL YEAR AND MARK OF THE MASTER EDUCATION: 1969- 7,46

 

(MAX MARK: 10,0)

 

(Note at this Master, of 5 years education: At the second year of this education, I obtained as general mark: 8,46, which was the biggest at my class, and so I obtained one money scholarship by the I.K.Y. (National Institute of grants), which I kept at all the rest years of my education)

 

RANK AT THE ARMY-RELEASING YEAR: Standard-Bearer/January 1972

 

•        BOOKS WRITING: (Through my master education, elaborating the Teachers texts)

 

Steam Turbines, Machines of Internal Combustion, Electronics,

 

Aerodynamics, Exercises of Nuclear Technology

 

•        FOREIGN LANGUAGES: English – Well

 

•        EMPLOYMENT AT THE INDUSTRY:

 

1. 1/2/1972 – 28/3/1972: 2 months

 

Construction of Diesel Machines “MALKOTSIS”

 

2. 11/5/1972 – 15/6/1973: 13 months

 

Steel Foundry “G. NIKOLAKOPOULOS” (Chief of the Technical service)

 

3. 11/7/1973 – 4/6/1975: 23 months

 

Beer Industry “KAROLOS FIX” (Chief of the Technical service)

 

4. 6/6/1975 – 30/12/1977: 18 months

 

Dairy Industry “DELTA” (Assistant of the Technical service Chief)

 

5. 1/1/1978 – 21/2/1989: 11 years

 

Multinational Company “UNILEVER” (Detergents, ALGIDA-Ice

 

creams,foodstuffs)

 

(Chief of the Factory Maintenance and new jobs)

 

6. 22/2/1989 – 1/3/1992: 3 years

 

Elaboration of building iron bars “ERGON” (Chief of the R+D department)

 

7. 2/3/1992 – 31/1/2002: 10 years

 

Job contracting company “PARAMETROS” (Dairy Industries, at TURN-KEY

 

Mode)

 

(General study/design of all jobs as “DODONI” , “EPIRUS”)

 

8. 1/2/2002 – 25/1/2005: 3 years

 

Metallic Constructions Industry “EL.KAT” and several small Industries

 

(Consulting Services)

 

•        PENSION PERIOD

 

25/1/2005 till now (25/1/2019): 14 years

 

Employment with Projects of: Refrigeration (R717), Water Steam, Oil Refineries, etc.

 

The occupation with the equation: αννν have started at 2000, after the introduction at it from my uncle PANAGIS ANALYTIS (not now in life). 

The initial elaboration of the equation became with my brother CHARALAMBOS  ALIPRANTIS (not now in life) who was Professor of Mathematics at the Pardue University/USA

 

 

 

 

 

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