This text represents my work on the solutions of the Diophantine Equation:

α^ν = β^ν + γ^ν, for ν = 2, 4, 3, 5, 7, 11… (odd prime).

Recognizing that this difficult work has many errors, I will appreciate it very much, if in time, you send to me via the publisher, your comments and corrections, so that the next edition be better.

The proof of the non-existence of solutions of the above equation, for: ν = 3, 4, 5, 7, 11…, will be and proof of the relative FERMAT’S proposal with title:

“FERMAT’S  LAST  THEOREM” (F.L.T.).

S. Aliprantis

January 2021

PART 1

“The in use factorizations at the Diophantine Equation
α^v = β^ν + γ^ν, 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: α^ν = β^ν + γ^ν,
for v = 2, 3, 5, 7, 11, 13…: prime”

PART 3

“The Solutions of the Diophantine Equation:
α^ν = β^ν + γ^ν, for v = 2

PART 4

“The non-existence of Solutions of the Diophantine Equation
 α^ν = β^ν + γ^ν, for v = 4”

PART 5

 “The non-existence of Solutions of the Diophantine Equation
 α^ν = β^ν + γ^ν, for v = 3, 5, 7, 11, …: prime ≥ 3"

˜ The factorizations of the: α^ν = β^ν + γ^ν

“A”… β^ν = (α^ν – γ^ν) =
= (α – γ) · (α^ν-1 + α^ν-2 · γ¹ + … + γ^ν-1), ν≥2                          (1.5)

“Β”… γ^ν = (α^ν – β^ν) =
= (α – β) · (α^ν-1 + α^ν-2 · β¹ + … + β^ν-1), ν≥2                          (1.6)

“Γ”… α^ν = (β^ν + γ^ν) =
= (β + γ) · (β^ν-1 – β^ν-2 · γ¹ + … + γ^ν-1), ν≥3                          (1.7)

˜ Theorem in use at the factorizations

“From the Equation:

K^ν = Λ  Μ, with: (Λ, Μ) = 1, Λ>1, Μ>1, follow:                                (1.8)

K = K₁  K₂, (K₁, K₂) = 1, Λ = , Μ = ”

u Proof

Let the analysis of the K in prime numbers:

(For simplicity are considered only five prime numbers /
Not any problem for more / m∙n means: m with index n)

The K^ν = Λ  Μ becomes:

From the last equation follows that some prime numbers will go at the Λ and the rest will go at the M.

Also, we note that is impossible one prime number “power” to be splitted into 2 factors, as for example:

 
creating the situation:

where the Λ and the M have common factor the p₂,
and so: (Λ, Μ) ≠ 1.

So must be:

.

• Name: Spiros (of Dionissios) Aliprantis
• Year-Place of birth: 1944 - Cefalonia
• Lowest - Lower Educations: at Cefalonia
• Final year of the Lower Education: 1962
• Master Education: Mechanical and Electrical Engineer at the Athens National (“Metsovion”) Polytechnic School (Duration: 5 years)
• Rank at the Army Releasing - Year: Standard - Bearer - 1972
• Foreign languages: English - Well
• Employment at the industry, as Chief of the Technical service:

• Construction of Diesel Machines “MALKOTSIS”
• Steel Foundry “NIKOLAKOPOULOS”
• Beer industry “KAROLOS FIX”
• Dairy industry “DELTA”
• Multinational company “UNILEVER”
• Iron bars products for buildings “ERGON”
• Dairy factors constructions “PARAMETROS”
• Metallic constructions industry “EL.KAT”

The occupation with the equation: α^ν = β^ν + γ^ν, 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 Purdue University/USA.