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In this paper we give a proof for Beal’s conjecture . Since the discovery of the proof of Fermat’s last theorem by Andre Wiles, several questions arise on the correctness of Beal’s conjecture. By using a very rigorous method we come to the proof. Let G = {(x, y, z) ∈ N3 : min(x, y, z) ≥ 3} Ωn = {p ∈ P : p | n, p ƒ |zy − yz} ,
T = {(x, y, z) ∈ N3 : x ≥ 3, y ≥ 3, z ≥ 3}
∀(x, y, z) ∈ T consider the function fx,y,z be the function defined as :
fx,y,z : N3 → Z
(X, Y, Z) ›→ Xx + Y y − Zz
Denote by
Ex,y,z = {(X, Y, Z) ∈ N3 : fx,y,z(X, Y, Z) = 0}
and U = {(X, Y, Z) ∈ N3 : gcd(X, Y ) ≥ 2, gcd(X, Z) ≥ 2, gcd(Y, Z) ≥ 2} Let x = min(x, y, z) . The obtained result show that :if Ax + By = Cz has a solution and ΩA ƒ= ∅,
∀p ∈ ΩA ,
Q(B, C) =
x−1 y
[
j
Bj −
.zΣ
Cj]
j=1
has no solution in ( Z )2 \ {(0, 0)} Using this result we show that Beal’s conjecture is true
since
(x,y[,z)∈T
Ex,y,z ∩ U ƒ= ∅
Then (α, β, γ) N3 such that min(α, β, γ) 2 and Eα,β,γ U = The novel techniques use for the proof can be use to solve the variety of Diophantine equations . We provide also the solution to Beal’s equation . Our proof can provide an algorithm to generate solution to Beal’s equation
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