DISCOVERY ON BEAL’S CONJECTURE

STATISTIC AND ECONOMY

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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