การมีจริงของผลเฉลยของ X3+Y3 = 3Z3 และ X3+4Y3 = 1 ในจำนวนเต็มแบบเกาส์

Abstract

In this research, we study the existence of solutions of equations X3+Y3 = 3Z3 and X3+4Y3 = 1 in Gaussian integers Z[i] . There are 2 cases for the solution of the above equation: trivial solution and nontrivial solution. The study found that the equation X3+Y3 = 3Z3 has only trivial solution that is , (α,β,γ)=(δ,-δ,0) where δ ꞓ Z[i]. If the above equation has a nontrivial solution, then an elliptic curve y^2=x^3+3888 has rational point in Q^2 . However, we have shown that this curve has no rational point in Q^2 . Moreover, the equation X3+4Y3 = 1 has only trivial solution (1,0) .