不定方程x^2+4~k=y^9的整数解

The Integer Solution of Diophantine Equation x^2+4~k=y^9

利用初等数论及代数数论的方法研究了不定方程x^2+4~k=y^9(k=3,4,5)在Gauss整环中的可解性,证明该方程当k=4时仅有整数解(x,y)=(±16,2),而当k=3,5时无整数解.

In the report, the solvability of Diophantine equation x2 ＋4k = y9（ k = 3,4,5 ） in the Gauss integer ring was studied by using the methods of the elementary number theory and algebraic number theory, and it was proved that the equation has only integer solution （x, y） = （ ＋ 16,2） when k = 4 , however, it has no integer solution when k = 3,5.