On the simplest sextic fields and related Thue equations

الأصل · EN

We consider the parametric family of sextic Thue equations x⁶-2mx⁵y-5(m+3)x⁴y²-20x³y³+5mx²y⁴+2(m+3)xy⁵+y⁶=λ where m is an integer and λ is a divisor of 27(m²+3m+9). We show that the only solutions to the equations are the trivial ones with xy(x+y)(x-y)(x+2y)(2x+y)=0.