We study a family of quasi periodic -adic Ruban continued fractions in the -adic field and we give a criterion of a quadratic or transcendental -adic number which based on the -adic version of the subspace theorem due to Schlickewei.
On the assumption of a Riemann hypothesis for certain Hasse-Weil L-functions, it is shewn that a quaternary cubic form f(x) with rational integral coefficients and non-vanishing discriminant represents through integral vectors x almost all integers N having the (necessary) property that the equation f(x)=N is soluble in every p-adic field ℚₚ. The corresponding proposition for quinary forms is established unconditionally.
A proof is given that the system in the title has infinitely many solutions of the form , where and are rational numbers.