Universal octonary diagonal forms over some real quadratic fields. (Q1589754)

Let \(m=n^2-1\) be a positive square-free integer, \(K=\mathbb Q(\sqrt{m})\) and \({\mathcal O}_K\) be the ring of algebraic integers of \(K.\) Then \(\epsilon = n+\sqrt{m}\) is a fundamental unit of \({\mathcal O}_K\) and is totally positive. The main aim of the paper is to prove that the octonary diagonal form \(x_1^2+x_2^2+x_3^2+x_4^2+\epsilon(x_1^2+x_6^2+x_7^2+x_8^2)\) is universal over \({\mathcal O}_K.\)