On the resolution of index form equations in biquadratic number fields. I
For every algebraic number field K of degree n with integral basis \(\{1,w_ 2,...,w_ n\}\) one has \[ D_{K/{\mathbb{Q}}}(w_ 2X_ 2+...+w_ nX_ n)=(I(X_ 2,...,X_ n))^ 2D_{K/{\mathbb{Q}}}, \] where \(I(X_ 2,...,X_ n)\) is a homogeneous polynomial of degree n(n-1)/2 with coefficients in \({\mathbb{Z}}\), the so-called index form of the basis \(\{1,w_ 2,...,w_ n\}\). The authors consider the index form equation \(I(x_ 2,x_ 3,x_ 4)=\pm 1\) in \(x_ 2,x_ 3,x_ 4\in {\mathbb{Z}}\) in case that K is a totally real biquadratic number field. They present a method to find all small solutions of this equation, i.e. with \(| x_ i| \leq 10^ 7\) for \(i=2,3,4\). Their idea is to reduce the index form equation to an equation of the form (1) \(G_ n=y^ 2-d\), where \(\{G_ n\}\) is a binary recurrence sequence with rational integral terms and d is a constant, and to check for all small n whether (1) holds for some \(y\in {\mathbb{Z}}\). In this way, the authors find all small solutions of index form equations as above for all real quadratic extensions of \({\mathbb{Q}}(\sqrt 5)\) of discriminant 63000 and of \({\mathbb{Q}}(\sqrt{2})\) of discriminant 39000. [For part II, see the following review.]
scientific article; zbMATH DE number 4198081
scientific article