Simplifying method for algebraic approximation of certain algebraic numbers (Q1929775)

Motivated by the famous Ljunggren's (written erroneously Ljungerren in the Introduction of the paper) Diophantine equation \(x^2+1=2y^4\), the solution of which is straightforwardly implied by the solution of the Thue equation \(x^4-4x^3y-6x^2y^2+4xy^3+y^4=\pm 1\), the authors study the more general Thue equation \[ f(x,y)=tx^4-4sx^3y-6tx^2y^2+4sxy^3+ty^4=N, \quad \tag{*} \] where \(s,t\) are positive integer parameters. The aforementioned equation \(x^2+1=2y^4\) was solved first by \textit{W. Ljunggren}, in [Avh. Norske Vid. Akad. Oslo 1942, No. 5, 1--27 (1942; Zbl 0027.01103)] by a notoriously complicated method (using algebraic number theory). The challenge for a simpler method of solution was successfully met in the 1990s by a number of authors via either linear forms in logarithms (Baker's method) or Padé approximation. The paper makes reference to the relevant literature; it escapes to the attention of the authors a totally different short solution by this reviewer, as an example of application of the \textit{elliptic logarithm method} to a general class of quartic Diophantine equations [``Solving elliptic diophantine equations by estimating linear forms in elliptic logarithms. The case of quartic equations'', Acta Arith. 75, No. 2, 165--190 (1996; Zbl 0858.11016)]. The solution offered in this paper is indeed simple, in that it applies rather elementary tools in order to find a good and effective rational approximation to a root \(\theta\) of the quartic polynomial \(f(x,1)\). There are two main results in the paper: {Theorem 1. } For \(s,t\) positive integers with \(s>32t^3\), \(\theta\) a root of the polynomial \(f(x,1)\) and any integers \(p,q\), the following inequality holds: \(|\theta -p/q|>1/(c|q|^{\lambda})\), where \(c\) and \(\lambda\) are explicitly stated (in the paper) constants depending on \(s,t\). I find this theorem interesting both for its elementary (more or less) method of proof and its thoroughly explicit form. Theorem 2. If \((x,y)\) is an integer solution to \((*)\), then \(|y|\) is explicitly bounded by an explicitly stated (in the paper) upper bound, depending on \(s,t\) and \(N\). The proof of Theorem 2 is based on Theorem 1, and the solution of Ljunggren's equation \(x^2+1=2y^4\) is easily implied by Theorem 2. Two typo remarks. (1) The name `Padé' is erroneously written `Padè'. (2) Throughout the paper, the ring of integers are occasionally denoted by either \(Z\) or by \(\mathbb{Z}\).