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Solving a family of Thue equations with an application to the equation \(x^2-Dy^4=1\). (English) Zbl 1155.11318

Let \(D\) be a positive nonsquare integer. The authors study the Diophantine equation \(X^ 2-DY^ 4=1\) in positive integers \(X\) and \(Y\) and refine a theorem of W. Ljunggren [Skr. Norske Vid.-Akad. Oslo I 1936, no. 12, 1–73; Zbl 0016.00802]. Let \((T_ 1,U_ 1)\) be the smallest integer solution to the Pell equation \(X^ 2-DY^ 2=1\). For \(k\geq 1\), let \(T_ k+U_ k\sqrt D={(T_ 1+U_ 1\sqrt D)^ k}\) represent all positive integer solutions to the Pell equation. The authors prove:
There are at most two positive integer solutions \((X,Y)\) to the equation \(X^ 2-DY^ 4=1\). If two solutions \(Y_ 1<Y_ 2\) exist, then \(Y_ 1^ 2=U_ 1\) and \(Y_ 2^ 2=U_ 2\), except only if \(D=1785\) or \(D=16\cdot 1785\), in which case \(Y_ 1^ 2=U_ 1\) and \(Y_ 2^ 2=U_ 4\). If only one positive integer solution \((X,Y)\) exists, then \(Y^ 2=U_ l\) where \(U_ 1=lv^ 2\) for some squarefree integer \(l\), and either \(l=1\), \(l=2\), or \(l=p\) for some prime \(p\equiv 3\pmod 4\).
The problem is reduced to solving the family of Thue equations \(x^ 4+4tx^ 3y-6tx^ 2y^ 2-4t^ 2xy^ 3+t^ 2y^ 4=t_ 0^ 2\), where \(t_ 0\) divides \(t\) and \(t_ 0\leq \sqrt t\), for a positive integer \(t\). However, it is not required to solve this family completely, but only for solutions whose quotient \(x/y\) is near to \(\beta^ {(3)}\) or \(\beta^ {(4)}\), where \(\beta^ {(j)}\), \(j=1,\dots,4\), denote the roots of the univariate polynomial corresponding to the Thue equation in a particular order defined in the paper. For these two roots, an effective measure of irrationality can be proved by Thue’s hypergeometric method.

MSC:

11D59 Thue-Mahler equations
11D25 Cubic and quartic Diophantine equations

Citations:

Zbl 0016.00802
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