Bugeaud, Yann; Mignotte, Maurice; Siksek, Samir; Stoll, Michael; Tengely, Szabolcs Integral points on hyperelliptic curves. (English) Zbl 1168.11026 Algebra Number Theory 2, No. 8, 859-885 (2008). Let \(C\) be a hyperelliptic curve defined by the equation \(Y^2 = f(X)\), where \(f(X)\) is an irreducible polynomial of \({\mathbb Z}[X]\) of degree \(\geq 5\). We denote by \(J\) the Jacobian of \(C\). Suppose that the following assertions are satisfied:(1) A rational point on \(C\) is known.(2) A Mordell-Weil basis for \(J( {\mathbb Q})\) is known. (3) The canonical height \(\hat{h} : J({\mathbb Q}) \rightarrow {\mathbb R} \) is explicitly computable and we have explicit bounds for the difference \[ \mu_1 \leq h(D)-\hat{h}(D) \leq \mu_1^{\prime}, \] where \(h\) is an appropriate normalized logarithmic height on \(J\).In the paper under review, the authors present a new method for explicitly computing the integral points of \(C\). Using the assumptions (1) and (2), they give a completely explicit upper bound for the integral points of \(C\). Next, under the assumption (3), they show that a combination of this bound with a powerful refinement of the Mordell-Weil sieve, is capable of determining all the integral points of \(C\). As an illustration of this method the authors determine the integral points of the hyperelliptic curves \(Y^2-Y = X^5-X\) and \({Y\choose 2} ={X\choose 5}\). Reviewer: Dimitros Poulakis (Thessaloniki) Cited in 4 ReviewsCited in 24 Documents MSC: 11G30 Curves of arbitrary genus or genus \(\ne 1\) over global fields 11J86 Linear forms in logarithms; Baker’s method Keywords:hyperelliptic curve; Jacobian; height; Mordell-Weil group; Mordell-Weil sieve; descent PDFBibTeX XMLCite \textit{Y. Bugeaud} et al., Algebra Number Theory 2, No. 8, 859--885 (2008; Zbl 1168.11026) Full Text: DOI arXiv Online Encyclopedia of Integer Sequences: Complete list of integers x > 1 such that x^2 - x = y^q - y, where q is an odd prime and y is a prime power.