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The equations \(3x^2 - 2 = y^2\) and \(8x^2 - 7 = z^2\). (English) Zbl 0177.06802
The four numbers \(1,3,8,120\) have the property that the product of any two increased by \(1\) is a perfect square. Here a negative answer is given to the well-known problem whether there is any other positive integer that can replace 120 and still preserve the property [cf. J. H. van Lint, On a set of diophantine equations. Report 68-WSK-03, Eindhoven, The Netherlands; Technological University, Dept. of Math., 8 p. (1968; Zbl 0174.07901)]. The result is equivalent to the assertion that the only solutions in positive integers \(x, y, z\) of the equations (*) of the title are given by \(x=1\) and \(x=11\).
The proof depends on the work of a recent paper [A. Baker, Mathematika 15, 204–216 (1968; Zbl 0169.37802)] which enables an explicit upper bound to be established for the size of all the integer solutions of (*). The main theme of the present paper is to show that the remaining range can be covered without a prohibitive amount of computations. For this purpose a simple lemma an Diophantine approximation is used which reduces the work to one basic numerical verification; and this was successfully carried out by the Atlas Computer Laboratory. It is likely that many similar Diophantine problems can be resolved by the same techniques.
Reviewer: Alan Baker

MSC:
11D09 Quadratic and bilinear Diophantine equations
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