The equations \(3x^2 - 2 = y^2\) and \(8x^2 - 7 = z^2\).

*(English)*Zbl 0177.06802The 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.

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 |