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On the rational cuboids with a given face. (English) Zbl 1083.11023

A rational cuboid is a rectangular parallelepiped whose edges and face diagonals all have rational lengths. This is equivalent to the solution of the system of equations \[ x^2+y^2=l^2,\quad x^2+z^2=m^2,\quad y^2+z^2=n^2. \] In the paper under review, the authors develop a general theory to deal with the following problem: “For given positive integers \(a,b\) with \(a^2+b^2\) being a square and \(\text{gcd}(a,b)=1\), are there positive integers \(c,d\) such that both \(c^2+a^2 d^2\) and \(c^2+b^2d^2\) are squares?” More precisely, the author gives a sufficient condition for the problem being negative. As an application of his results obtains that the problem is negative for \((a,b)=(4,3)\), \((8,15)\), \((12,5)\), \((12,35)\), \((16,63)\), \((28,45)\), \((36,77)\), \((40,9)\), \((56,33)\), \((72,65)\) and affirmative for \((a,b)=(20,21)\), \((20,99)\), \((24,7)\), \((48,55)\), \((60,11)\), \((60,91)\), \((80,39)\), \((84,13)\).

MSC:

11D09 Quadratic and bilinear Diophantine equations
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References:

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