Chen, Yonggao; Guo, Shuguang On the rational cuboids with a given face. (English) Zbl 1083.11023 J. Number Theory 112, No. 2, 205-215 (2005). 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 \((a,b)=(20,21)\), \((20,99)\), \((24,7)\), \((48,55)\), \((60,11)\), \((60,91)\), \((80,39)\), \((84,13)\). Reviewer: Dimitros Poulakis (Thessaloniki) Cited in 2 Documents MSC: 11D09 Quadratic and bilinear Diophantine equations Keywords:Pythagorean triple; Diophantine equations PDFBibTeX XMLCite \textit{Y. Chen} and \textit{S. Guo}, J. Number Theory 112, No. 2, 205--215 (2005; Zbl 1083.11023) Full Text: DOI References: [1] Bremner, A., The rational cuboid and a quartic surface, Rocky Mountain J. Math., 18, 1, 105-121 (1988) · Zbl 0648.10011 [2] Colman, W. J.A., Some observations on the classical cuboid and its parametric solutions, Fibonacci Quart., 26, 4, 338-343 (1988) · Zbl 0661.10022 [3] Dickson, L. E., The History of the Theory of Numbers (1919), Carnegie Institute of Washington: Carnegie Institute of Washington Washington · JFM 47.0894.01 [4] Engel, M., Numerische Lösung eines Quaderproblems, Wiss. Z. Pädagog. Hochsch. Erfurt/Mühlhausen Math. Natur. Reihe, 22, 1, 78-86 (1986) [5] Korec, I., Nonexistence of a small perfect rational cuboid II, Acta Math. Univ. Comenian., 44/45, 39-48 (1984) · Zbl 0555.10006 [6] Narumiya, N.; Shiga, H., On certain rational cuboid problems, Nihonkai Math. J., 12, 1, 75-88 (2001) · Zbl 1022.11014 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.