Symmetry-Based Approach to the Problem of a Perfect Cuboid
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SYMMETRY-BASED APPROACH TO THE PROBLEM OF A PERFECT CUBOID R. A. Sharipov
UDC 511.528
Abstract. A perfect cuboid is a rectangular parallelepiped in which the lengths of all edges, the lengths of all face diagonals, and also the lengths of spatial diagonals are integers. No such cuboid has yet been found, but their nonexistence has also not been proved. The problem of a perfect cuboid is among unsolved mathematical problems. The problem has a natural S3 -symmetry connected to permutations of edges of the cuboid and the corresponding permutations of face diagonals. In this paper, we give a survey of author’s results and results of J. R. Ramsden on using the S3 symmetry for the reduction and analysis of the Diophantine equations for a perfect cuboid. Keywords and phrases: polynomial, Diophantine equation, perfect cuboid. AMS Subject Classification: 11D09, 11D41, 11D72
1. A brief historical survey. The problem of a perfect cuboid was first mentioned about 300 years ago in the book by Paul Halcke [14], which is a collection of recreational mathematical problems and puzzles. It is reduced to the following system of Diophantine equations: (x1 )2 + (x2 )2 − (d3 )2 = 0,
(x2 )2 + (x3 )2 − (d1 )2 = 0,
(x3 )2 + (x1 )2 − (d2 )2 = 0,
(x1 )2 + (x2 )2 + (x3 )2 = L2 .
(1.1)
Here x1 , x2 , and x3 are the lengths of the edges of the cuboid, d1 , d2 , and d3 are the lengths of the diagonals on the faces, and L is the length of the spatial diagonal. In a perfect cuboid, all these values must be integers. If only one of them is not an integer, then such a cuboid is said to be almost perfect. Almost perfect cuboids are known. A big list of almost perfect cuboids found by computer calculations is contained in the appendix to [43]. The first examples of almost perfect cubiods where the length of the spatial diagonal L is not an integer were found by Saunderson in [49]. They are parametrized by the Pythagoras triples of integers u2 + v 2 = w2 and are given by the formulas x1 = u4v 2 − w2 , x2 = v 4u2 − w2 , x3 = 4uvw. (1.2) L. Euler was interested in this problem after Saunderson. In [37], he found a parametric family of almost perfect cuboids expressed by the formulas x1 = 2mn 3m2 − n2 3n2 − m2 , (1.3) x2 = m2 − n2 m2 − 4mn + n2 m2 + 4mn + n2 , x3 = 8mn m4 − n4 . The Euler formulas (1.3) are embedded into the Saunderson formulas (1.2) by the substitution u = 2mn, v = m2 − n2 , w = m2 + n2 . Despite this circumstance and contrary to the historical justice, almost perfect cuboids with one noninteger parameter L are called Euler cubes or Euler bricks. After Euler, the problem on perfect cuboids was forgotten for a long time. Interest in it revived only in the 20th century. Pocklington [37] and Spohn [71] proved that the Saunderson series of almost Translated from Itogi Nauki i Tekhniki, Seriya Sovremennaya Matematika i Ee Prilozheniya. Tematicheskie Obzory, Vol. 152, Mathematical Physics, 2018.
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