On the equivalence of cuboid equations and their factor equations.

On the equivalence of cuboid equations
and their factor equations.

Ruslan Sharipov Bashkir State University, 32 Zaki Validi street, 450074 Ufa, Russia r-sharipovmail.ru

An Euler cuboid is a rectangular parallelepiped with integer edges and integer face diagonals. An Euler cuboid is called perfect if its space diagonal is also integer. Some Euler cuboids are already discovered. As for perfect cuboids, none of them is currently known and their non-existence is not yet proved. Euler cuboids and perfect cuboids are described by certain systems of Diophantine equations. These equations possess an intrinsic symmetry. Recently they were factorized with respect to this symmetry and the factor equations were derived. In the present paper the factor equations are shown to be equivalent to the original cuboid equations regarding the search for perfect cuboids and in selecting Euler cuboids.

:
11D41, 11D72, 13A50, 13F20

1. Introduction.

The search for perfect cuboids has the long history. The reader can follow this history since 1719 in the references [1--44]. In order to write the cuboidal Diophantine equations we use the following polynomials:

Here , , are edges of a cuboid, , , are its face diagonals, and is its space diagonal. An Euler cuboid is described by a system of three Diophantine equations. In terms of the polynomials 1.1 these equations are written as

In the case of a perfect cuboid the number of equations is greater by one, i. e. instead of the equations 1.2 we write the following system of four equations:

The permutation group acts upon the cuboid variables , , , , , , and according to the rules expressed by the formulas

The variables , , and , , are usually arranged into a matrix:

The rules 1.4 means that acts upon the matrix 1.5 by permuting its columns. Applying the rules 1.4 to the polynomials 1.1, we derive

The polynomials , , , in 1.1 are treated as elements of the polynomial ring . For the sake of brevity we denote

where is the matrix given by the formula 1.5.

Definition Definition 1.1

A polynomial is called multisymmetric if it is invariant with respect to the action 1.4 of the group .

Multisymmetric polynomials constitute a subring in the ring 1.7. We denote this subring through . The formulas 1.6 show that the polynomial belongs to the subring , i. e. it is multisymmetric, while the polynomials , , are not multisymmetric. Nevertheless, the system of equations 1.2 in whole is invariant with respect to the action of the group . The same is true for the system of equations 1.3.

The polynomials , , generate an ideal in the ring . It is natural to call it the cuboid ideal and denote this ideal through

Similarly, one can define the perfect cuboid ideal

Each polynomial equation with follows from the equations 1.2. Therefore such an equation is called a cuboid equation. Similarly, each polynomial equation with follows from the equations 1.3. Such an equation is called a perfect cuboid equation.

The symmetry approach to the equations 1.2 and 1.3 initiated in [45] leads to studying the following ideals in the ring of multisymmetric polynomials:

Definition Definition 1.2

A polynomial equation of the form with or with is called an factor equation for the Euler cuboid equations 1.2 or for the perfect cuboid equations 1.3 respectively.

The ideal from 1.10 was studied in [46] (there it was denoted through ). The polynomial used as a generator in 1.9 is multisymmetric in the sense of the definition 1.1. Therefore it belongs to the ideal . In [46] this polynomial was denoted through :

In addition to 1.11 in [46] the following seven polynomials were considered:

Theorem 1.1

The ideal from 1.10 is finitely generated within the ring . Eight polynomials 1.11, 1.12, 1.13, 1.14, 1.15, 1.16, 1.17, and 1.18 belong to the ideal and constitute a basis of this ideal.

The theorem 1.1 was proved in [46]. The ideal in 1.8 is similar to the ideal . There is the following theorem describing this ideal.

Theorem 1.2

The ideal from 1.10 is finitely generated within the ring . Seven polynomials 1.12, 1.13, 1.14, 1.15, 1.16, 1.17, and 1.18 belong to the ideal and constitute a basis of this ideal.

The theorem 1.2 can be proved in a way similar to the proof of the theorem 1.1 in [46]. I do not give the proof of the theorem 1.2 here for the sake of brevity.

Relying on the theorem 1.2 and using the polynomials 1.12, 1.13, 1.14, 1.15, 1.16, 1.17, 1.18, we write the system of seven factor equations

The factor equations 1.19 correspond to the case of Euler cuboids. Similarly, in the case of perfect cuboids, relying on the theorem 1.1 and using the polynomials given by the formulas 1.11, 1.12, 1.13, 1.14, 1.15, 1.16, 1.17, 1.18, we write the following system of eight factor equations:

The structure of the polynomials , , , , , , , in 1.11, 1.12, 1.13, 1.14, 1.15, 1.16, 1.17, 1.18 is so that each solution of the equations 1.2 is a solution for the equations 1.19. Similarly, each solution of the equations 1.3 is a solution for the equations 1.20. The main goal of this paper is to prove converse propositions. They are given by the following two theorems.

Theorem 1.3

Each integer or rational solution of the factor equations 1.19 such that , , , , , and is an integer or rational solution for the equations 1.2.

Theorem 1.4

Each integer or rational solution of the factor equations 1.20 such that , , , , , and is an integer or rational solution for the equations 1.3.

2. The analysis of the factor equations.

Let’s consider the factor equations 1.19 associated with Euler cuboids. Due to 1.12, 1.13, 1.14, 1.15, 1.16, 1.17, and 1.18 the factor equations 1.19 can be united into a single matrix equation

In order to study the equations 2.1 we denote through the transposed matrix

If we have a solution of the equation 2.1 which is not a solution for the initial system of cuboid equations 1.2, then the equations 1.2 should not be fulfilled simultaneously. Therefore we have the vectorial inequality

Applying 2.3 to 2.1, we derive that the columns of the matrix in 2.1 are linearly dependent. Then the rows of in 2.2 are also linearly dependent, i. e.

The condition 2.4 leads to several special cases which are considered below one by one. In addition to we define the following two matrices:

The matrices and in 2.5 are submatrices of the matrix .

3. The case .

The first column of the matrix 2.2 is nonzero. Therefore . Now we consider the case where . In this case each column of the matrix is proportional to its first column. In particular, this yields

The equations 3.1 lead to the equalities

Applying 3.2 to the formulas 1.1, we derive

Then we substitute 3.3 into 1.12. As a result we get

The relationships 3.4 mean that if the equations 1.19 are fulfilled, then in the case of the equations 1.2 are also fulfilled.

Theorem 3.1

Each solution of the equations 1.19 corresponding to the case is a solution for the equations 1.2.

Theorem 3.2

Each solution of the equations 1.20 corresponding to the case is a solution for the equations 1.3.

Due to 1.11 the equation in 1.3 coincides with the equation in 1.20. For this reason the theorem 3.2 is immediate from the theorem 3.1.

4. The case and .

The condition for the matrix in 2.5 means that the third column of the matrix 2.2 is proportional to the first column of this matrix. The condition for the matrix in 2.5 means that the first and the second columns of the matrix 2.2 are linearly independent. Other columns are expressed as linear combinations of these two columns. As a result we can write

It is easy to see that the conditions 4.1 are sufficient for to provide the condition , which is in agreement with 2.4.

The second equality in 4.1 is very important. It means that , , and , are roots of the following quadratic equation:

The quadratic equation 4.2 has at most two roots. Let’s denote them and . Then we have the following subcases derived from and :

The numbers and in the formulas 4.3 are arbitrary two numbers not coinciding with each other: . They are integer numbers in the case of integer solutions and they are rational numbers in the case of rational solutions.

The three cases in 4.3 are similar to each other. Without loss of generality we can consider only one of them, e. g. the first one. Then from 4.3 we derive

Due to the relationships 4.4 and 4.5 the matrix equation 2.1 reduces to

The matrix equality 4.6 means that instead of the seven equations 1.19 we have two equations and . Substituting , , and into these two equations, we derive

The equations 4.7 can be written in the following way:

Now it is easy to see that the equations 4.8 can be satisfied by three integer or rational numbers , , and if and only if all of them are zero. Substituting into 4.1 and 4.3, we get

The equalities 4.9 contradict the condition for the matrix in 2.5. This contradiction yields the following two theorems.

Theorem 4.1

The factor equations 1.19, as well as the original equations 1.2, have no integer or rational solution in the case of and .

Theorem 4.2

The factor equations 1.20, as well as the original equations 1.3, have no integer or rational solution in the case of and .

5. The case and .

The condition for the matrix in 2.5 means that the second column of the matrix 2.2 is proportional to the first column of this matrix. The condition for the matrix in 2.5 means that the first and the third columns of the matrix 2.2 are linearly independent. Other columns are expressed as linear combinations of these two columns. As a result we can write the relationships similar to the relationships 4.1:

The conditions 5.1 are sufficient for to provide the condition .

Like in the case of 4.1, the second condition 5.1 mean that , , and are roots of the quadratic equation similar to 4.2:

The quadratic equation 5.2 has at most two roots. Let’s denote them and . Then we have the following subcases derived from and :

The numbers and in the formulas 5.3 are arbitrary two integer or rational numbers not coinciding with each other: .

The three cases in 5.3 are similar to each other. Without loss of generality we can consider only one of them, e. g. the first one. Then from 5.3 we derive

Due to the relationships 5.4 and 5.5 the matrix equation 2.1 reduces to

The matrix equality 5.6 means that instead of the seven equations 1.19 we have two equations and . Substituting , , and into these two equations, we derive

The second equation 5.7 can be written in the following form:

The equation 5.8 can be satisfied by two integer or rational numbers and if and only if both of them are zero. Substituting into the first equation 5.7, we get . Substituting into 5.1 and 5.3, we get

The equalities 5.9 contradict the condition for the matrix in 2.5. This contradiction yields the following two theorems.

Theorem 5.1

The factor equations 1.19, as well as the original equations 1.2, have no integer or rational solution in the case of and .

Theorem 5.2

The factor equations 1.20, as well as the original equations 1.3, have no integer or rational solution in the case of and .

6. The case and .

In this case the columns of both matrices and in 2.5 are linearly independent. Hence each column of the matrix 2.2 can be expressed as a linear combination of the first and the second columns of this matrix or as a linear combination of the first and the third columns of this matrix. In particular, we have

The relationships 6.1 mean that , , and , , are roots of two quadratic equations coinciding with 5.2 and 4.2 respectively. As a result we distinguish three subcases 4.3 with and three subcases 5.3 with . The first subcase 4.3 should be paired with the first subcase 5.3, the second subcase 4.3 should be paired with the second subcase 5.3, and the third subcase 4.3 should be paired with the third subcase 5.3. Otherwise we would have , which contradicts the condition 2.4.

Due to the pairing of subcases we have three subcases instead of nine ones, which are a priori possible. These three subcases are similar to each other. Therefore without loss of generality we can consider only one subcase, e. g. the following one:

Here and . The relationships 6.2 lead to the relationships 4.4, 4.5, 5.4, 5.5 and then to the equations 4.6 and 5.6. The matrix equations 4.6 and 5.6 mean that instead of the seven equations 1.19 we have two equations and . Substituting , , , and into these two equations, we derive

The second equation 6.3 can be written in the following way:

The equation 6.4 can be satisfied by two integer or rational numbers and if and only if both of them are zero. Substituting into 6.3, we get . Then the equalities 6.2 are written as

The equalities 6.5 lead to the equalities and . The latter ones contradict the inequalities in the theorems 1.3 and 1.4. Therefore we can conclude this section with the following two theorems.

Theorem 6.1

The factor equations 1.19, as well as the original equations 1.2, have no integer or rational solutions such that , , , , , and in the case of and .

Theorem 6.2

The factor equations 1.20, as well as the original equations 1.3, have no integer or rational solutions such that , , , , , and in the case of and .

7. The ultimate result and conclusions.

The four cases considered in sections 3, 4, 5, and 6 exhaust all options compatible with the inequality 2.4. For this reason the theorems 1.3 and 1.4 follow from the theorems 3.1, 4.1, 5.1, 6.1 and the theorems 3.2, 4.2, 5.2, 6.2 respectively. The theorems 1.3 and 1.4 constitute the main result of this paper. The theorem 1.4 means that the factor equations 1.20 are equally admissible for seeking perfect cuboids or for proving their non-existence as the original equations 1.3. As for the factor equations 1.19, due to the theorem 1.3 they are equally admissible for selecting Euler cuboids as the original equations 1.2.

References

  • 1 , Euler brick, Wikipedia, Wikimedia Foundation Inc..
  • 2 Halcke P., Deliciae mathematicae oder mathematisches Sinnen-Confect, N. Sauer, 1719.
  • 3 Saunderson N., Elements of algebra, Vol. 2, Cambridge Univ. Press, 1740.
  • 4 Euler L., Vollständige Anleitung zur Algebra, 3 Theile, Kaiserliche Akademie der Wissenschaften, 1770-1771.
  • 5 Pocklington H. C., Some Diophantine impossibilities, Proc. Cambridge Phil. Soc. 17 (1912), 108–121.
  • 6 Dickson L. E, History of the theory of numbers, Vol. 2: Diophantine analysis, Dover, 2005.
  • 7 Kraitchik M., On certain rational cuboids, Scripta Math. 11 (1945), 317–326.
  • 8 Kraitchik M., Théorie des Nombres, Tome 3, Analyse Diophantine et application aux cuboides rationelles, Gauthier-Villars, 1947.
  • 9 Kraitchik M., Sur les cuboides rationelles, Proc. Int. Congr. Math. 2 (1954), 33–34.
  • 10 Bromhead T. B., On square sums of squares, Math. Gazette 44 (1960), no. 349, 219–220.
  • 11 Lal M., Blundon W. J., Solutions of the Diophantine equations , , , Math. Comp. 20 (1966), 144–147.
  • 12 Spohn W. G., On the integral cuboid, Amer. Math. Monthly 79 (1972), no. 1, 57-59.
  • 13 Spohn W. G., On the derived cuboid, Canad. Math. Bull. 17 (1974), no. 4, 575-577.
  • 14 Chein E. Z., On the derived cuboid of an Eulerian triple, Canad. Math. Bull. 20 (1977), no. 4, 509–510.
  • 15 Leech J., The rational cuboid revisited, Amer. Math. Monthly 84 (1977), no. 7, 518–533. , see also Erratum.
  • 16 Leech J., Five tables relating to rational cuboids, Math. Comp. 32 (1978), 657–659.
  • 17 Spohn W. G., Table of integral cuboids and their generators, Math. Comp. 33 (1979), 428–429.
  • 18 Lagrange J., Sur le dérivé du cuboide Eulérien, Canad. Math. Bull. 22 (1979), no. 2, 239–241.
  • 19 Leech J., A remark on rational cuboids, Canad. Math. Bull. 24 (1981), no. 3, 377–378.
  • 20 Korec I., Nonexistence of small perfect rational cuboid, Acta Math. Univ. Comen. 42/43 (1983), 73–86.
  • 21 Korec I., Nonexistence of small perfect rational cuboid II, Acta Math. Univ. Comen. 44/45 (1984), 39–48.
  • 22 Wells D. G., The Penguin dictionary of curious and interesting numbers, Penguin publishers, 1986.
  • 23 Bremner A., Guy R. K., A dozen difficult Diophantine dilemmas, Amer. Math. Monthly 95 (1988), no. 1, 31–36.
  • 24 Bremner A., The rational cuboid and a quartic surface, Rocky Mountain J. Math. 18 (1988), no. 1, 105–121.
  • 25 Colman W. J. A., On certain semiperfect cuboids, Fibonacci Quart. 26 (1988), no. 1, 54–57. , Some observations on the classical cuboid and its parametric solutions, see also.
  • 26 Korec I., Lower bounds for perfect rational cuboids, Math. Slovaca 42 (1992), no. 5, 565–582.
  • 27 Guy R. K., Is there a perfect cuboid? Four squares whose sums in pairs are square. Four squares whose differences are square, Unsolved Problems in Number Theory, 2nd ed., Springer-Verlag, 1994, pp. 173–181.
  • 28 Rathbun R. L., Granlund T., The integer cuboid table with body, edge, and face type of solutions, Math. Comp. 62 (1994), 441–442.
  • 29 Van Luijk R., On perfect cuboids, Doctoraalscriptie, Mathematisch Instituut, Universiteit Utrecht, 2000.
  • 30 Rathbun R. L., Granlund T., The classical rational cuboid table of Maurice Kraitchik, Math. Comp. 62 (1994), 442–443.
  • 31 Peterson B. E., Jordan J. H., Integer hexahedra equivalent to perfect boxes, Amer. Math. Monthly 102 (1995), no. 1, 41–45.
  • 32 Rathbun R. L., The rational cuboid table of Maurice Kraitchik, e-print math.HO/0111229 in Electronic Archive http://arXiv.org.
  • 33 Hartshorne R., Van Luijk R., Non-Euclidean Pythagorean triples, a problem of Euler, and rational points on K3 surfaces, e-print math.NT/0606700 in Electronic Archive http://arXiv.org.
  • 34 Waldschmidt M., Open diophantine problems, e-print math.NT/0312440 in Electronic Archive http://arXiv.org.
  • 35 Ionascu E. J., Luca F., Stanica P., Heron triangles with two fixed sides, e-print math.NT/0608 185 in Electronic Archive http://arXiv.org.
  • 36 Ortan A., Quenneville-Belair V., Euler’s brick, Delta Epsilon, McGill Undergraduate Mathematics Journal 1 (2006), 30-33.
  • 37 Knill O., Hunting for Perfect Euler Bricks, Harvard College Math. Review 2 (2008), no. 2, 102. , see also http://www.math.harvard.edu/~knill/various/eulercuboid/index.html.
  • 38 Sloan N. J. A, Sequences A031173, A031174, and A031175, On-line encyclopedia of integer sequences, OEIS Foundation Inc..
  • 39 Stoll M., Testa D., The surface parametrizing cuboids, e-print arXiv:1009.0388 in Electronic Archive http://arXiv.org.
  • 40 Sharipov R. A., A note on a perfect Euler cuboid., e-print arXiv:1104.1716 in Electronic Archive http://arXiv.org.
  • 41 Sharipov R. A., Perfect cuboids and irreducible polynomials, Ufa Mathematical Journal 4, (2012), no. 1, 153–160. , see also e-print arXiv:1108.5348 in Electronic Archive http://arXiv.org.
  • 42 Sharipov R. A., A note on the first cuboid conjecture, e-print arXiv:1109.2534 in Electronic Archive http://arXiv.org.
  • 43 Sharipov R. A., A note on the second cuboid conjecture. Part I, e-print arXiv:1201.1229 in Electronic Archive http://arXiv.org.
  • 44 Sharipov R. A., A note on the third cuboid conjecture. Part I, e-print arXiv:1203.2567 in Electronic Archive http://arXiv.org.
  • 45 Sharipov R. A., Perfect cuboids and multisymmetric polynomials, e-print arXiv:1205.3135 in Electronic Archive http://arXiv.org.
  • 46 Sharipov R. A., On an ideal of multisymmetric polynomials associated with perfect cuboids, e-print arXiv:1206.6769 in Electronic Archive http://arXiv.org.
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