A strategy of numeric search for perfect cuboids in the case of the second cuboid conjecture.

A strategy of numeric search for perfect cuboids in the case of the second cuboid conjecture

A perfect cuboid is a rectangular parallelepiped whose edges, whose face diagonals, and whose space diagonal are of integer lengths. The problem of finding such cuboids or proving their non-existence is not solved thus far. The second cuboid conjecture specifies a subclass of perfect cuboids described by one Diophantine equation of tenth degree and claims their non-existence within this subclass. Regardless of proving or disproving this conjecture in the present paper the Diophantine equation associated with it is studied and is used in order to build an optimized strategy of computer-assisted search for perfect cuboids within the subclass covered by the second cuboid conjecture.

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11D41, 11D72, 30B10, 30E10, 30E15

1. Introduction.

For the history and various approaches to the problem of perfect cuboids the reader is referred to [?]. In this paper we resume the research initiated in [?]. The papers [?] deal with another approach based on so-called multisymmetric polynomials. In this paper we do not touch this approach.

Perfect cuboids are described by six Diophantine equations. These equations are immediate from the Pythagorean theorem:

The variables , , in 1.1 stand for three edges of a cuboid, the variables , , correspond to its face diagonals, and represents its space diagonal.

In [43] an algebraic parametrization for the Diophantine equations 1.1 was suggested. It uses four rational variables , , , and :

The variables , , in 1.2 are different from the original ones which are used in [43], here we use and instead of and , and we use instead of .

Only two of the four variables , , , and are independent. The variables and are taken for independent ones. Then the variable is expressed through and as a solution of the following algebraic equation:

Once the variable is expressed as a function of and by solving the equation 1.3, the variable is given by the formula

The equation 1.3, along with the formula 1.4, produces two algebraic functions

Substituting 1.5 into 1.2, we get six algebraic functions

which are linear with respect to . The functions 1.6 satisfy the cuboid equations 1.1 identically with respect to , , and . This fact is presented by the following theorem (see Theorem 5.2 in [43]).

Theorem 1.1

A perfect cuboid does exist if and only if there are three rational numbers , , and satisfying the equation 1.3 and obeying four inequalities , , , and .

The rational numbers , , and cam be brought to a common denominator:

Substituting 1.7 into 1.3, one easily derives the Diophantine equation

Theorem 1.1 then is reformulated in the following form (see Theorem 4.1 in [44]).

Theorem 1.2

A perfect cuboid does exist if and only if for some positive coprime integer numbers , , and the Diophantine equation 1.8 has a positive solution obeying the inequalities , , , and .

In [44] the Diophantine equation was treaded as a polynomial equation for , while , , and were considered as parameters. As a result in [44] several special cases of the equation 1.8 were specified. They are introduced through the following relationships for the parameters , , and :

The cases 2, 5, and 6 are trivial. They produce no perfect cuboids (see [44]). The case 1 corresponds to the first cuboid conjecture (see [44]). It is less trivial, but it produces no perfect cuboids either (see [45]). The cases 2 and 4 correspond to the second cuboid conjecture (see [44] and [46]). The case, where none of the conditions 1.9 is fulfilled, corresponds to the third cuboid conjecture (see [44] and [47]).

In this paper we consider the cases 3 and 4 associated with the second cuboid conjecture. In the case 3 the equality is resolved by substituting

Here are two positive coprime integers. Upon substituting 1.10 into the equation 1.8 it reduces to the equation

(see [45]), where is the following polynomial of tenth degree:

The case 4 is similar. In this case the equality is resolved by substituting

Upon substituting 1.13 into the equation 1.8 it reduces to the equation

The roots , , , and of the equations 1.11 and 1.14 do not produce perfect cuboids (see Theorem 1.2). Upon splitting off the linear factors from 1.11 and 1.14 we get the equation

Conjecture 1.1

For any positive coprime integers the polynomial in 1.12 is irreducible in the ring .

Conjecture 1.1 is known as the second cuboid conjecture. It was formulated in [44]. In particular it claims that the equation 1.15 has no integer roots for any positive coprime integers . We do not try to prove or disprove Conjecture 1.1 in this paper. Instead, we study real positive roots of the equation 1.15 in the case where is much larger than . Using asymptotic expansions for the roots of the equation 1.15 as , below we build an optimized strategy of computer-assisted search for perfect cuboids in the realm of Conjecture 1.1.

2. Asymptotic expansions for roots of the polynomial equation.

Note that the polynomial in 1.12 is even. Along with each root it has the opposite root . We use the condition

in order to divide the roots of the equation 1.15 into two groups. We denote through , , , , the roots that obey the conditions 2.1. Then , , , , are opposite roots of the equation 1.15:

Typically, asymptotic expansions for roots of a polynomial equation look like power series (see [61]). In our case we have the expansions

The coefficient in 2.3 should be nonzero: .

Let’s substitute 2.3 into the equation 1.15. For this purpose we represent the polynomial from 1.12 formally as the sum

Each nonzero term in 2.4, i. e. a term with the nonzero coefficient

Taking into account 2.5, the equation 1.15 is written as

The equality 2.6 should be fulfilled identically with respect to the variable . Since , a necessary condition for that is the coincidence of exponents of at least two summands of the form in the leading order with respect to the variable . This yields the equalities

The maximality of the exponent in 2.7 means that all exponents are not greater than , i. e. we have the following inequality:

Lemma 2.1

The coincidence in the formula 2.7 is impossible.

Demonstration Proof

Indeed, due to 2.8 the coincidence would mean . But the sum 2.6 has no two summands with simultaneously coinciding indices and . Lemma 2.1 is proved.∎

Let’s treat and as coordinates of a point on the coordinate plane. Since and are integer, such a point belongs to the integer grid, being its node. The numbers , and , from 2.7 mark two nodes of this grid. These are the points and in Fig. 2.1. Due to Lemma 2.1 from the equality 2.7 we derive the following formula for the exponent :

The right hand side of the formula 2.9 up to the sign coincides with the slope of the straight line connecting the nodes and in Fig. 2.1:


The nodes and correspond to some nonzero summands in the sum 2.4 being a formal presentation of the polynomial 1.12. They are selected by the maximality condition for the parameter . The maximum is taken over all summands in the sum 2.4 for a fixed value of .

Lemma 2.2

The exponent in the asymptotic expansion 2.3 is determined by the slope of a straight line connecting some two nodes of the integer grid associated with some two nonzero terms in the polynomial 2.4.

Let be some node of the integer grid in Fig. 2.1 associated with some nonzero summand of the sum 2.4 and different from the nodes and . Its coordinates and satisfy the inequality 2.8. From 2.7 and 2.8 one derives the inequality

Let’s write this inequality as follows:

In Fig. 3.1 three positions of the node relative to the node are shown. The node can be located to the left of the node , to the right of the node , or on the same vertical line with the node . In the first case . In the second case . And finally, in the third case .

In the first case, i. e. if , from 2.11 we derive

The right hand side of the inequality 2.12 up to the sign coincides with the slope of the line . Applying 2.10, we get the inequality . Inverting signs, we write this inequality in the following form:

In the second case, i. e. if , from 2.11 we derive

By analogy with 2.13 the inequality 2.14 is transformed to

And finally, in the third case, i. e. if , from the inequality 2.11 we derive

The inequality 2.16 is equivalent to the following inequality:

Each of the inequalities 2.13, 2.15, and 2.17 in its case means that the point is located not above the line . This fact is formulated as a lemma.

Lemma 2.3

All nodes of the integer grid associated with nonzero summands in the polynomial 2.4 are located not above the line on which the nodes implementing the maximum of the parameter are located.

In order to apply Lemmas 2.1, 2.2, and 2.3 let’s mark all of the nodes associated with the polynomial 1.12 on the coordinate plane.


Definition Definition 2.1

For any polynomial of two variables the convex hull of all integer nodes on the coordinate plane associated with monomials of this polynomial is called the Newton polygon of .

Remark. Note that in our case the polynomial 1.12 depend on three variables , , and . However, we treat as a parameter and consider as a polynomial of two variables when applying Definition 2.1 to it.

The Newton polygon of the polynomial 1.12 is shown in Fig. 2.2. Its boundary consists of two parts — the upper part and the lower part. The upper parts is drawn in green, the lower part is drawn in red. In Fig. 2.2 the nodes on the upper boundary of the Newton polygon are denoted according to the formula 2.4. The coefficients in 1.12 associated with these nodes are given by the formulas

Theorem 2.1

The values of exponents in the expansion 2.3 for roots of the equation 1.15 are determined according to the formula , where stands for slopes of segments of the polygonal line being the upper boundary of the Newton polygon in Fig. 2.2.

Theorem 2.1 is immediate from Lemmas 2.2 and 2.3. The formula in this theorem follows from the formula 2.10. In our particular case we have

The options 2.19 are derived from Fig. 2.2 due to the above theorem.

3. Leading terms in asymptotic expansions.

The term obtained upon expanding brackets in 2.3 is called the leading term of the asymptotic expansion 2.3. Three options for the value of are given by the formula 2.19. Let’s consider each of these options separately.

The case . This case corresponds to the horizontal segment on the upper boundary of the Newton polygon in Fig. 2.2. This segment comprises three nodes , , and . Therefore, substituting the expansion 2.3 with into the equation 1.15, we get the following equation for :

Taking into account 2.18, the equation 3.1 is transformed to

The equation 3.2 has two real roots

each of which is of multiplicity . The condition 2.1 excludes the root from 3.3. The remain is one root of multiplicity :

The asymptotic expansion 2.3 corresponding to 3.4 is

The case . This case corresponds to the short slant segment in the upper boundary of the Newton polygon in Fig. 2.2. It comprises two nodes and . Therefore, substituting the expansion 2.3 with into the equation 1.15, we get the following equation for :

The common divisor can be factored out from the equation 3.6. Since , we can remove this common divisor. Then the equation takes the form

Taking into account 2.18, the equation 3.7 is transformed to

The quadratic equation 3.8 has two simple root

The condition 2.1 excludes the root from 3.9. Therefore as a remain we have only one root, which is of multiplicity :

The asymptotic expansion 2.3 corresponding to 3.10 is

The case . This case corresponds to the long slant segment in the upper boundary of the Newton polygon in Fig. 3.2. It comprises three nodes , , and . Therefore, substituting the expansion 2.3 with into the equation 1.15, we get the following equation for :

The common divisor is factored out from the equation 3.12. Since , we can remove this common divisor. Then the equation takes the form

Taking into account 2.18, the equation 3.13 is transformed to

The quartic equation 3.14 has four roots. All of them are complex:

Here . The roots 3.16 are excluded by the condition 2.1. The remain is two root 3.15 of multiplicity . They yield the following asymptotic expansions:

The results 3.5, 3.11, 3.17 are summed up in the following theorem.

Theorem 3.1

For sufficiently large positive values of the parameter , i. e. for , the tenth-degree equation 1.15 has five roots of multiplicity satisfying the condition 2.1. Three of them , , and are real roots. Their asymptotics as are given by the formulas

The rest two roots and of the equation 1.15 are complex. Their asymptotics as are given by the formulas

The complex roots 3.19 do not provide perfect cuboids. However, below they are important for determining the exact number of real roots.

4. Asymptotic estimates for real roots.

According to the formula 3.18 the roots and are not growing as . For this reason we do not need to calculate in 3.5 for them. But we need to find estimates for remainder terms and in the formulas

as . Our goal is to obtain estimates of the form

In order to get such estimates we substitute

into the equation 1.15. Then we perform another substitution into the equation obtained as a result of substituting 4.3 into 1.15:

Upon two substitutions 4.3 and 4.4 and upon removing denominators the equation 1.15 is written as a polynomial equation in the new variables and . It is a peculiarity of this equation that it can be written as

Here is a polynomial given by an explicit formula. The formula for is rather huge. Therefore it is placed to the ancillary file strategy_formulas.txt in a machine-readable form.

Let and let the parameter run over the interval from to :

From and from 4.4 we derive the estimate . Using this estimate and using the inequalities 4.6, by means of direct calculations one can derive the following estimate for the modulus of the function :

For fixed and the estimate 4.7 means that the left hand side of the equation 4.5 is a continuous function of taking the values within the range from to while is in the interval 4.6. The right hand side of 4.5 is also a continuous function of . It decreases from to in the interval 4.6. Therefore somewhere in the interval 4.6 there is at least one root of the equation 4.5.

The parameter is related to the initial variable by means of the formula 4.3. The inequalities 4.5 for imply the following inequalities for :

The inequalities 4.8 and the above considerations prove the following theorem.

Theorem 4.1

For each there is at least one real root of the equation 1.15 satisfying the inequalities 4.8.

The above considerations can be repeated for the case where the parameter runs over the interval from to . In this case due to 4.3 from

we derive the inequalities

for the variable and hence we obtain the following theorem.

Theorem 4.2

For each there is at least one real root of the equation 1.15 satisfying the inequalities 4.9.

Now let’s proceed to the growing root of the equation 1.15 (see Theorem 3.1). Upon refining the asymptotic formula 3.18 for looks like

The formula 4.10 is in agreement with the expansion 3.11. It means that

Like in 4.2, our goal here is to obtain estimates of the form

In order to get such estimates we substitute

into the equation 1.15. Immediately after that we perform the substitution 4.4 into the equation obtained by substituting 4.12 into 1.15. As a result of two substitutions 4.12 and 4.4 upon eliminating denominators the equation 1.15 is written as a polynomial equation in the new variables and . It looks like

Here is a polynomial of three variables. The explicit formula for is rather huge. Therefore it is placed to the ancillary file strategy_formulas.txt in a machine-readable form.

Let and let the parameter run over the interval from to :

From and from 4.4 we derive the estimate . Using this estimate and using the inequalities 4.14, by means of direct calculations one can derive the following estimate for the modulus of the function :

For fixed and the estimate 4.15 means that the left hand side of the equation 4.13 is a continuous function of taking the values within the range from to while is in the interval 4.14. The right hand side of the equation 4.13 is also a continuous function of . It decreases from to in the interval 4.14. Therefore somewhere in the interval 4.14 there is at least one root of the polynomial equation 4.13.

The parameter is related to the initial variable by means of the formula 4.12. The inequalities 4.14 for imply the following inequalities for :

The inequalities 4.16 and the above considerations prove the following theorem.

Theorem 4.3

For each there is at least one real root of the equation 1.15 satisfying the inequalities 4.16.

Theorems 4.1, 4.2, and 4.3 solve the problem of obtaining estimates of the form 4.2 and 4.11 for the remainder terms in the refined asymptotic expansions 4.1 and 4.10 for .

5. Asymptotic estimates for complex roots.

Let’s proceed to complex roots of the equation 1.15. Upon refining the asymptotic formula 3.19 for the complex root is written as

The formula 5.1 is in agreement with the first expansion 3.17. It means that and . Like in the formula 4.2 and in the formula 4.11, our goal here is to obtain estimates of the form

In order to get such estimates we substitute

into the equation 1.15. Immediately after that we perform the substitution 4.4 into the equation obtained by substituting 5.3 into 1.15. As a result of two substitutions 5.3 and 4.4 upon eliminating denominators the equation 1.15 is written as a polynomial equation in the new variables and . It looks like

Here is a polynomial of three variables with purely real coefficients. The explicit formula for is rather huge. Therefore it is placed to the ancillary file strategy_formulas.txt in a machine-readable form.

Let and let the parameter run over the interval from to :

From and from 4.4 we derive the estimate . Using this estimate and using the inequalities 5.5, by means of direct calculations one can derive the following estimate for the modulus of the function :

For fixed and the estimate 5.6 means that the left hand side of the equation 5.4 is a continuous function of taking its values within the range from to while runs over the interval 5.5. The right hand side of the equation 5.4 is also a continuous function of . It increases from to in the interval 5.5. Therefore somewhere in the interval 5.5 there is at least one root of the polynomial equation 5.4.

The parameter is related to the initial variable by means of the formula 5.3. Therefore the inequalities 5.5 for imply the following inequalities for :

The inequalities 5.7 and the above considerations prove the following theorem.

Theorem 5.1

For each there is at least one purely imaginary root of the equation 1.15 satisfying the inequalities 5.7.

The complex root is similar to the root . Upon refining the asymptotic formula 3.19 for the complex root is written as

The formula 5.8 is in agreement with the second expansion 3.17. It means that and . Like in the formulas 4.2, 4.11, and 4.2, our goal here is to obtain estimates of the form

In order to get such estimates we substitute

into the equation 1.15. Immediately after that we perform the substitution 4.4 into the equation obtained by substituting 5.10 into 1.15. As a result of two substitutions 5.10 and 4.4 upon eliminating denominators the equation 1.15 is written as a polynomial equation in the new variables and . It looks like

Here is a polynomial of three variables. The explicit formula for is rather huge. Therefore it is placed to the ancillary file strategy_formulas.txt in a machine-readable form.

Let and let the parameter run over the interval from to (see 5.5). From and from 4.4 we derive the estimate . Using this estimate and using the inequalities 5.5, by means of direct calculations one can derive the following estimate for the modulus of the function :

For fixed and the estimate 5.12 means that the left hand side of the equation 5.11 is a continuous function of taking its values within the range from to while runs over the interval 5.5. The right hand side of the equation 5.12 is also a continuous function of . It increases from to in the interval 5.5. Therefore somewhere in the interval 5.5 there is at least one root of the polynomial equation 5.12.

The parameter is related to the initial variable by means of the formula 5.10. Therefore the inequalities 5.5 for imply the following inequalities for :

The inequalities 5.13 and the above considerations prove the following theorem.

Theorem 5.2

For each there is at least one purely imaginary root of the equation 1.15 satisfying the inequalities 5.13.

Theorems 5.1 and 5.2 solve the problem of obtaining estimates of the form 5.2 and 5.9 for the remainder terms in the refined asymptotic expansions 5.1 and 5.8 for . Along with Theorems 4.1, 4.2, and 4.3, they separate the roots , , , , of the equation 1.15 from each other for sufficiently large and provide rather precise intervals for their location.

6. Non-intersection of asymptotic intervals.

Theorems 4.1, 4.2, 4.3, 5.1, 5.2 define five asymptotic intervals 4.8, 4.9, 4.16, 5.7, and 5.13 for . It is easy to see that the intervals 4.8 and 4.9 do not intersect. For the other pairs of intervals among 4.8, 4.9, 4.16, 5.7, 5.13 this is not so obvious. Therefore we need some elementary lemmas.

Lemma 6.1

For the asymptotic intervals 4.8, 4.9, 4.16, 5.7, and 5.13 do not comprise the origin.

Demonstration Proof

Indeed, from for the left endpoint of the interval 4.8 we derive

The left endpoint of the interval 4.9 is obviously positive: