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For three centuries, no one has been able to prove the existence or otherwise of a perfect cuboid (PC) – a rectangular having integer (rational): three sides, three face (surface) diagonals and a space (body) . If among the seven:

a, b, c, dab, dbc, dac, ds only one is irrational, this is called a nearly-perfect cuboid (NPC). The NPC are divided into three types:

1. The space diagonal ds is irrational,

2. One of the sides a, b, c is irrational,

3. One of the face diagonals dab, dbc, dac is irrational.

In this paper we consider the third problem and show how parametric formulae for such NPC can be constructed. PC problem is equivalent to the existence of an integer (rational) solution of the Diophantine system:

2 2 2 a + b = dab, 2 2 2 b + c = dbc, 2 2 2 a + c = dac, 2 2 2 2 a + b + c = ds .

In [1] are found complete parametrizations for NPC (only one face diag- onal dab is irrational) by using a pair of solutions of the congruent number equation. Given parametrizations are not rational and depend on two pa- rameters.

2 In [2–3] are summarized the literature written about PC. In numerous papers, only a few are related to the rational parametrizations [4–9]. In this paper we use the same notations as used in [1].

2 Basic Algebra

Theorem 3 in [1] establishes PC equivalent equation:

1 − γ2 2 1 − β2 2 1 − α2 2 + = , (1)  2γ   2β   2α  where α, β, γ are nontrivial rational numbers, i.e. from set

Q \{0; ± 1}.

After simplification:

2 2 α 2 1 − (αβ) 1 − 1 − γ 2  β     = α . 2γ 4(αβ) · β 

By using new notations: α ξ ≡ αβ, ζ ≡ ; (2) β

PC equation takes the form

1 − γ2 2 (1 − ξ2)(1 − ζ2) = . (3)  2γ  4ξζ

It is clear that the product (ratio) ξζ is square (= ) and

3 ξ, ζ ∈ Q \{0; ± 1}, ξ =6 ζ.

From (2) and (3) we obtain:

Theorem 1. The existence of Perfect cuboid is equivalent to the existence of nontrivial different ξ and ζ rational numbers satisfying three conditions:

ξζ = , (4) (1 − ξ2)(1 − ζ2)= , (5) 2 2 (1 − ξ )(1 − ζ )+4ξζ = . (6)

Let’s discuss the identity like condition (5):

2 (1 − T 2) 1 − (4T 3 − 3T )2 = (1 − T 2)(1 − 4T 2) . (7)   

3 We try to replace ξ and ζ with T and 4T − 3T , respectively. From condition (4), it is possible if

2 4T − 3=  . (8)

The complete rational parametrization of (8) is

t2 +3 T = , 4t where t is an arbitrary (nonzero) rational number. We obtain the rational parametrization of conditions (4) and (5) by for- mulae:

4 t2 +3 t2 +3 t2 − 3 2 ξ = , ζ = . (9) 4t 4t  2t 

Of course (9) represents the incomplete parametrization of (4) and (5), be- cause identity (7) does not describe all rational solutions of (5).

3 The First Parametrization

PC equation (1) is obtained from the system [1]:

2 2 ds 1+ α dbc 1 − α = , = , a 2α a 2α 2 2 dac 1+ β c 1 − β = , = , (10) a 2β a 2β 2 2 dab 1+ γ b 1 − γ = , = . a 2γ a 2γ

From (2) follows

2 2 ξ α = ξζ, β = , (11) ζ and if (4) and (5) are satisfied, then the ratios

ds dbc dac c b , , , , a a a a a are rationals. So,

5 Theorem 2. Nearly-perfect cuboid (only one face diagonal is irrational) is obtained by nontrivial different ξ and ζ rational numbers satisfying two con- ditions:

ξζ = , (1 − ξ2)(1 − ζ2)= .

From (9) and (11):

t4 − 9 2t α = , β = . (12) 8t2 t2 − 3

By substituting (12) in (10) we get the first parametrization for NPC:

I parametrization

2 4 a = 16t (t − 9), 4 2 4 2 b =(t − 10t + 9)(t +2t + 9), c =4t(t2 + 3)(t4 − 10t2 + 9);

2 4 2 dac =4t(t + 3)(t − 2t + 9), 4 4 dbc =(t − 1)(t − 81), 8 4 ds = t + 46t + 81.

6 4 The Second Parametrization

In [1] three PC equations are obtained so, naturally we expect other parametriza- tions. Consider the PC equation from Theorem 1 in [1]:

2α 2 2γ 2 2β 2 + = . (13) 1+ α2  1+ γ2  1+ β2 

After simplification:

α 2 4β2 1 − (αβ)2 1 − 2γ 2  β   = . 1+ γ2  (1 + α2)2(1 + β2)2

PC equation (13) is obtained from the following system [1]:

2 a 2α dbc 1 − α = 2 , = 2 , ds 1+ α ds 1+ α 2 b 1 − β dac 2β = 2 , = 2 , (14) ds 1+ β ds 1+ β 2 c 2γ dab 1 − γ = 2 , = 2 . ds 1+ γ ds 1+ γ

By substituting (12) in (14), we obtain the second parametrization for NPC:

7 II parametrization

2 4 4 2 a = 16t (t − 9)(t − 2t + 9), 4 2 8 4 b =(t − 10t + 9)(t + 46t + 81), c =4t(t2 − 3)(t4 − 10t2 + 9)(t4 +2t2 + 9);

2 8 4 dac =4t(t − 3)(t + 46t + 81), 4 2 8 4 dbc =(t − 2t + 9)(t − 82t + 81), 4 2 8 4 ds =(t − 2t + 9)(t + 46t + 81).

5 The Third Parametrization

From Theorem 2 in [1] PC equation is 2γ 2 2β 2 2α 2 + = . (15) 1 − γ2  1 − β2  1 − α2 

After simplification:

2 2 2 β 2γ 2 4α 1 − (αβ) 1 −    =  α . 1 − γ2  (1 − α2)2(1 − β2)2

In this case insert

t2 +3 β t2 +3 t2 − 3 2 αβ = , = 4t α 4t  2t  8 into the generating system [1] for PC equation (15):

2 ds 1+ α dbc 2α = , = , a 1 − α2 a 1 − α2 2 dac 1+ β c 2β = , = , a 1 − β2 a 1 − β2 2 dab 1+ γ b 2γ = , = . a 1 − γ2 a 1 − γ2

We receive the third parametrization for NPC:

III parametrization

a =(t4 − 1)(t4 − 81), b =4t(t2 − 3)(t4 +2t2 + 9), 2 4 c = 16t (t − 9);

8 4 dac = t + 46t + 81, 2 4 2 dbc =4t(t − 3)(t + 10t + 9), 4 2 4 2 ds =(t − 2t + 9)(t + 10t + 9).

9 6 The Basis of a Computer Search

The goal of this paper is to find a perfect cuboid. The obtained parametriza- tions can form the basis for a computer search. The first, second and third parametrizations give for diagonal dab conditions of rationality, respectively:

2 2 16t2(t4 − 9) + (t4 − 10t2 + 9)(t4 +2t2 + 9) = ,   h i 2 2 4 4 2 h16t (t − 9)(t − 2t + 9)i + 2 4 2 8 4  +h(t − 10t + 9)(t + 46t + 81)i = ,

2 2 (t4 − 1)(t4 − 81) + 4t(t2 − 3)(t4 +2t2 + 9) = .   h i

If, by using a computer, it is possible to find the nontrivial (=6 0, ±1, ±3) rational t, for which one of the given expressions is square, then PC prob- lem has a solution. Otherwise from Theorem 1 it must be proved that two nontrivial different rational numbers which satisfied the (4)–(6) conditions do not exist.

References

1. Meskhishvili M., Perfect cuboid and congruent number equation solu- tions. http://arxiv.org/pdf/1211.6548v2.pdf

2. van Luijk R., On perfect cuboids. Doctoraalscriptie, Universiteit Utrecht, 2000. http://www.math.leidenuniv.nl/reports/ 2001-12.shtml

3. Meskhishvili M., Three-Century problem. Tbilisi, 2013.

10 4. Leech J., The rational cuboid revisited. Amer. Math. Monthly 84 (1977), No. 7, 518–533.

5. MacLeod A. J., Parametric expressions for a “nearly-perfect” cuboid. http://ru.scribd.com/doc/58729272/Parametric-Equations- for-Nearly-Perfect-Cubiods

6. Bromhead T., On square sums of squares. Math. Gaz. 44 (1960), 219–220.

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Author’s address: Georgian-American High School, 18 Chkondideli Str., Tbilisi 0180, Georgia. E-mail: [email protected]

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