Editing Euler brick (section)

== Perfect cuboid ==
{{unsolved|mathematics|Does a perfect cuboid exist?}}
A '''perfect cuboid''' (also called a '''perfect Euler brick''' or '''perfect box''') is an Euler brick whose [[space diagonal]] also has integer length.  In other words, the following equation is added to the system of [[Diophantine equation]]s defining an Euler brick:

:<math>a^2 + b^2 + c^2 = g^2,</math>

where {{math|''g''}} is the space diagonal.  {{As of|2020|March|df=}}, no example of a perfect cuboid had been found and no one has proven that none exist.<ref>{{Cite journal |last1=Ivanov |first1=A. A. |last2=Skopin |first2=A. V. |title=On sets with integer n-distances |journal=Journal of Mathematical Sciences |volume=251 |issue=4 |pages=548–556 |date=March 2020 |doi=10.1007/s10958-020-05159-4 |url=https://link.springer.com/article/10.1007/s10958-020-05159-4 |access-date=October 11, 2024}}</ref>
[[File:Euler_brick_perfect.svg|right|thumb|Euler brick with edges {{math|''a'', ''b'', ''c''}} and face diagonals {{math|''d'', ''e'', ''f''}}]]

Exhaustive computer searches show that, if a perfect cuboid exists,
* the odd edge must be greater than 2.5 × 10<sup>13</sup>,<ref name=Matson>{{cite web |first=Robert D. |last=Matson |title=Results of a Computer Search for a Perfect Cuboid |url=http://unsolvedproblems.org/S58.pdf |date= January 18, 2015|work=unsolvedproblems.org |accessdate=February 24, 2020}}</ref>
* the smallest edge must be greater than {{val|5e11}},<ref name=Matson/> and
* the space diagonal must be greater than 9 × 10<sup>15</sup>.<ref name=Belogourov>Alexander Belogourov, Distributed search for a perfect cuboid, https://www.academia.edu/39920706/Distributed_search_for_a_perfect_cuboid</ref>

Some facts are known about properties that must be satisfied by a ''primitive'' perfect cuboid, if one exists, based on [[modular arithmetic]]:<ref>M. Kraitchik, On certain Rational Cuboids, Scripta Mathematica, volume 11 (1945).</ref>

* One edge, two face diagonals and the space diagonal must be odd, one edge and the remaining face diagonal must be divisible by 4, and the remaining edge must be divisible by 16.
* Two edges must have length divisible by 3 and at least one of those edges must have length divisible by 9.
* One edge must have length divisible by 5.
* One edge must have length divisible by 7.
* One edge must have length divisible by 11.
* One edge must have length divisible by 19.
* One edge or space diagonal must be divisible by 13.
* One edge, face diagonal or space diagonal must be divisible by 17.
* One edge, face diagonal or space diagonal must be divisible by 29.
* One edge, face diagonal or space diagonal must be divisible by 37.

In addition:

* The space diagonal is neither a [[prime power]] nor a [[Semiprime|product of two primes]].<ref name=Korec/>{{rp|p. 579}}
* The space diagonal can only contain prime divisors that are congruent to 1 modulo 4.<ref name=Korec>I. Korec, Lower bounds for Perfect Rational Cuboids, Math. Slovaca, 42 (1992), No. 5, p. 565-582.</ref>{{rp|p. 566}}<ref>Ronald van Luijk, On Perfect Cuboids, June 2000</ref>

===Heronian triangles===
If a perfect cuboid exists with edges <math>a, b, c</math>, corresponding face diagonals <math>d, e, f</math>, and space diagonal <math>g</math>, then the following [[Heronian triangle|Heronian triangles]] exist:
* A Heronian triangle with side lengths <math>(d^2, e^2, f^2)</math>, an area of <math>abcg</math>, and rational angle bisectors.<ref name="Luca">Florian Luca (2000). "Perfect Cuboids and Perfect Square Triangles". ''Mathematics Magazine'', 73(5), 400–401.</ref>
* An acute Heronian triangle with side lengths <math>(af, be, cd)</math> and an area of <math>\frac{abcg}{2}</math>.
* Obtuse Heronian triangles with side lengths <math>(bf, ae, gd)</math>, <math>(ad, cf, ge)</math>, and <math>(ce, bd, gf)</math>, each with an area of <math>\frac{abcg}{2}</math>.
* Right Heronian triangles with side lengths <math>(ab, cg, ef)</math>, <math>(ac, bg, df)</math>, and <math>(bc, ag, de)</math>, each with an area of <math>\frac{abcg}{2}</math>.

===Cuboid conjectures===
Three '''cuboid conjectures''' are three [[mathematics|mathematical]] propositions claiming [[irreducible polynomial|irreducibility]] of three univariate [[polynomial]]s with [[integer]] [[coefficient]]s depending on several integer parameters. The conjectures are related to the [[#Perfect cuboid|perfect cuboid]] problem.<ref name=shr_01>{{cite journal |author=Sharipov R.A. |title=Perfect cuboids and irreducible polynomials|journal=Ufa Math Journal|year=2012 |volume=4 |issue=1 |pages=153&ndash;160|arxiv=1108.5348|bibcode=2011arXiv1108.5348S}}</ref><ref name=shr_02>{{cite journal |author=Sharipov R.A. |title=Asymptotic approach to the perfect cuboid problem|journal=Ufa Math Journal|year=2015 |volume=7 |issue=3 |pages=100&ndash;113|doi=10.13108/2015-7-3-95 |doi-access=free }}</ref> Though they are not equivalent to the perfect cuboid problem, if all of these three conjectures  are valid, then no perfect cuboids exist. They are neither proved nor disproved.

'''Cuboid conjecture 1.''' ''For any two positive [[coprime]] integer numbers <math>a \neq u</math> the eighth degree polynomial''
{{NumBlk|:|<math>
P_{au}(t)=t^8+6\,(u^2-a^2)\,t^6+(a^4-4\,a^2\,u^2+u^4)\,t^4-6\,a^2\,u^2\,(u^2-a^2)\,t^2+u^4\,a^4</math>|{{EquationRef|1}}}}
''is irreducible over the [[ring (mathematics)|ring]] of integers <math>\mathbb Z</math>''.

'''Cuboid conjecture 2.''' ''For any two positive coprime integer numbers <math>p \neq q</math> the tenth-degree polynomial''

{{NumBlk|:|<math>\begin{align}
Q_{pq}(t)= {} & t^{10}+(2q^2+p^2)(3q^2-2p^2)t^8 \\[4pt]
& {} +(q^8+10p^2q^6+4p^4q^4-14p^6q^2+p^8)t^6\\[4pt]
& {} -p^2 q^2(q^8-14p^2q^6+4p^4q^4+10p^6\,q^2+p^8)t^4 \\[4pt]
& {} -p^6\,q^6\,(q^2+2\,p^2)\,(-2\,q^2+3\,p^2)\,t^2\\[4pt]
& {} -q^{10}\,p^{10}
\end{align}
</math>|{{EquationRef|2}}}}

''is irreducible over the ring of integers <math>\mathbb Z</math>''.

'''Cuboid conjecture 3.''' ''For any three positive coprime integer numbers <math>a</math>, <math>b</math>, <math>u</math> such that none of the conditions''

{{NumBlk|:|<math>\begin{array}{lcr}
\text{1)}\qquad a=b;\qquad\qquad & \text{3)}\qquad b\,u=a^2;\qquad\qquad &\text{5)}\qquad a=u;\\
\text{2)}\qquad a=b=u;\qquad\qquad &\text{4)}\qquad a\,u=b^2;\qquad\qquad &\text{6)}\qquad b=u
\end{array}</math>|{{EquationRef|3}}}}

''are fulfilled, the twelfth-degree polynomial''

{{NumBlk|:|<math>\begin{align}
P_{abu}(t) = {} & t^{12}+(6u^2-2a^2-2b^2)t^{10} \\
& {} + (u^4+b^4+a^4+4a^2u^2+4b^2u^2-12b^2 a^2)t^8 \\
& {} + (6a^4u^2+6u^2b^4-8a^2b^2u^2-2u^4a^2-2u^4b^2-2a^4b^2-2b^4a^2)t^6 \\
& {} + (4u^2b^4a^2+4a^4u^2b^2-12u^4a^2b^2+u^4a^4+u^4b^4+a^4b^4)t^4 \\
& {} + (6a^4u^2b^4-2u^4a^4b^2-2u^4a^2b^4)t^2+u^4a^4b^4.
\end{align}</math>|{{EquationRef|4}}}}

''is irreducible over the ring of integers <math>\mathbb Z</math>''.