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 equations defining an Euler brick: :a^2 + b^2 + c^2 = g^2, where is the space diagonal. , no example of a perfect cuboid had been found and no one has proven that none exist. Exhaustive computer searches show that, if a perfect cuboid exists, • the odd edge must be greater than 2.5 × 1013, • the smallest edge must be greater than , Some facts are known about properties that must be satisfied by a
primitive perfect cuboid, if one exists, based on
modular arithmetic: • 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
product of two primes.
Heronian triangles If a perfect cuboid exists with edges a, b, c, corresponding face diagonals d, e, f, and space diagonal g, then the following
Heronian triangles exist: • A Heronian triangle with side lengths (d^2, e^2, f^2), an area of abcg, and rational angle bisectors. • An acute Heronian triangle with side lengths (af, be, cd) and an area of \frac{abcg}{2}. • Obtuse Heronian triangles with side lengths (bf, ae, gd), (ad, cf, ge), and (ce, bd, gf), each with an area of \frac{abcg}{2}. • Right Heronian triangles with side lengths (ab, cg, ef), (ac, bg, df), and (bc, ag, de), each with an area of \frac{abcg}{2}.
Cuboid conjectures Three
cuboid conjectures are three
mathematical propositions claiming
irreducibility of three univariate
polynomials with
integer coefficients depending on several integer parameters. The conjectures are related to the
perfect cuboid problem. 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 a \neq u the eighth degree polynomial {{NumBlk|:| 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|}}
is irreducible over the ring of integers \mathbb Z.
Cuboid conjecture 2. For any two positive coprime integer numbers p \neq q the tenth-degree polynomial {{NumBlk|:|\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} |}}
is irreducible over the ring of integers \mathbb Z.
Cuboid conjecture 3. For any three positive coprime integer numbers a, b, u such that none of the conditions {{NumBlk|:|\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}|}}
are fulfilled, the twelfth-degree polynomial {{NumBlk|:|\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}|}}
is irreducible over the ring of integers \mathbb Z. == Almost-perfect cuboids ==