Cuboid conjectures

Three cuboid conjectures are three mathematical propositions claiming irreducibility of three univariate polynomials with integer coefficients depending on several integer parameters. They are neither proved nor disproved.

The first cuboid conjecture

Cuboid conjecture 1. For any two positive coprime integer numbers $\displaystyle a\neq u$ the eighth degree polynomial

$P_{au}(t)=t^8+6\,(u^2-a^2)\,t^6+(a^4-4\,a^2\,u^2+u^4)\,t^4-6\,a^2\,u^2\,(u^2-a^2)\,t^2+u^4\,a^4$

(1)

is irreducible over the ring of integers $\displaystyle\mathbb Z$.

The second cuboid conjecture

Cuboid conjecture 2. For any two positive coprime integer numbers $\displaystyle p\neq q$ the tenth-degree polynomial

${\displaystyle {\begin{aligned}Q_{pq}(t)={}&t^{10}+(2q^{2}+p^{2})(3q^{2}-2p^{2})t^{8}\\[4pt]&{}+(q^{8}+10p^{2}q^{6}+4p^{4}q^{4}-14p^{6}q^{2}+p^{8})t^{6}\\[4pt]&{}-p^{2}q^{2}(q^{8}-14p^{2}q^{6}+4p^{4}q^{4}+10p^{6}\,q^{2}+p^{8})t^{4}\\[4pt]&{}-p^{6}\,q^{6}\,(q^{2}+2\,p^{2})\,(-2\,q^{2}+3\,p^{2})\,t^{2}\\[4pt]&{}-q^{10}\,p^{10}\end{aligned}}}$

(2)

is irreducible over the ring of integers $\displaystyle\mathbb Z$.

The third cuboid conjecture

Cuboid conjecture 3. For any three positive coprime integer numbers $\displaystyle a$, $\displaystyle b$, $\displaystyle u$ such that none of the conditions

$\begin{array}{lcr} \text{1)}\qquad a=b;\qquad\qquad & \text{3)}\qquad b\,u=a^2;\qquad\qquad &\text{5)}\qquad a=u;\\ \text{2)}\qquad a=b=u;\qquad\qquad &\text{4)}\qquad a\,u=b^2;\qquad\qquad &\text{6)}\qquad b=u \end{array}$

(3)

is fulfilled the twelfth degree polynomial

${\displaystyle {\begin{aligned}P_{abu}(t)={}&t^{12}+(6u^{2}-2a^{2}-2b^{2})t^{10}\\&{}+(u^{4}+b^{4}+a^{4}+4a^{2}u^{2}+4b^{2}u^{2}-12b^{2}a^{2})t^{8}\\&{}+(6a^{4}u^{2}+6u^{2}b^{4}-8a^{2}b^{2}u^{2}-2u^{4}a^{2}-2u^{4}b^{2}-2a^{4}b^{2}-2b^{4}a^{2})t^{6}\\&{}+(4u^{2}b^{4}a^{2}+4a^{4}u^{2}b^{2}-12u^{4}a^{2}b^{2}+u^{4}a^{4}+u^{4}b^{4}+a^{4}b^{4})t^{4}\\&{}+(6a^{4}u^{2}b^{4}-2u^{4}a^{4}b^{2}-2u^{4}a^{2}b^{4})t^{2}+u^{4}a^{4}b^{4}.\end{aligned}}}$

(4)

is irreducible over the ring of integers $\displaystyle\mathbb Z$.

Background

The conjectures 1, 2, and 3 are related to the perfect cuboid problem.^[1][2] Though they are not equivalent to the perfect cuboid problem, if all of these three conjectures are valid, then no perfect cuboids exist.

References

1. Sharipov R.A. (2012). "Perfect cuboids and irreducible polynomials". Ufa Math Journal. 4 (1): 153–160. arXiv:1108.5348.
2. Sharipov R.A. (2015). "Asymptotic approach to the perfect cuboid problem". Ufa Math Journal. 7 (3): 100–113.