  Nobody has discovered any perfect cuboid and there is no formula to deliver
all possible Euler bricks. During investigations of famous open problems
related to the perfect cuboid and Euler brick; I have found some new important
conjectures on Pythagorean triples and biquadratic Diophantine equation [4]
which are reduced & quivalence form for perfect cuboid and Euler brick
problems. The details of the conjectures have been provided in Sections 2-3. If
any perfect cuboid exists it will be only among the solutions of six
conjectures and all the Euler bricks are only among the solutions of next three
conjectures [4]. For example, if any odd $n\in \mathbb{N}$ satisfy
$n=e^2-f^2=g^2-h^2=k^2-l^2$ and $e^2f^2=g^2h^2+k^2l^2$; then we can discover a
perfect cuboid of type 1 as $\{e^2-f^2,2gh,2kl,g^2+h^2,k^2+l^2,2ef,e^2+f^2\}$
having $(e^2-f^2,2gh,2kl)$ having $(e^2-f^2,2gh,2kl)$ as its edges;
$(g^2+h^2,k^2+l^2,2ef)$ as its face diagonals and $e^2+f^2$ as its body
diagonal where $e,f,g,h,k,l~(>1)\in \mathbb{N}$.
