Multiple and Complete New Important Conjectures on Perfect Cuboid and Euler Brick

Abstract

Nobody has discovered any perfect cuboid and there is no formula to deliver all possible Euler bricks. During investigations of famous open problems regarding the perfect cuboid and Euler brick; I have found new important conjectures on Pythagorean triples and biquadratic Diophantine equations [4] which are reduced $\&$ complete 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 $(e^2-f^2,2gh,2kl)$ 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 $e,f,g,h,k,l~(>1)\in \mathbb{N}$. Equivalently, biquadratic Diophantine equation conjectures have been introduced for these perfect cuboid conjectures. For the benefit of readers, along with the original contribution for new important conjectures on perfect cuboid and Euler brick problems; brief review related to Pythagorean Triple, perfect cuboid and Euler brick problems as well as on Diophantine Equation and Biquadratic Diophantine Equation; studied in the past by previous researchers, have been discussed in the paper.