Matrix Solutions to the Diophantine equation $X^4+Y^4=2Z^4$

Abstract

The present study explores integer and rational matrix solutions for the Diophantine equation $X^4+Y^4=2Z^4$, establishing that infinitely many solutions exist over different matrix rings, including $M_2(\mathbb{Z})$, $M_3(\mathbb{Z})$, $M_4(\mathbb{Z}).$ Furthermore, expanding the scope beyond the integer domain, the analysis demonstrates that an infinite number of matrix solutions can be found in degree-4 and degree-16 field extensions, specifically within $M_4(\mathbb{Q}(\sqrt[4]{l}))$ and $M_4(\mathbb{Q}(\sqrt[4]{l_1},\sqrt[4]{l_2}))$ when the rational numbers involved lack perfect fourth powers and are not perfect squares.