We don't know whether it's possible to construct a complete mathematical framework that fully explains why certain simple equations have no solutions in whole numbers at all.
open
Global / Unspecified, Global
WS01349
Some equations are proven to have no whole-number solutions through specific clever arguments, but a fully general and unified explanatory framework covering every such case remains an unresolved number-theoretic goal. This continues to drive research into Diophantine equations.