Observations of the Erdős–Straus conjecture

Using the structural properties of the solutions to the Erdős–Straus Conjecture, we prove that a solution exists for all even values of \( n \), as well as for all values of the form \( n = 4m + 3 \), and provide explicit decompositions for these cases. For values of the form \( n = 4m + 1 \), we establish a general decomposition when \( m \) is odd. The case for even \( m \) remains unresolved, with its limitations clearly characterized.