The Diophantine Equations 6 x + 4y = z2 and 24x + 4y = z2

This research aims to study and find all solutions of two Diophantine equations 6 x + 4y = z2 and 24x + 4y = z2 where x, y and z are non-negative integers, by using elementary concepts of number theory and Mihăilescu’s theorem. The research results found that the Diophantine equation 6 x + 4y = z2 has the unique non-negative integer solution (x, y, z) = (2,3,10). The Diophantine equation 24x + 4y = z2 has exactly two non-negative integer solutions (x, y, z), which are (1, 0, 5) and (2, 5, 40).