In this paper we find non-negative integer solutions for exponential
Diophantine equations of the type pβ 3x+py=z2, where p is a prime
number. We prove that such equation has a unique solution
(x,y,z)=(log3β(pβ2),0,pβ1) if 2ξ =pβ‘2(mod3) and (x,y,z)=(0,1,2) if p=2. We also display the infinite solution
set of that equation in the case p=3. Finally, a brief discussion of the case
pβ‘1(mod3) is made, where we display an equation that does not have a
non-negative integer solution and leave some open questions. The proofs are
based on the use of the properties of the modular arithmetic