Let \(c_1,c_2,c_3\) be nonzero integers such that \(c_1+c_2+c_3=0\). We consider the mixed power equation \(c_1(p_1^2+p_1'^3)+c_2(p_2^2+p_2'^3)+c_3(p_3^2+p_3'^3)=0\) where \(p_1,p_2,p_3\) belong to a certain set \({\mathcal {A}}\) of primes and \(p_1',p_2',p_3'\) belong to another set \({\mathcal {A}}'\) of primes. We prove a Roth-type result that whenever the densities of \({\mathcal {A}}\) and
... [Show full abstract] \({\mathcal {A}}'\) satisfy a certain lower bound, then the above equation has nontrivial solutions. The same method can be generalized to deduce analogous results for other equations involving mixed powers of higher degrees.