Representing integers as sums of mixed powers of primes

We establish two new Waring–Goldbach type representations: every sufficiently large odd integer n can be expressed as n = p 1 2 + p 2 2 + p 3 3 + p 4 3 + p 5 5 + p 6 6 + p 7 c , where each p i is prime and c ∈ { 6 , 7 } .