On Sums of Two Squares, Two Cubes, and Higher Powers of Primes

We show that every sufficiently large odd integer n can be represented as aligned n&= p₁² + p₂² + p₃³ + p₄³ + p₅ʳ + p₆ˢ + p₇ᵗ, aligned n = p 1 2 + p 2 2 + p 3 3 + p 4 3 + p 5 r + p 6 s + p 7 t , where each pᵢ p i is prime, and r , s , t are fixed integers satisfying either $$(r,s,t)=(3,4,t)$$ ( r , s , t ) = ( 3 , 4 , t ) with t ≥ 5 t ≥ 5 , or 3 ≤ r ≤ s ≤ 6 3 ≤ r ≤ s ≤ 6 with t ≥ 4 t ≥ 4 and (r,s)≠ (3,3),(3,4) ( r , s ) ≠ ( 3 , 3 ) , ( 3 , 4 ) .