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

Source abstract

Abstract We show that every sufficiently large odd integer n can be represented as $$\begin{aligned} n&= p_1^2 + p_2^2 + p_3^3 + p_4^3 + p_5^r + p_6^s + p_7^t, \end{aligned}$$ n = p 1 2 + p 2 2 + p 3 3 + p 4 3 + p 5 r + p 6 s + p 7 t , where each $$p_i$$ 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 $$t \ge 5$$ t ≥ 5 , or $$3 \le r \le s \le 6$$ 3 ≤ r ≤ s ≤ 6 with $$t \ge 4$$ t ≥ 4 and $$(r,s)\ne (3,3),(3,4)$$ ( r , s ) ≠ ( 3 , 3 ) , ( 3 , 4 ) .