Waring--Goldbach Problem: One square, four fifth powers and twenty-two seventh powers
We prove that for every sufficiently large odd integer $n$, the equation $$ n = p_1^2 + p_2^5 + p_3^5 + p_4^5 + p_5^5 + p_6^7 + \cdots + p_{27}^7 $$ has a solution with all $p_j$ prime.
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