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In this paper, we show that every pair of sufficiently large even integers can be represented as a pair of eight prime cubes and k powers of 
$2$. In particular, we prove that 
$k=335$ is admissible, which improves the previous result.
In this paper, we show that every pair of large positive even integers can be represented in the form of a pair of Goldbach–Linnik equations, that is, linear equations in two primes and 
$k$ powers of two. In particular, 
$k=34$ powers of two suffice, in general, and 
$k=18$ under the generalised Riemann hypothesis. Our result sharpens the number of powers of two in previous results, which gave 
$k=62$, in general, and 
$k=31$ under the generalised Riemann hypothesis.
