Published online by Cambridge University Press: 19 October 2016
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.