On a problem on partitions into distinct primes

Source abstract

For a positive integer $n$, we write $L(n)$ in place of the maximum number of distinct primes whose sum is $n$, assuming there is a way of writing $n$ as a sum of distinct primes, with $L(n) = 0$ otherwise. Again if $n$ admits a partition into distinct primes, then we let $K(n)$ denote the number of parts in the unique partition that has a largest product, with $K(n) = 0$ otherwise. In the latest editions of Guy's classic text on unsolved problems in number theory, the problem as to whether or not $L(n) - K(n)$ is unbounded was left open, and it appears that this has remained open. We solve this problem in the affirmative. Our proof relies, to a large extent, on a consequence of Vinogradov's theorem together with a 1948 result due to Sierpiński on the existence of primes whose gaps to the immediately preceding and following primes both exceed any prescribed positive bound.