On a problem on partitions into distinct primes
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.
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BibTeX
@misc{campbell2026problem,
title = {{On a problem on partitions into distinct primes}},
author = {John M. Campbell},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.19534},
eprint = {2610.19534},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.19534},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - On a problem on partitions into distinct primes
AU - John M. Campbell
PY - 2026
DA - 2026/10/14
PB - arXiv
DO - 10.48550/arxiv.2610.19534
UR - https://arxiv.org/abs/2610.19534
AB - 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.
ER -