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Prime Plus a Non-Square-Free Integer: An Elementary Approach
Abstract
Lee and O'Clarey recently conjectured that every integer n>24 can be written as the sum of a prime and a positive integer that is not square-free. They proved this for odd n, and for all n>24 assuming the generalised Riemann hypothesis for Dirichlet L-functions. We prove their conjecture unconditionally for every integer n>24 that is not divisible by 997#, where 997#=∏p≤ 997p. In particular, any counterexample must be divisible by every prime up to $997$, and hence exceeds 10415. The proof uses an elementary congruence argument together with finite computation.
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BibTeX
@misc{campbell2026prime,
title = {{Prime Plus a Non-Square-Free Integer: An Elementary Approach}},
author = {Peter J. Campbell},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.14714},
eprint = {2609.14714},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.14714},
}
RIS
TY - UNPB
TI - Prime Plus a Non-Square-Free Integer: An Elementary Approach
AU - Peter J. Campbell
PY - 2026
DA - 2026/09/13
PB - arXiv
DO - 10.48550/arxiv.2609.14714
UR - https://arxiv.org/abs/2609.14714
AB - Lee and O'Clarey recently conjectured that every integer n>24 can be written as the sum of a prime and a positive integer that is not square-free. They proved this for odd n, and for all n>24 assuming the generalised Riemann hypothesis for Dirichlet L-functions. We prove their conjecture unconditionally for every integer n>24 that is not divisible by 997#, where 997#=∏_p≤ 997p. In particular, any counterexample must be divisible by every prime up to $997$, and hence exceeds 10^415. The proof uses an elementary congruence argument together with finite computation.
ER -