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Tight universality of m-gonal forms with minimal criterion sets
Abstract
For integers m≥3 and n≥1, an m-gonal form is called tight 𝒯(n)-universal if it represents exactly the positive integers 𝒯(n)={ n, n+1, n+2, … }. In this paper, we study the minimal criterion set CS(m,n) for tight 𝒯(n)-universality. Our main result determines CS(m,n) for all n ≥8, 3 ≤ m ≤ ⌊ (3n+1)/2 ⌋. More precisely, CS(m,n)= { n, n+1, …, 2n−1 }, & m=5,newline { n, n+1, …, 2n }, & m≠5. We also establish the corresponding tight 𝒯(n)-universality results and show that the upper bound on m is optimal.
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BibTeX
@misc{kim2026tight,
title = {{Tight universality of $m$-gonal forms with minimal criterion sets}},
author = {Byeong Moon Kim and Ji Young Kim},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.12779},
eprint = {2609.12779},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.12779},
}
RIS
TY - UNPB
TI - Tight universality of m-gonal forms with minimal criterion sets
AU - Byeong Moon Kim
AU - Ji Young Kim
PY - 2026
DA - 2026/10/15
PB - arXiv
DO - 10.48550/arxiv.2609.12779
UR - https://arxiv.org/abs/2609.12779
AB - For integers m≥3 and n≥1, an m-gonal form is called tight 𝒯(n)-universal if it represents exactly the positive integers 𝒯(n)={ n, n+1, n+2, … }. In this paper, we study the minimal criterion set CS(m,n) for tight 𝒯(n)-universality. Our main result determines CS(m,n) for all n ≥8, 3 ≤ m ≤ ⌊ (3n+1)/2 ⌋. More precisely, CS(m,n)= { n, n+1, …, 2n−1 }, & m=5,newline { n, n+1, …, 2n }, & m≠5. We also establish the corresponding tight 𝒯(n)-universality results and show that the upper bound on m is optimal.
ER -