{"paper":{"title":"Rainbow numbers for $x_1+x_2=kx_3$ in $\\mathbb{Z}_n$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Erin Bevilacqua, J\\\"urgen Kritschgau, Michael Tait, Michael Young, Samuel King, Suzannah Tebon","submitted_at":"2018-09-12T17:36:43Z","abstract_excerpt":"In this work, we investigate the fewest number of colors needed to guarantee a rainbow solution to the equation $x_1 + x_2 = k x_3$ in $\\mathbb{Z}_n$. This value is called the Rainbow number and is denoted by $rb(\\mathbb{Z}_n, k)$ for positive integer values of $n$ and $k$. We find that $rb(\\mathbb{Z}_p, 1) = 4$ for all primes greater than $3$ and that $rb(\\mathbb{Z}_n, 1)$ can be deterimined from the prime factorization of $n$. Furthermore, when $k$ is prime, $rb(\\mathbb{Z}_n, k)$ can be determined from the prime factorization of $n$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1809.04576","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}