{"paper":{"title":"Residue Classes Having Tardy Totients","license":"","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Florian Luca, John Friedlander","submitted_at":"2007-09-19T16:10:51Z","abstract_excerpt":"We show, in an effective way, that there exists a sequence of congruence classes $a_k\\pmod {m_k}$ such that the minimal solution $n=n_k$ of the congruence $\\phi(n)\\equiv a_k\\pmod {m_k}$ exists and satisfies $\\log n_k/\\log m_k\\to\\infty $ as $k\\to\\infty$. Here, $\\phi(n)$ is the Euler function. This answers a question raised in \\cite{FS}. We also show that every congruence class containing an even integer contains infinitely many values of the Carmichael function $\\lambda(n)$ and the least such $n$ satisfies $n\\ll m^{13}$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0709.3056","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"}