{"paper":{"title":"Prime chains and Pratt trees","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.PR"],"primary_cat":"math.NT","authors_text":"Florian Luca, Kevin Ford, Sergei V. Konyagin","submitted_at":"2009-04-02T21:46:53Z","abstract_excerpt":"We study the distribution of prime chains, which are sequences p_1,...,p_k of primes for which p_{j+1}\\equiv 1\\pmod{p_j} for each j. We give estimates for the number of chains with p_k\\le x (k variable), and the number of chains with p_1=p and p_k \\le px. The majority of the paper concerns the distribution of H(p), the length of the longest chain with p_k=p, which is also the height of the Pratt tree for p. We show H(p)\\ge c\\log\\log p and H(p)\\le (\\log p)^{1-c'} for almost all p, with c,c' explicit positive constants. We can take, for any \\epsilon>0, c=e-\\epsilon assuming the Elliott-Halbersta"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0904.0473","kind":"arxiv","version":4},"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"}