{"paper":{"title":"On the fixed points of the map $x \\mapsto x^x$ modulo a prime, II","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Adam Tyler Felix, P\\\"ar Kurlberg","submitted_at":"2016-07-18T04:25:20Z","abstract_excerpt":"We study number theoretic properties of the map $x \\mapsto x^{x} \\mod{p}$, where $x \\in \\{1,2,\\ldots,p-1\\}$, and improve on some recent upper bounds, due to Kurlberg, Luca, and Shparlinski, on the number of primes $p < N$ for which the map only has the trivial fixed point $x=1$.\n  A key technical result, possibly of independent interest, is the existence of subsets $\\mathscr{N}_{q} \\subset \\{2,3,\\ldots,q-1\\}$ such that almost all $k$-tuples of distinct integers $n_{1}, n_{2},\\ldots,n_{k} \\in \\mathscr{N}_q$ are multiplicatively independent (if $k$ is not too large), and $|\\mathscr{N}_q| = q \\cd"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1607.04948","kind":"arxiv","version":2},"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"}