{"paper":{"title":"A new theorem on the prime-counting function","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.NT","authors_text":"Zhi-Wei Sun","submitted_at":"2014-09-19T14:57:22Z","abstract_excerpt":"For $x>0$ let $\\pi(x)$ denote the number of primes not exceeding $x$. For integers $a$ and $m>0$, we determine when there is an integer $n>1$ with $\\pi(n)=(n+a)/m$. In particular, we show that for any integers $m>2$ and $a\\le\\lceil e^{m-1}/(m-1)\\rceil$ there is an integer $n>1$ with $\\pi(n)=(n+a)/m$. Consequently, for any integer $m>4$ there is a positive integer $n$ with $\\pi(mn)=m+n$. We also pose several conjectures for further research; for example, we conjecture that for each $m=1,2,3,\\ldots$ there is a positive integer $n$ such that $m+n$ divides $p_m+p_n$, where $p_k$ denotes the $k$-th"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1409.5685","kind":"arxiv","version":6},"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"}