{"paper":{"title":"Conjectures on representations involving primes","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Zhi-Wei Sun","submitted_at":"2012-11-07T16:22:04Z","abstract_excerpt":"We pose 100 new conjectures on representations involving primes or related things, which might interest number theorists and stimulate further research. Below are five typical examples: (i) For any positive integer $n$, there exists $k\\in\\{0,\\ldots,n\\}$ such that $n+k$ and $n+k^2$ are both prime. (ii) Each integer $n>1$ can be written as $x+y$ with $x,y\\in\\{1,2,3,\\ldots\\}$ such that $x+ny$ and $x^2+ny^2$ are both prime. (iii) For any rational number $r>0$, there are distinct primes $q_1,\\ldots,q_k$ with $r=\\sum_{j=1}^k1/(q_j-1)$. (iv) Every $n=4,5,\\ldots$ can be written as $p+q$, where $p$ is "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1211.1588","kind":"arxiv","version":29},"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"}