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(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 "},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1211.1588","kind":"arxiv","version":29},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2012-11-07T16:22:04Z","cross_cats_sorted":[],"title_canon_sha256":"424f70f855bf0f0e60c1a7d0d39219ceef6bbcff6f99408d86160f79b8981989","abstract_canon_sha256":"35ae2b12ca27a82b69ec95d96e530a0e63c43d1d02425a616865f2846ce67ade"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:29:11.200824Z","signature_b64":"3JXMwAWiEoRvKfXsPrh7hm6zPZogx5ycd/IzJbpxVxlNriEUmtLn3zV7rLbGSl9JZ7gpy5t6m1KQtHzkj1/LCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"fb288c754f463cd9f65ed297f9e78e8608dc9eb390c03233e307258a556cfdf1","last_reissued_at":"2026-05-18T00:29:11.200245Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:29:11.200245Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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 $r=\\sum_{j=1}^k1/(q_j-1)$. 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