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In this note we present an alternative approach grounded in the theory of Diophantine equations towards a proof of the conjecture for all $k \\ge 2$."},"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":"0908.0554","kind":"arxiv","version":25},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2009-08-05T18:33:44Z","cross_cats_sorted":["cs.DS"],"title_canon_sha256":"befe3ec0e86afd28abf5c899792f93fc29099adc52dd623c9b634178d18a799d","abstract_canon_sha256":"085d7410afbb86bf9bb04e6c689d8d9ea3309040e683ecef400a9426bc420728"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:20:05.545226Z","signature_b64":"NwEG6IC79mGBrew0P1pRUEIj8CcXIa6GOztLYWrTz1GBPlxWRCn3AFdkCy7zc43s6nGzc0VN2xiuwPDlOKB1Bg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"f8332cfa883d4ff6baa952066205d1565f21c8b562d5ecb2514bc7ee5d1be0f5","last_reissued_at":"2026-05-18T04:20:05.544807Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:20:05.544807Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On integers as the sum of a prime and a $k$-th power","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.DS"],"primary_cat":"math.NT","authors_text":"Aran Nayebi","submitted_at":"2009-08-05T18:33:44Z","abstract_excerpt":"Let $\\mathcal{R}_k(n)$ be the number of representations of an integer $n$ as the sum of a prime and a $k$-th power. 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