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We study the representations of an integer $n$ in the form $$ p_1 + p_2 + ... + p_k = n, $$ where $p_i$ is a prime from the Beatty sequence $$ \\mathcal B_i = \\left\\{n \\in \\mathbb N : n = [ \\alpha_i m + \\beta_i ] \\text{for some} m \\in \\mathbb Z \\right\\}. $$"},"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":"0706.0943","kind":"arxiv","version":1},"metadata":{"license":"","primary_cat":"math.NT","submitted_at":"2007-06-07T04:14:11Z","cross_cats_sorted":[],"title_canon_sha256":"4570195e9ac11a22b3723f500ef901b6cf1c93e86e7472198adca69134d53413","abstract_canon_sha256":"ee0243370b676654fba515907eb02fc34526a8369955bce2f9c4bf6b38bd1a66"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:42:02.850403Z","signature_b64":"poR/LLWm1PvKeZkmaGfGa/D/4f3s4wiesVeEdaCbg2kdGe91zPQzW3DSfgtpgPSj14JaJDfeIetV0d1NuHdKBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"2d58862a44ccfddb3589c84b557ffae9efbd4e8ca128dc06f450dc6cf103843d","last_reissued_at":"2026-05-18T04:42:02.849979Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:42:02.849979Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On sums of primes from Beatty sequences","license":"","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Angel V Kumchev","submitted_at":"2007-06-07T04:14:11Z","abstract_excerpt":"Let $k \\ge 2$ and $\\alpha_1, \\beta_1, ..., \\alpha_k, \\beta_k$ be reals such that the $\\alpha_i$'s are irrational and greater than 1. Suppose further that some ratio $\\alpha_i/\\alpha_j$ is irrational. We study the representations of an integer $n$ in the form $$ p_1 + p_2 + ... + p_k = n, $$ where $p_i$ is a prime from the Beatty sequence $$ \\mathcal B_i = \\left\\{n \\in \\mathbb N : n = [ \\alpha_i m + \\beta_i ] \\text{for some} m \\in \\mathbb Z \\right\\}. $$"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0706.0943","kind":"arxiv","version":1},"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"0706.0943","created_at":"2026-05-18T04:42:02.850039+00:00"},{"alias_kind":"arxiv_version","alias_value":"0706.0943v1","created_at":"2026-05-18T04:42:02.850039+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0706.0943","created_at":"2026-05-18T04:42:02.850039+00:00"},{"alias_kind":"pith_short_12","alias_value":"FVMIMKSEZT65","created_at":"2026-05-18T12:25:55.427421+00:00"},{"alias_kind":"pith_short_16","alias_value":"FVMIMKSEZT65WNMJ","created_at":"2026-05-18T12:25:55.427421+00:00"},{"alias_kind":"pith_short_8","alias_value":"FVMIMKSE","created_at":"2026-05-18T12:25:55.427421+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/FVMIMKSEZT65WNMJZBFVK7725H","json":"https://pith.science/pith/FVMIMKSEZT65WNMJZBFVK7725H.json","graph_json":"https://pith.science/api/pith-number/FVMIMKSEZT65WNMJZBFVK7725H/graph.json","events_json":"https://pith.science/api/pith-number/FVMIMKSEZT65WNMJZBFVK7725H/events.json","paper":"https://pith.science/paper/FVMIMKSE"},"agent_actions":{"view_html":"https://pith.science/pith/FVMIMKSEZT65WNMJZBFVK7725H","download_json":"https://pith.science/pith/FVMIMKSEZT65WNMJZBFVK7725H.json","view_paper":"https://pith.science/paper/FVMIMKSE","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=0706.0943&json=true","fetch_graph":"https://pith.science/api/pith-number/FVMIMKSEZT65WNMJZBFVK7725H/graph.json","fetch_events":"https://pith.science/api/pith-number/FVMIMKSEZT65WNMJZBFVK7725H/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/FVMIMKSEZT65WNMJZBFVK7725H/action/timestamp_anchor","attest_storage":"https://pith.science/pith/FVMIMKSEZT65WNMJZBFVK7725H/action/storage_attestation","attest_author":"https://pith.science/pith/FVMIMKSEZT65WNMJZBFVK7725H/action/author_attestation","sign_citation":"https://pith.science/pith/FVMIMKSEZT65WNMJZBFVK7725H/action/citation_signature","submit_replication":"https://pith.science/pith/FVMIMKSEZT65WNMJZBFVK7725H/action/replication_record"}},"created_at":"2026-05-18T04:42:02.850039+00:00","updated_at":"2026-05-18T04:42:02.850039+00:00"}