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We prove that an integer is $-2$-happy if and only if it is congruent to 1 modulo 3 and that it is $-3$-happy if and only if it is odd. Defining a $d$-sequence to be an arithmetic sequence with constant difference $d$ and setting $d = \\gcd(2,b - 1)$, we prove that if $b \\leq -3$ odd or $b \\in \\{-4,-6,-8,-10\\}$, the"},"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":"1705.04648","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2017-05-12T16:39:52Z","cross_cats_sorted":[],"title_canon_sha256":"928c6d7c549b563bef4183b65b9219080b711ff0f222f76e9f124962ff7eeabe","abstract_canon_sha256":"8654064844550705b8eeffa098ac75e8bf0945988f685ca4ec848d0df7a123eb"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:44:37.574866Z","signature_b64":"uNwsKLsQAkfLzfGAwOqWoi0H15thLXjgaZuiDr1x+dVoosXdLDuhSFsPRPSXQylkn8Dq/06isQPzRc4N/NlVDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"682af06972017df9db04dc38ebbccf24654645c2ed397b20daa4e9e82db03a97","last_reissued_at":"2026-05-18T00:44:37.574434Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:44:37.574434Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Sequences of Consecutive Happy Numbers in Negative Bases","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Helen G. Grundman, Pamela E. Harris","submitted_at":"2017-05-12T16:39:52Z","abstract_excerpt":"For $b\\leq -2$ and $e \\geq 2$, let $S_{e,b}:\\mathbb{Z}\\to\\mathbb{Z}_{\\geq 0}$ be the function taking an integer to the sum of the $e$-powers of the digits of its base $b$ expansion. An integer $a$ is a $b$-happy number if there exists $k\\in\\mathbb{Z}^+$ such that $S_{2,b}^k(a) = 1$. We prove that an integer is $-2$-happy if and only if it is congruent to 1 modulo 3 and that it is $-3$-happy if and only if it is odd. Defining a $d$-sequence to be an arithmetic sequence with constant difference $d$ and setting $d = \\gcd(2,b - 1)$, we prove that if $b \\leq -3$ odd or $b \\in \\{-4,-6,-8,-10\\}$, the"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1705.04648","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":"1705.04648","created_at":"2026-05-18T00:44:37.574505+00:00"},{"alias_kind":"arxiv_version","alias_value":"1705.04648v1","created_at":"2026-05-18T00:44:37.574505+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1705.04648","created_at":"2026-05-18T00:44:37.574505+00:00"},{"alias_kind":"pith_short_12","alias_value":"NAVPA2LSAF67","created_at":"2026-05-18T12:31:31.346846+00:00"},{"alias_kind":"pith_short_16","alias_value":"NAVPA2LSAF67TWYE","created_at":"2026-05-18T12:31:31.346846+00:00"},{"alias_kind":"pith_short_8","alias_value":"NAVPA2LS","created_at":"2026-05-18T12:31:31.346846+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/NAVPA2LSAF67TWYE3Q4OXPGPER","json":"https://pith.science/pith/NAVPA2LSAF67TWYE3Q4OXPGPER.json","graph_json":"https://pith.science/api/pith-number/NAVPA2LSAF67TWYE3Q4OXPGPER/graph.json","events_json":"https://pith.science/api/pith-number/NAVPA2LSAF67TWYE3Q4OXPGPER/events.json","paper":"https://pith.science/paper/NAVPA2LS"},"agent_actions":{"view_html":"https://pith.science/pith/NAVPA2LSAF67TWYE3Q4OXPGPER","download_json":"https://pith.science/pith/NAVPA2LSAF67TWYE3Q4OXPGPER.json","view_paper":"https://pith.science/paper/NAVPA2LS","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1705.04648&json=true","fetch_graph":"https://pith.science/api/pith-number/NAVPA2LSAF67TWYE3Q4OXPGPER/graph.json","fetch_events":"https://pith.science/api/pith-number/NAVPA2LSAF67TWYE3Q4OXPGPER/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/NAVPA2LSAF67TWYE3Q4OXPGPER/action/timestamp_anchor","attest_storage":"https://pith.science/pith/NAVPA2LSAF67TWYE3Q4OXPGPER/action/storage_attestation","attest_author":"https://pith.science/pith/NAVPA2LSAF67TWYE3Q4OXPGPER/action/author_attestation","sign_citation":"https://pith.science/pith/NAVPA2LSAF67TWYE3Q4OXPGPER/action/citation_signature","submit_replication":"https://pith.science/pith/NAVPA2LSAF67TWYE3Q4OXPGPER/action/replication_record"}},"created_at":"2026-05-18T00:44:37.574505+00:00","updated_at":"2026-05-18T00:44:37.574505+00:00"}