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More precisely, for an irreducible fraction $\\frac{r}s>0$, they constructed coprime polynomials $\\mathcal{R}_{\\frac{r}s}(q),~ \\mathcal{S}_{\\frac{r}s}(q) \\in {\\mathbb Z}[q]$ with $\\mathcal{R}_{\\frac{r}s}(1)=r,~\\mathcal{S}_{\\frac{r}s}(1)=s$. Their theory has a rich background and many applications. By definition, if $r \\equiv r' \\pmod{s}$, then $\\mathcal{S}_{\\frac{r}s}(q)=\\mathcal{S}_{\\frac{r'}s}(q)$. We show that $rr'{\\equiv} -1 \\pmod{s}$ implies $\\mathcal{S}_{\\frac{r}s}(q)=\\mathcal{S}_{\\frac{r'}s}(q)$, and i"},"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":"2403.08446","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2024-03-13T12:00:31Z","cross_cats_sorted":["math.QA"],"title_canon_sha256":"313576c161a4c5b97a3f20de82305981ff2b3fa2855d39982a5bfb1db8555ef2","abstract_canon_sha256":"e265e10b0c7c79f821c0c621c7ecbc053ec9693bdaf8f01cdaca3afceac0d0ee"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T09:42:23.771771Z","signature_b64":"30HCI75c2Je+5f75oXGL25bzvQTctZGTXaMm/nEz/dUxpZp5KP5FChyjmx6hoFAqDlfXI6aDNIH68PVbKSJsCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"1b6e03a52f6e7910c57f53e7ea655e8a26330955cedae63770102f6d9b490305","last_reissued_at":"2026-07-05T09:42:23.771211Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T09:42:23.771211Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Arithmetic on $q$-deformed rational numbers","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.QA"],"primary_cat":"math.CO","authors_text":"Kengo Miyamoto, Kohji Yanagawa, Michihisa Wakui, Takeyoshi Kogiso, Xin Ren","submitted_at":"2024-03-13T12:00:31Z","abstract_excerpt":"Recently, Morier-Genoud and Ovsienko introduced a $q$-deformation of rational numbers. More precisely, for an irreducible fraction $\\frac{r}s>0$, they constructed coprime polynomials $\\mathcal{R}_{\\frac{r}s}(q),~ \\mathcal{S}_{\\frac{r}s}(q) \\in {\\mathbb Z}[q]$ with $\\mathcal{R}_{\\frac{r}s}(1)=r,~\\mathcal{S}_{\\frac{r}s}(1)=s$. Their theory has a rich background and many applications. By definition, if $r \\equiv r' \\pmod{s}$, then $\\mathcal{S}_{\\frac{r}s}(q)=\\mathcal{S}_{\\frac{r'}s}(q)$. We show that $rr'{\\equiv} -1 \\pmod{s}$ implies $\\mathcal{S}_{\\frac{r}s}(q)=\\mathcal{S}_{\\frac{r'}s}(q)$, and i"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2403.08446","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2403.08446/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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":"2403.08446","created_at":"2026-07-05T09:42:23.771276+00:00"},{"alias_kind":"arxiv_version","alias_value":"2403.08446v2","created_at":"2026-07-05T09:42:23.771276+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2403.08446","created_at":"2026-07-05T09:42:23.771276+00:00"},{"alias_kind":"pith_short_12","alias_value":"DNXAHJJPNZ4R","created_at":"2026-07-05T09:42:23.771276+00:00"},{"alias_kind":"pith_short_16","alias_value":"DNXAHJJPNZ4RBRL7","created_at":"2026-07-05T09:42:23.771276+00:00"},{"alias_kind":"pith_short_8","alias_value":"DNXAHJJP","created_at":"2026-07-05T09:42:23.771276+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":1,"internal_anchor_count":0,"sample":[{"citing_arxiv_id":"2603.04295","citing_title":"Plane geometry of $q$-rationals and Springborn Operations","ref_index":13,"is_internal_anchor":false}]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/DNXAHJJPNZ4RBRL7KPT6UZK6RI","json":"https://pith.science/pith/DNXAHJJPNZ4RBRL7KPT6UZK6RI.json","graph_json":"https://pith.science/api/pith-number/DNXAHJJPNZ4RBRL7KPT6UZK6RI/graph.json","events_json":"https://pith.science/api/pith-number/DNXAHJJPNZ4RBRL7KPT6UZK6RI/events.json","paper":"https://pith.science/paper/DNXAHJJP"},"agent_actions":{"view_html":"https://pith.science/pith/DNXAHJJPNZ4RBRL7KPT6UZK6RI","download_json":"https://pith.science/pith/DNXAHJJPNZ4RBRL7KPT6UZK6RI.json","view_paper":"https://pith.science/paper/DNXAHJJP","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2403.08446&json=true","fetch_graph":"https://pith.science/api/pith-number/DNXAHJJPNZ4RBRL7KPT6UZK6RI/graph.json","fetch_events":"https://pith.science/api/pith-number/DNXAHJJPNZ4RBRL7KPT6UZK6RI/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/DNXAHJJPNZ4RBRL7KPT6UZK6RI/action/timestamp_anchor","attest_storage":"https://pith.science/pith/DNXAHJJPNZ4RBRL7KPT6UZK6RI/action/storage_attestation","attest_author":"https://pith.science/pith/DNXAHJJPNZ4RBRL7KPT6UZK6RI/action/author_attestation","sign_citation":"https://pith.science/pith/DNXAHJJPNZ4RBRL7KPT6UZK6RI/action/citation_signature","submit_replication":"https://pith.science/pith/DNXAHJJPNZ4RBRL7KPT6UZK6RI/action/replication_record"}},"created_at":"2026-07-05T09:42:23.771276+00:00","updated_at":"2026-07-05T09:42:23.771276+00:00"}