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Suppose that $a_1,a_2,a_3$ are rational p-adic integers, $P=-2a_1^3+9a_1a_2-27a_3$, $Q=(a_1^2-3a_2)^3$ and $PQ(P^2-Q)(P^2-3Q)(P^2-4Q)\\not\\equiv 0\\pmod p$. In this paper we show that the number of solutions of the congruence $x^3+a_1x^2+a_2x+a_3\\equiv 0\\pmod p$ depends only on $\\sum_{k=1}^{[p"},"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":"1310.6721","kind":"arxiv","version":7},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2013-10-24T19:36:53Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"2b353820da0bdb3e5b6bf8ffcad4b345f0c2b1839b99abf27e1a748cc5c6387d","abstract_canon_sha256":"dec4c2c55f06cec058dc7280235efaa5cf578ec1bb2b5ee0e91ff056b2dcc1d5"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:06:42.595403Z","signature_b64":"yOIlFVC/x4k/JcEZ5MJ0R6Z18KgPtVTFkDrd6pBeYwJz37uDBHctbMHIwN2e85hSECL6NtLBM7EiSwy504xADQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"b0c29c4946680ec55aed02c292132dc96d4d15983a73e1d67671174fa6fe8ba3","last_reissued_at":"2026-05-18T03:06:42.594958Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:06:42.594958Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Cubic congruences and sums involving $\\binom{3k}k$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.NT","authors_text":"Zhi-Hong Sun","submitted_at":"2013-10-24T19:36:53Z","abstract_excerpt":"Let $p$ be a prime greater than $3$ and let $a$ be a rational p-adic integer. In this paper we try to determine $\\sum_{k=1}^{[p/3]}\\binom{3k}ka^k\\pmod p$, and real the connection between cubic congruences and the sum $\\sum_{k=1}^{[p/3]}\\binom{3k}ka^k$, where $[x]$ is the greatest integer not exceeding $x$. Suppose that $a_1,a_2,a_3$ are rational p-adic integers, $P=-2a_1^3+9a_1a_2-27a_3$, $Q=(a_1^2-3a_2)^3$ and $PQ(P^2-Q)(P^2-3Q)(P^2-4Q)\\not\\equiv 0\\pmod p$. 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