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Built on the work of Morton, in the paper we prove the uniform congruence: $$&\\sum_{x=0}^{p-1}\\Big(\\frac{x^3+mx+n}p\\Big) \\equiv {-(-3m)^{\\frac{p-1}4} \\sum_{k=0}^{p-1}\\binom{-\\frac 1{12}}k\\binom{-\\frac 5{12}}k (\\frac{4m^3+27n^2}{4m^3})^k\\pmod p&\\t{if $4\\mid p-1$,} \\frac{2m}{9n}(\\frac{-3m}p)(-3m)^{\\frac{p+1}4} \\sum_{k=0}^{p-1}\\binom{-\\frac 1{12}}k\\binom{-\\frac 5{12}}k (\\frac{4m^3+27n^2}{4m^3})^k\\pmod p&\\text{if $4\\mid p-3$,}$$ where $(\\frac ap)$ is the Legendre symbol. We also establish many congruences for $x\\pmod p$, where $x$ is given "},"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":"1202.1237","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2012-02-06T18:41:27Z","cross_cats_sorted":[],"title_canon_sha256":"0e65d3f6c77cb35e57b1dfc7ff217ed7081e0d83ba55cc8b5fc0a03283d5ec20","abstract_canon_sha256":"f70742ff946017cf68fa54492c7b7ba483cd7eaf2e31e998d532392611b97cd9"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:02:26.707881Z","signature_b64":"o5mgEB/dvJOEbozo8nvepYZ159sWBB9dL/8/6gIGwyW19A2jRVn+FRv7biQOOyFApBE8d2s2AleUlIrdMvmGAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"67d6e55423123570696e3726a3f27f4e50b9bcd565b2642e875e8e42422f344e","last_reissued_at":"2026-05-18T04:02:26.707423Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:02:26.707423Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Jacobsthal sums, Legendre polynomials and binary quadratic forms","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Zhi-Hong Sun","submitted_at":"2012-02-06T18:41:27Z","abstract_excerpt":"Let $p>3$ be a prime and $m,n\\in\\Bbb Z$ with $p\\nmid mn$. Built on the work of Morton, in the paper we prove the uniform congruence: $$&\\sum_{x=0}^{p-1}\\Big(\\frac{x^3+mx+n}p\\Big) \\equiv {-(-3m)^{\\frac{p-1}4} \\sum_{k=0}^{p-1}\\binom{-\\frac 1{12}}k\\binom{-\\frac 5{12}}k (\\frac{4m^3+27n^2}{4m^3})^k\\pmod p&\\t{if $4\\mid p-1$,} \\frac{2m}{9n}(\\frac{-3m}p)(-3m)^{\\frac{p+1}4} \\sum_{k=0}^{p-1}\\binom{-\\frac 1{12}}k\\binom{-\\frac 5{12}}k (\\frac{4m^3+27n^2}{4m^3})^k\\pmod p&\\text{if $4\\mid p-3$,}$$ where $(\\frac ap)$ is the Legendre symbol. 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