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This is a further extension of a congruence of Glaisher."},"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":"math/0502187","kind":"arxiv","version":4},"metadata":{"license":"","primary_cat":"math.NT","submitted_at":"2005-02-09T20:21:42Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"c0da663a0d350544ada797da8dc139d4ff81b0708fd0e85f59a7c40e42577b7f","abstract_canon_sha256":"3d525ae86222385657441663c3cda4f4706185c8bd4a2aeb5f890f868ce7a3ab"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-04T15:03:27.624618Z","signature_b64":"S+Zl1iVw/FQUgZoleAEABsPzfyLV2qTBHPBpcjPQCACFK21vTEh9SFNByCtcmjA/Tb6sQ1yXjPDOldAQ/tTEAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"5362e33b86c3c4369b38aa282084b79319f2314ee4c425ac519e50e43c1fad72","last_reissued_at":"2026-07-04T15:03:27.624201Z","signature_status":"signed_v1","first_computed_at":"2026-07-04T15:03:27.624201Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Congruences for sums of binomial coefficients","license":"","headline":"","cross_cats":["math.CO"],"primary_cat":"math.NT","authors_text":"Roberto Tauraso, Zhi-Wei Sun","submitted_at":"2005-02-09T20:21:42Z","abstract_excerpt":"Let q>1 and m>0 be relatively prime integers. We find an explicit period $\\nu_m(q)$ such that for any integers n>0 and r we have $[n+\\nu_m(q),r]_m(a)=[n,r]_m(a) (mod q)$ whenever a is an integer with $\\gcd(1-(-a)^m,q)=1$, or a=-1 (mod q), or a=1 (mod q) and 2|m, where $[n,r]_m(a)=\\sum_{k=r(mod m)}\\binom{n}{k}a^k$. 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