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We prove a new bound for multiple Gauss sums and, as an application, improve previous results in the Birch--Goldbach problem. Let $F_1, \\ldots, F_R \\in \\mathbb{Z}[x_1, \\ldots, x_s]$ be forms with differing degrees, with $D$ being the highest degree, and let $\\boldsymbol{F} = (F_1, \\ldots, F_R)$ be nonsingular. We prove that the system $\\boldsymbol{F}(\\boldsymbol{x})=\\mathbf{0}$ is solvable in primes provided that $s \\geq D^2 4^{D+2} R^5$."},"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":"2604.03347","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2026-04-03T10:38:32Z","cross_cats_sorted":[],"title_canon_sha256":"4e628f608e030f9b97f28822412875183c276145b12904b389819edc7b750c47","abstract_canon_sha256":"0aa73b3a500a57e6f4ff7ea7b4872172029ab445ce414544126752e5521dac7b"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-20T00:03:10.455225Z","signature_b64":"5qZnCWLp1lEubh4A1nyEWPUUAtXKM0CTPlXAVeO6vwqDz8IiBBDTW2isIGV6U80SAKhgRGut2sIjjBDF0LX0Cg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"bec3e4c1d4f53dbec2af546142f0c06551e6b6f6868e0ebb035e4c4442464e61","last_reissued_at":"2026-05-20T00:03:10.454398Z","signature_status":"signed_v1","first_computed_at":"2026-05-20T00:03:10.454398Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Multiple Gauss sums","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"A bound on multiple Gauss sums proves that nonsingular form systems have prime solutions when the number of variables meets or exceeds D squared times 4 to the D plus 2 times R to the fifth.","cross_cats":[],"primary_cat":"math.NT","authors_text":"Jianya Liu, Sizhe Xie","submitted_at":"2026-04-03T10:38:32Z","abstract_excerpt":"A multiple Gauss sum is a complete multiple exponential sum twisted by Dirichlet characters. We prove a new bound for multiple Gauss sums and, as an application, improve previous results in the Birch--Goldbach problem. Let $F_1, \\ldots, F_R \\in \\mathbb{Z}[x_1, \\ldots, x_s]$ be forms with differing degrees, with $D$ being the highest degree, and let $\\boldsymbol{F} = (F_1, \\ldots, F_R)$ be nonsingular. We prove that the system $\\boldsymbol{F}(\\boldsymbol{x})=\\mathbf{0}$ is solvable in primes provided that $s \\geq D^2 4^{D+2} R^5$."},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"We prove that the system F(x)=0 is solvable in primes provided that s ≥ D² 4^{D+2} R^5, where F consists of R nonsingular forms of differing degrees with maximum degree D.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The system of forms is nonsingular; the proof relies on this algebraic condition to control the singular series or major arcs in the analytic argument.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"New bound on multiple Gauss sums improves the Birch-Goldbach result: nonsingular systems of R forms of max degree D in s variables have prime solutions when s ≥ D² 4^{D+2} R^5.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"A bound on multiple Gauss sums proves that nonsingular form systems have prime solutions when the number of variables meets or exceeds D squared times 4 to the D plus 2 times R to the fifth.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"c656bfd670fc82f83cb035952694746defcbd137bc16b48c24715db7f6dfd4c5"},"source":{"id":"2604.03347","kind":"arxiv","version":2},"verdict":{"id":"865f62c4-6b00-456e-8aad-a7e98e3d6473","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-13T18:46:56.776255Z","strongest_claim":"We prove that the system F(x)=0 is solvable in primes provided that s ≥ D² 4^{D+2} R^5, where F consists of R nonsingular forms of differing degrees with maximum degree D.","one_line_summary":"New bound on multiple Gauss sums improves the Birch-Goldbach result: nonsingular systems of R forms of max degree D in s variables have prime solutions when s ≥ D² 4^{D+2} R^5.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The system of forms is nonsingular; 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