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Second, we show that the above Diophantine system has an integer parametric solution for $f(X)=X(X+a)$ with nonzero integers $a$, if there are integers $m,n,k$ such that \\[\\begin{cases} \\begin{split} (n^2-m^2) (4mnk(k+a+1) + a(m^2+2mn-n^2)) &\\equiv0\\pmod{(m^2+n^2)^2},\\\\ (m^2+2mn-n^2) ((m^2-2mn-n^2)k(k+a+1) - 2amn) &\\equiv0 \\pmod{(m^2+n^2)^2}, \\end{split} \\end{cases}\\] where $k\\equiv0\\pmod{4}$ when"},"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":"1706.03433","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2017-06-12T01:38:36Z","cross_cats_sorted":[],"title_canon_sha256":"290792415adc80d236565720853959e4613c6509b0694ca0c37c6f4345e98a5f","abstract_canon_sha256":"639f6f38b6561a053bbdaf58493672314e745e58e627be7ae154f96b3b9aae74"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:42:35.695033Z","signature_b64":"m1VpjOrFEMKNpnaNKxkGi6H+PWQHuVR0Sh2seH/ZalPM6FcgX3AY/gJjMH967A7NW2LRv4SCRZWEPD15IkwhAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"0e7282a30a66bd22483a677666d0534d6bb4c755850c867bfba9b46152db511c","last_reissued_at":"2026-05-18T00:42:35.694496Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:42:35.694496Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Arithmetic properties of polynomials","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Yong Zhang, Zhongyan Shen","submitted_at":"2017-06-12T01:38:36Z","abstract_excerpt":"In this paper, first, we prove that the Diophantine system \\[f(z)=f(x)+f(y)=f(u)-f(v)=f(p)f(q)\\] has infinitely many integer solutions for $f(X)=X(X+a)$ with nonzero integers $a\\equiv 0,1,4\\pmod{5}$. Second, we show that the above Diophantine system has an integer parametric solution for $f(X)=X(X+a)$ with nonzero integers $a$, if there are integers $m,n,k$ such that \\[\\begin{cases} \\begin{split} (n^2-m^2) (4mnk(k+a+1) + a(m^2+2mn-n^2)) &\\equiv0\\pmod{(m^2+n^2)^2},\\\\ (m^2+2mn-n^2) ((m^2-2mn-n^2)k(k+a+1) - 2amn) &\\equiv0 \\pmod{(m^2+n^2)^2}, \\end{split} \\end{cases}\\] where $k\\equiv0\\pmod{4}$ when"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1706.03433","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"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":"1706.03433","created_at":"2026-05-18T00:42:35.694577+00:00"},{"alias_kind":"arxiv_version","alias_value":"1706.03433v1","created_at":"2026-05-18T00:42:35.694577+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1706.03433","created_at":"2026-05-18T00:42:35.694577+00:00"},{"alias_kind":"pith_short_12","alias_value":"BZZIFIYKM26S","created_at":"2026-05-18T12:31:08.081275+00:00"},{"alias_kind":"pith_short_16","alias_value":"BZZIFIYKM26SESB2","created_at":"2026-05-18T12:31:08.081275+00:00"},{"alias_kind":"pith_short_8","alias_value":"BZZIFIYK","created_at":"2026-05-18T12:31:08.081275+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/BZZIFIYKM26SESB2M53GNUCTJV","json":"https://pith.science/pith/BZZIFIYKM26SESB2M53GNUCTJV.json","graph_json":"https://pith.science/api/pith-number/BZZIFIYKM26SESB2M53GNUCTJV/graph.json","events_json":"https://pith.science/api/pith-number/BZZIFIYKM26SESB2M53GNUCTJV/events.json","paper":"https://pith.science/paper/BZZIFIYK"},"agent_actions":{"view_html":"https://pith.science/pith/BZZIFIYKM26SESB2M53GNUCTJV","download_json":"https://pith.science/pith/BZZIFIYKM26SESB2M53GNUCTJV.json","view_paper":"https://pith.science/paper/BZZIFIYK","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1706.03433&json=true","fetch_graph":"https://pith.science/api/pith-number/BZZIFIYKM26SESB2M53GNUCTJV/graph.json","fetch_events":"https://pith.science/api/pith-number/BZZIFIYKM26SESB2M53GNUCTJV/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/BZZIFIYKM26SESB2M53GNUCTJV/action/timestamp_anchor","attest_storage":"https://pith.science/pith/BZZIFIYKM26SESB2M53GNUCTJV/action/storage_attestation","attest_author":"https://pith.science/pith/BZZIFIYKM26SESB2M53GNUCTJV/action/author_attestation","sign_citation":"https://pith.science/pith/BZZIFIYKM26SESB2M53GNUCTJV/action/citation_signature","submit_replication":"https://pith.science/pith/BZZIFIYKM26SESB2M53GNUCTJV/action/replication_record"}},"created_at":"2026-05-18T00:42:35.694577+00:00","updated_at":"2026-05-18T00:42:35.694577+00:00"}