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It is shown that if $(a(p_j),b(p_j)) \\in N \\times N$ is a solution of the $j'$th equation one has the inequality $\\frac{p_j}{100}\\leq A(p_j) B(p_j) \\leq \\frac{121}{10^4} p_j$, where $A(p_j)\\equiv a(p_j)+1, B(p_j)\\equiv b(p_j)+1\\,$ and $p_{j}$ is a natural number ending in 1, such that $\\{A(p_1)\\geq 4, B(p_1)\\geq 8\\}$, $\\{A(p_2) \\geq 2, B(p_2)\\geq 2\\}$, and $\\{A(p_3) \\geq 10, B(p_3)\\geq 10\\}$ hold, respec"},"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":"1808.06145","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2018-08-18T23:13:04Z","cross_cats_sorted":[],"title_canon_sha256":"8444d3ce1175efc7ad82bd575b3b9dce969afef3f594eff5124058d68f756b13","abstract_canon_sha256":"7dba543449683dea209052223f5804497601ab61163484b96b0905bdb43f360e"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:07:45.904869Z","signature_b64":"4pmFEDwS4r9fl0whLi7ib4q6tX/e/FeQGnbCnuXTORNkV92mV5Y8W3uowxzPkGx2GQP06EvcAknI0dsaxYBTCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"978492dc98aee5cbd12f5b563a248fd92296094c14528d6ab17ab5e6c1fec7de","last_reissued_at":"2026-05-18T00:07:45.903916Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:07:45.903916Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Some results on natural numbers represented by quadratic polynomials in two variables","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"B. M. Cerna Magui\\~na, H. Blas, V. H. L\\'opez Sol\\'is","submitted_at":"2018-08-18T23:13:04Z","abstract_excerpt":"We consider a set of equations of the form $p_j (x,y) = (10 x+m_j)(10 y + n_j),\\,\\,x\\geq 0, y\\geq0$, $j=1,2,3$, such that $\\{m_1=7, n_1=3\\}$, $\\{m_2=n_2=9\\}$ and $\\{m_3=n_3=1\\}$, respectively. It is shown that if $(a(p_j),b(p_j)) \\in N \\times N$ is a solution of the $j'$th equation one has the inequality $\\frac{p_j}{100}\\leq A(p_j) B(p_j) \\leq \\frac{121}{10^4} p_j$, where $A(p_j)\\equiv a(p_j)+1, B(p_j)\\equiv b(p_j)+1\\,$ and $p_{j}$ is a natural number ending in 1, such that $\\{A(p_1)\\geq 4, B(p_1)\\geq 8\\}$, $\\{A(p_2) \\geq 2, B(p_2)\\geq 2\\}$, and $\\{A(p_3) \\geq 10, B(p_3)\\geq 10\\}$ hold, respec"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1808.06145","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":"1808.06145","created_at":"2026-05-18T00:07:45.904081+00:00"},{"alias_kind":"arxiv_version","alias_value":"1808.06145v1","created_at":"2026-05-18T00:07:45.904081+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1808.06145","created_at":"2026-05-18T00:07:45.904081+00:00"},{"alias_kind":"pith_short_12","alias_value":"S6CJFXEYV3S4","created_at":"2026-05-18T12:32:50.500415+00:00"},{"alias_kind":"pith_short_16","alias_value":"S6CJFXEYV3S4XUJP","created_at":"2026-05-18T12:32:50.500415+00:00"},{"alias_kind":"pith_short_8","alias_value":"S6CJFXEY","created_at":"2026-05-18T12:32:50.500415+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/S6CJFXEYV3S4XUJPLNLDUJEP3E","json":"https://pith.science/pith/S6CJFXEYV3S4XUJPLNLDUJEP3E.json","graph_json":"https://pith.science/api/pith-number/S6CJFXEYV3S4XUJPLNLDUJEP3E/graph.json","events_json":"https://pith.science/api/pith-number/S6CJFXEYV3S4XUJPLNLDUJEP3E/events.json","paper":"https://pith.science/paper/S6CJFXEY"},"agent_actions":{"view_html":"https://pith.science/pith/S6CJFXEYV3S4XUJPLNLDUJEP3E","download_json":"https://pith.science/pith/S6CJFXEYV3S4XUJPLNLDUJEP3E.json","view_paper":"https://pith.science/paper/S6CJFXEY","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1808.06145&json=true","fetch_graph":"https://pith.science/api/pith-number/S6CJFXEYV3S4XUJPLNLDUJEP3E/graph.json","fetch_events":"https://pith.science/api/pith-number/S6CJFXEYV3S4XUJPLNLDUJEP3E/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/S6CJFXEYV3S4XUJPLNLDUJEP3E/action/timestamp_anchor","attest_storage":"https://pith.science/pith/S6CJFXEYV3S4XUJPLNLDUJEP3E/action/storage_attestation","attest_author":"https://pith.science/pith/S6CJFXEYV3S4XUJPLNLDUJEP3E/action/author_attestation","sign_citation":"https://pith.science/pith/S6CJFXEYV3S4XUJPLNLDUJEP3E/action/citation_signature","submit_replication":"https://pith.science/pith/S6CJFXEYV3S4XUJPLNLDUJEP3E/action/replication_record"}},"created_at":"2026-05-18T00:07:45.904081+00:00","updated_at":"2026-05-18T00:07:45.904081+00:00"}