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If one relaxes the problem and only considers the first two conditions, an infinite series of solutions is known: $m=2^k-2$, $n=(m+1)^2-1=2^k \\cdot m$ for all integers $k\\geq 2$. One additional solution is also known: $m=75=3\\cdot 5^2$ and $n=1215=3^5 \\cdot 5$ with $m+1=76=2^2\\cdot 19$ and $n+1=1216=2^6 \\cdot 19$. No other solutions with $n<2^{32}\\approx 4.3\\"},"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":"2506.01099","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.NT","submitted_at":"2025-06-01T17:48:06Z","cross_cats_sorted":[],"title_canon_sha256":"242b95ed703c4fdce830ecb954c9fd4707e44982f604f44fe2447ba3bb9946f8","abstract_canon_sha256":"05dd4071bf2b5b7fa652a659fce96e3d406c92419696d0ed454005a3e5672ac0"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T11:13:46.384818Z","signature_b64":"Zne0H5e5xKbxe4yDxmI+jhXfXFR4Q3B4B7SGkJtKlmPN25DYX1Eb84SxzixptXuOuXPjuPsJ8GB9mZ1bZUKpCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"7e20f04446c9a1840c23e42fbb1b5c78a9031b88137d6bb4cfbc91bf1a6bfe50","last_reissued_at":"2026-07-05T11:13:46.384369Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T11:13:46.384369Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On one of Erd\\H{o}s' Problems -- An Efficient Search for Benelux Pairs","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Christian Hercher","submitted_at":"2025-06-01T17:48:06Z","abstract_excerpt":"Erd\\H{o}s asked for positive integers $m<n$, such that $m$ and $n$ have the same set of prime factors, $m+1$ and $n+1$ have the same set of prime factors, and $m+2$ and $n+2$ have the same set of prime factors. No such integers are known. If one relaxes the problem and only considers the first two conditions, an infinite series of solutions is known: $m=2^k-2$, $n=(m+1)^2-1=2^k \\cdot m$ for all integers $k\\geq 2$. One additional solution is also known: $m=75=3\\cdot 5^2$ and $n=1215=3^5 \\cdot 5$ with $m+1=76=2^2\\cdot 19$ and $n+1=1216=2^6 \\cdot 19$. No other solutions with $n<2^{32}\\approx 4.3\\"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2506.01099","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2506.01099/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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":"2506.01099","created_at":"2026-07-05T11:13:46.384429+00:00"},{"alias_kind":"arxiv_version","alias_value":"2506.01099v1","created_at":"2026-07-05T11:13:46.384429+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2506.01099","created_at":"2026-07-05T11:13:46.384429+00:00"},{"alias_kind":"pith_short_12","alias_value":"PYQPARCGZGQY","created_at":"2026-07-05T11:13:46.384429+00:00"},{"alias_kind":"pith_short_16","alias_value":"PYQPARCGZGQYIDBD","created_at":"2026-07-05T11:13:46.384429+00:00"},{"alias_kind":"pith_short_8","alias_value":"PYQPARCG","created_at":"2026-07-05T11:13:46.384429+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/PYQPARCGZGQYIDBD4QX3WG24PC","json":"https://pith.science/pith/PYQPARCGZGQYIDBD4QX3WG24PC.json","graph_json":"https://pith.science/api/pith-number/PYQPARCGZGQYIDBD4QX3WG24PC/graph.json","events_json":"https://pith.science/api/pith-number/PYQPARCGZGQYIDBD4QX3WG24PC/events.json","paper":"https://pith.science/paper/PYQPARCG"},"agent_actions":{"view_html":"https://pith.science/pith/PYQPARCGZGQYIDBD4QX3WG24PC","download_json":"https://pith.science/pith/PYQPARCGZGQYIDBD4QX3WG24PC.json","view_paper":"https://pith.science/paper/PYQPARCG","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2506.01099&json=true","fetch_graph":"https://pith.science/api/pith-number/PYQPARCGZGQYIDBD4QX3WG24PC/graph.json","fetch_events":"https://pith.science/api/pith-number/PYQPARCGZGQYIDBD4QX3WG24PC/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/PYQPARCGZGQYIDBD4QX3WG24PC/action/timestamp_anchor","attest_storage":"https://pith.science/pith/PYQPARCGZGQYIDBD4QX3WG24PC/action/storage_attestation","attest_author":"https://pith.science/pith/PYQPARCGZGQYIDBD4QX3WG24PC/action/author_attestation","sign_citation":"https://pith.science/pith/PYQPARCGZGQYIDBD4QX3WG24PC/action/citation_signature","submit_replication":"https://pith.science/pith/PYQPARCGZGQYIDBD4QX3WG24PC/action/replication_record"}},"created_at":"2026-07-05T11:13:46.384429+00:00","updated_at":"2026-07-05T11:13:46.384429+00:00"}