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We prove that the following are equivalent:\n  (a) There are a real $\\theta > 1$ and infinitely many indices $n$ for which the number of distinct prime factors of $s_n$ is greater than the super-logarithm of $n$ to base $\\theta$.\n  (b) There do not exist non-zero integers $a_0,b_0,\\ldots,a_\\ell,b_\\ell $ such that $s_{2n}=\\prod_{i=0}^\\ell a_i^{(2n)^i}$ and $s_{2n-1}=\\prod_{i=0}^\\ell b_i^{(2"},"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":"1511.08784","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2015-11-27T20:20:31Z","cross_cats_sorted":[],"title_canon_sha256":"bf7818327400f3ed9f621df23207c456afb9673f922481773bfda735502ab54b","abstract_canon_sha256":"2520d6692643821038826a5de41b8927d9a10b0e606f13372630f2ecc4453b3b"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:16:09.866643Z","signature_b64":"ki/A8mticaqCoZnYlHJxg9bP7PY/TuK5PTDiK3bAS6i9ZTJJe9FDBQAhGkj72KDUl0PPAEB0mR3wzUg2S/wrBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"037f4acfd674db51bc43617aa1ce61ec613673c327a82ddaa7fef7b869af2a2d","last_reissued_at":"2026-05-18T00:16:09.865968Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:16:09.865968Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On the number of distinct prime factors of a sum of super-powers","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Paolo Leonetti, Salvatore Tringali","submitted_at":"2015-11-27T20:20:31Z","abstract_excerpt":"Given $k, \\ell \\in {\\bf N}^+$, let $x_{i,j}$ be, for $1 \\le i \\le k$ and $0 \\le j \\le \\ell$, some fixed integers, and define, for every $n \\in {\\bf N}^+$, $s_n := \\sum_{i=1}^k \\prod_{j=0}^\\ell x_{i,j}^{n^j}$. We prove that the following are equivalent:\n  (a) There are a real $\\theta > 1$ and infinitely many indices $n$ for which the number of distinct prime factors of $s_n$ is greater than the super-logarithm of $n$ to base $\\theta$.\n  (b) There do not exist non-zero integers $a_0,b_0,\\ldots,a_\\ell,b_\\ell $ such that $s_{2n}=\\prod_{i=0}^\\ell a_i^{(2n)^i}$ and $s_{2n-1}=\\prod_{i=0}^\\ell b_i^{(2"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1511.08784","kind":"arxiv","version":3},"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":"1511.08784","created_at":"2026-05-18T00:16:09.866068+00:00"},{"alias_kind":"arxiv_version","alias_value":"1511.08784v3","created_at":"2026-05-18T00:16:09.866068+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1511.08784","created_at":"2026-05-18T00:16:09.866068+00:00"},{"alias_kind":"pith_short_12","alias_value":"AN7UVT6WOTNV","created_at":"2026-05-18T12:29:10.953037+00:00"},{"alias_kind":"pith_short_16","alias_value":"AN7UVT6WOTNVDPCD","created_at":"2026-05-18T12:29:10.953037+00:00"},{"alias_kind":"pith_short_8","alias_value":"AN7UVT6W","created_at":"2026-05-18T12:29:10.953037+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/AN7UVT6WOTNVDPCDMF5KDTTB5R","json":"https://pith.science/pith/AN7UVT6WOTNVDPCDMF5KDTTB5R.json","graph_json":"https://pith.science/api/pith-number/AN7UVT6WOTNVDPCDMF5KDTTB5R/graph.json","events_json":"https://pith.science/api/pith-number/AN7UVT6WOTNVDPCDMF5KDTTB5R/events.json","paper":"https://pith.science/paper/AN7UVT6W"},"agent_actions":{"view_html":"https://pith.science/pith/AN7UVT6WOTNVDPCDMF5KDTTB5R","download_json":"https://pith.science/pith/AN7UVT6WOTNVDPCDMF5KDTTB5R.json","view_paper":"https://pith.science/paper/AN7UVT6W","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1511.08784&json=true","fetch_graph":"https://pith.science/api/pith-number/AN7UVT6WOTNVDPCDMF5KDTTB5R/graph.json","fetch_events":"https://pith.science/api/pith-number/AN7UVT6WOTNVDPCDMF5KDTTB5R/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/AN7UVT6WOTNVDPCDMF5KDTTB5R/action/timestamp_anchor","attest_storage":"https://pith.science/pith/AN7UVT6WOTNVDPCDMF5KDTTB5R/action/storage_attestation","attest_author":"https://pith.science/pith/AN7UVT6WOTNVDPCDMF5KDTTB5R/action/author_attestation","sign_citation":"https://pith.science/pith/AN7UVT6WOTNVDPCDMF5KDTTB5R/action/citation_signature","submit_replication":"https://pith.science/pith/AN7UVT6WOTNVDPCDMF5KDTTB5R/action/replication_record"}},"created_at":"2026-05-18T00:16:09.866068+00:00","updated_at":"2026-05-18T00:16:09.866068+00:00"}