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We show that there exist numbers $G(d,k)$ and $N(d,k)$ such that for $n> N(d,k)$ the answer is yes if and only if $n\\equiv 0\\quad \\pmod {G(d,k)}$. Furthermore, a formula for $G(d,k)$ is given, showing that e.g. $G(d,k)=1$ if $k\\ge \\left\\lfloor\\frac{d+1}{2}\\right\\rfloor$ or if both $d$ and $k$ are even, and also in some other cases (meaning that all numbers beyond $N(d,k)$ occur as the number of $k$-faces of some simple $d$-polytope).\n  This question has previously been studied","authors_text":"Anders Bj\\\"orner, Svante Linusson","cross_cats":[],"headline":"","license":"","primary_cat":"math.CO","submitted_at":"1996-12-11T00:00:00Z","title":"The number of faces of a simple polytope"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/9612218","kind":"arxiv","version":1},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:9dddb02af404ab434ebadc76e84b01a5943909bd25158f6d1c422704ffe3ceb7","target":"record","created_at":"2026-05-18T01:05:37Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"68354e20753c7c88c876d284007bfa6a4d92353e887d22fb8c38f4a0e1768bea","cross_cats_sorted":[],"license":"","primary_cat":"math.CO","submitted_at":"1996-12-11T00:00:00Z","title_canon_sha256":"eb0cf081185b3a02dadf75c615dbcab3a94d8f604d7d155e0f61cc63d1ad11e4"},"schema_version":"1.0","source":{"id":"math/9612218","kind":"arxiv","version":1}},"canonical_sha256":"f980cce0526c86902b9bb7c9e14e43fef55fd8e33d3a622d79b9659eac658436","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"f980cce0526c86902b9bb7c9e14e43fef55fd8e33d3a622d79b9659eac658436","first_computed_at":"2026-05-18T01:05:37.347836Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:05:37.347836Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"CSDo8ekoG2uBTB14XP15JKNAgi4IhXpQvWrKfURrv/TMV9h7EBxnJaJyFtfIJmx0L07+1jAjf3xOJXjydTs3Cg==","signature_status":"signed_v1","signed_at":"2026-05-18T01:05:37.348380Z","signed_message":"canonical_sha256_bytes"},"source_id":"math/9612218","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:9dddb02af404ab434ebadc76e84b01a5943909bd25158f6d1c422704ffe3ceb7","sha256:077c6c3523d3b8bd84c5cf8af35ad1f74073b71f5d714a7c117978e0e6431760"],"state_sha256":"9e674dffe538f6c8535c456fceaa98ae7689d71983277188023987a7fd54abde"}