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In this paper we investigate the smallest base $q_2$ of $\\B_2$, and show that if $M=2m$ the smallest base $$q_2 =\\frac{m+1+\\sqrt{m^2+2m+5}}{2},$$ and if $M=2m-1$ the smallest base $q_2$ is the appropriate root of $$ x^4=(m-1)\\,x^3+2 m\\, x^2+m \\,x+1. $$ Moreover, for $M=2$ we show that $q_2$ is also the smallest base of $\\B_k$ for all $k\\ge 3$. This turns out to be different from that for $M=1$.","authors_text":"Derong Kong, Wenxia Li, Yuru Zou","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2015-07-29T13:21:29Z","title":"Smallest bases of expansions with multiple digits"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1507.08135","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:acc3fa90e9b18d45469e893a3b7638eb8c0c1ccadc6ddd3386381684fd536421","target":"record","created_at":"2026-05-18T01:36:08Z","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":"055c0f9f0c11942c4fa81a69167ca475f1613e8b8aeab276bcd2163a2e24cd10","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2015-07-29T13:21:29Z","title_canon_sha256":"0f6c0462dfdd2cb09c8ec5ef0a1f1ac14d1f7c36c8e02ab2c0d34f3b06d02a77"},"schema_version":"1.0","source":{"id":"1507.08135","kind":"arxiv","version":1}},"canonical_sha256":"0d95141deb45e84c6cf7b241fbc1d639bf9ee6ea10ad588b386ae21ccd2409c8","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"0d95141deb45e84c6cf7b241fbc1d639bf9ee6ea10ad588b386ae21ccd2409c8","first_computed_at":"2026-05-18T01:36:08.256979Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:36:08.256979Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"UMk2tueztt7cKtCP+F163kyemYVgcjRvfRam9onKt6aUTl6KYCLFJons8hMyuthQ/5Jc3zMJb2ocoJUgDssaCA==","signature_status":"signed_v1","signed_at":"2026-05-18T01:36:08.257455Z","signed_message":"canonical_sha256_bytes"},"source_id":"1507.08135","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:acc3fa90e9b18d45469e893a3b7638eb8c0c1ccadc6ddd3386381684fd536421","sha256:8a7d53282fdbcea3e8b55936382627dad247e8345a0bedc2ce3b8318a083d5d9"],"state_sha256":"9699fb0c58c906d81b0f7fb5ed94f62d66e705fa63a99381337e4f7a2c16432f"}