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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 $M=1$."},"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":"1507.08135","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2015-07-29T13:21:29Z","cross_cats_sorted":[],"title_canon_sha256":"0f6c0462dfdd2cb09c8ec5ef0a1f1ac14d1f7c36c8e02ab2c0d34f3b06d02a77","abstract_canon_sha256":"055c0f9f0c11942c4fa81a69167ca475f1613e8b8aeab276bcd2163a2e24cd10"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:36:08.257455Z","signature_b64":"UMk2tueztt7cKtCP+F163kyemYVgcjRvfRam9onKt6aUTl6KYCLFJons8hMyuthQ/5Jc3zMJb2ocoJUgDssaCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"0d95141deb45e84c6cf7b241fbc1d639bf9ee6ea10ad588b386ae21ccd2409c8","last_reissued_at":"2026-05-18T01:36:08.256979Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:36:08.256979Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Smallest bases of expansions with multiple digits","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Derong Kong, Wenxia Li, Yuru Zou","submitted_at":"2015-07-29T13:21:29Z","abstract_excerpt":"Given two positive integers $M$ and $k$, let $\\B_k$ be the set of bases $q>1$ such that there exists a real number $x$ having exactly $k$ different $q$-expansions over the alphabet $\\{0,1,\\cdots,M\\}$. 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 $k\\ge 3$. 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