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Cooper, Eichhorn, and O'Bryant [1] have shown that the (upper) density of B is at most 1/4, and it is conjectured that B has density 0. This note uses results of Gauss on sums of 3 squares to show that the subset of B consisting of all n not congruent to 15 mod 16 has density 0. The final section gives some comput"},"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":"1009.3985","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2010-09-21T02:48:31Z","cross_cats_sorted":[],"title_canon_sha256":"454b736da0b6614aa0c7a4ab68d861f3041393a66eabaf991d08dbd903825281","abstract_canon_sha256":"ae13cb3c6ba1ca1f4a33ba64ea0b519e9c1c14d03e9fcc8ad851f203e9ce8da3"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:40:34.776554Z","signature_b64":"6KxiMj7sRgnO7KGrPXAQ4XUToZuCs53hOMgsghWOyhNRTJp4dtEHaDlBc+veCNBmw2vGqiic075V9PIBcGWDBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"ebee2a65b57abb34db38400812ca0b8fdde17f172a26c0a66e2f324caf592601","last_reissued_at":"2026-05-18T04:40:34.775973Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:40:34.775973Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Disquisitiones Arithmeticae and online sequence A108345","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Paul Monsky","submitted_at":"2010-09-21T02:48:31Z","abstract_excerpt":"Let g be the element that is the sum of x^(n^2) for n >= 0 of A=Z/2[[x]], and let B consist of all n for which the coefficient of x^n in 1/g is 1. (The elements of B are the entries 0, 1, 2, 3, 5, 7, 8, 9, 13, ... in A108345; see The On-Line Encyclopedia of Integer Sequences (OEIS).) Cooper, Eichhorn, and O'Bryant [1] have shown that the (upper) density of B is at most 1/4, and it is conjectured that B has density 0. This note uses results of Gauss on sums of 3 squares to show that the subset of B consisting of all n not congruent to 15 mod 16 has density 0. 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