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More generally, we show that all solutions to equations of the shape $$ Y^2 = t^2 + M \\cdot q^m + N \\cdot q^n, $$ where $q$ is an odd prime, $n > m > 0$ and $t^2, |M|, N < q$, either arise from \"obvious\" polynomial families or satisfy $m \\leq 3$. Our arguments rely upon Pad\\'e approximants to the binomial function, considered $q$-adically."},"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":"1610.09830","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2016-10-31T08:56:28Z","cross_cats_sorted":[],"title_canon_sha256":"9e628308156fe9c38af2959d4efee5318b8de10c6944f3057d95468a347e1d3e","abstract_canon_sha256":"94eb2406e6902004a8d8ff4bc0a678b7ff2fb76d2ee312872d52b7c4accac082"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:00:47.107556Z","signature_b64":"K+ztKb3cu6gBVdp6qAjaQp4WfYArKGCWFqcaO0/ZR42iBpAjwksOR/Nmn6jlOfAH4ifqDlxuna39wQIvB6u5Bw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"5f1bbfc0ec605579e760f48ad916acaf55f5e0bb32cbb12c34f1bbcaf1d28f3a","last_reissued_at":"2026-05-18T01:00:47.107144Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:00:47.107144Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Squares with three nonzero digits","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Adrian-Maria Scheerer, Michael A. 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