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If $F$ is a bivariate polynomial for which there exist infinite $(F,m)$-Diophantine sets, then there is a complete qualitative characterization of all such polynomials $F$. Otherwise, various finiteness results are known. We prove that given a finite set of distinct integers $ S$ of size $n$, there are infinitely many bivariate polynomials $F$ such that $ S$ is an $(F,2)$-Diophantine set. In addition, we show"},"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":"1708.08525","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2017-08-28T21:03:50Z","cross_cats_sorted":[],"title_canon_sha256":"361b8945b27b3841cf125d55c0473ba4db2f6870e36184ac45202ac7c7ee7a33","abstract_canon_sha256":"b31c22f68704c74293fb093c7aa0dd7ea0de8f48e5d47a5aa94d65b9636122cd"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:10:19.299081Z","signature_b64":"9Cfl2+AcUmsHxMqZ4faOTWQLiIMdcN3Xf6X1xklLrJDCw73HGzgjiVkCmcZmFFCB3itlfeV/3qhoMkZkmqQYBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"d426eb93d796ecb10d1828036eb1d80965ba0566b788ff15cc9cba40ce1074cd","last_reissued_at":"2026-05-18T00:10:19.298575Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:10:19.298575Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On large $F$-Diophantine sets","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Mohammad Sadek, Nermine El-Sissi","submitted_at":"2017-08-28T21:03:50Z","abstract_excerpt":"Let $F\\in\\mathbb{Z}[x,y]$ and $m\\ge2$ be an integer. A set $A\\subset \\mathbb{Z}$ is called an $(F,m)$-Diophantine set if $F(a,b)$ is a perfect $m$-power for any $a,b\\in A$ where $a\\ne b$. If $F$ is a bivariate polynomial for which there exist infinite $(F,m)$-Diophantine sets, then there is a complete qualitative characterization of all such polynomials $F$. Otherwise, various finiteness results are known. We prove that given a finite set of distinct integers $ S$ of size $n$, there are infinitely many bivariate polynomials $F$ such that $ S$ is an $(F,2)$-Diophantine set. 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