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(Such conditions are known under an additional assumption that $a^{q+1}=1$.) In this paper, we prove the following: (i) If $f$ is a PP of ${\\Bbb F}_{q^2}$, then $\\text{gcd}(r,q+1)>1$ and $(-a)^{(q+1)/\\text{gcd}(r,q+1)}\\ne 1$. (ii) For a fixed $r>2$ and subject to the conditions that $q+1\\equiv 0\\pmod r$ and $a^{q+1}\\ne 1$, there are only fin"},"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":"1510.00437","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2015-10-01T22:10:54Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"976369ab0ea5cafeb74ca979a18c409a9657975d1daf9366b161b37efaf25f56","abstract_canon_sha256":"65c9d2f7aff7ee9bdf3055be845c4a853b60a3772225936d8284428c08bf757c"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:31:13.085371Z","signature_b64":"teDgGN0Kr013qLMAikHg/pePIzBdhTjCxr7fRXTF4oMroK2vMNM/1q/qDEws8W8ZgBwJkeSIfBUBxHEpb+8eAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"22ab71094409fb052dfacec48bee6cde39ba106f30ad80331ae11b90731b3d57","last_reissued_at":"2026-05-18T01:31:13.084625Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:31:13.084625Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Permutation Polynomials of $\\Bbb F_{q^2}$ of the form $a{\\tt X}+{\\tt X}^{r(q-1)+1}$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.NT","authors_text":"Xiang-dong Hou","submitted_at":"2015-10-01T22:10:54Z","abstract_excerpt":"Let $q$ be a prime power, $2\\le r\\le q$, and $f=a{\\tt X}+{\\tt X}^{r(q-1)+1}\\in\\Bbb F_{q^2}[{\\tt X}]$, where $a\\ne 0$. The conditions on $r,q,a$ that are necessary and sufficient for $f$ to be a permutation polynomial (PP) of ${\\Bbb F}_{q^2}$ are not known. (Such conditions are known under an additional assumption that $a^{q+1}=1$.) In this paper, we prove the following: (i) If $f$ is a PP of ${\\Bbb F}_{q^2}$, then $\\text{gcd}(r,q+1)>1$ and $(-a)^{(q+1)/\\text{gcd}(r,q+1)}\\ne 1$. 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