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Carlitz found formulas for~$N_q$ when $k_1=\\dots=k_n=m_1=\\dots=m_n=1$, $k=2$, $n=3$ or $4$, $p>2$; and when ${m_1=\\dots=m_n=2}$, $k=k_1=\\dots=k_n=1$, $n=3$ or $4$, $p>2$. In earlier papers, we studied the above equation with $k_1=\\dots=k_n=1$ and obtained some generalizations of Carlitz's results. Recently, Pan, Zhao and Cao considered the case of arbitrary positive integers $k_1,\\dots,k_n$ and proved the formula $N"},"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":"1609.04807","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2016-09-15T19:59:24Z","cross_cats_sorted":[],"title_canon_sha256":"23c5683b4beb26e4b5dc5e5aad49909140a27597f682763d284aa80be6185172","abstract_canon_sha256":"00f72a624cf9572830f8419cca6c45c20bbba4d0449e749529e6b7f62d5e2ec3"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:23:57.073130Z","signature_b64":"4+Ju0i9+P5xyyeKbwrQyeWEVnRwSa1TZwErABdq57oDCFKA01dxyQ3A5z4LOFXsAXvA6Xs7pwngImxnizB1mDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"112cc6a2b2417eb0a1d2d7b580739d16af5dbf7cde7eb145e6f9aa4f1d465b18","last_reissued_at":"2026-05-18T00:23:57.072699Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:23:57.072699Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On two problems of Carlitz and their generalizations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Ioulia N. Baoulina","submitted_at":"2016-09-15T19:59:24Z","abstract_excerpt":"Let $N_q$ be the number of solutions to the equation $$ (a_1^{}x_1^{m_1}+\\dots+a_n^{}x_n^{m_n})^k=bx_1^{k_1}\\cdots x_n^{k_n} $$ over the finite field $\\mathbb F_q=\\mathbb F_{p^s}$. Carlitz found formulas for~$N_q$ when $k_1=\\dots=k_n=m_1=\\dots=m_n=1$, $k=2$, $n=3$ or $4$, $p>2$; and when ${m_1=\\dots=m_n=2}$, $k=k_1=\\dots=k_n=1$, $n=3$ or $4$, $p>2$. In earlier papers, we studied the above equation with $k_1=\\dots=k_n=1$ and obtained some generalizations of Carlitz's results. 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