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Define the permutation $\\tau_g:\\,{\\mathcal H}_p\\to{\\mathcal H}_p$ by $$ \\tau_g(b):=\\begin{cases} g^b,&\\text{if }g^b\\in{\\mathcal H}_p,\\\\ -g^b,&\\text{if }g^b\\not\\in{\\mathcal H}_p,\\\\ \\end{cases} $$ for each $b\\in{\\mathcal H}_p$, where ${\\mathcal H}_p=\\{1,2,\\ldots,(p-1)/2\\}$ is viewed as a subset of ${\\mathbb F}_p$. In this paper, we investigate the sign of $\\tau_g$. For example, if $p\\equiv 5\\pmod{8}$, then $$ (-1)^{|\\tau_g|}=(-1)^{\\frac{1}{4}(h(-4p)+"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1810.11642","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2018-10-27T13:40:22Z","cross_cats_sorted":[],"title_canon_sha256":"af3c5b4705054d4237676b9e8503094341a927f5c1abf90a538e910f23a112b5","abstract_canon_sha256":"3d9972eaef763b2fb824b4022ecf0624a62533870670eb67465d61b6d20b888e"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:02:07.925725Z","signature_b64":"XA7hkAfIZuxVKizaGe9ykvsFOJc9hURh4nJBSsI6QN7mxGTkiGnHFxUFaUIxIUb4TWbMnNU84D8GN7vGrvZiDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"38876ac09e03792393c6acf230e7b32801c742e3f70d44aaa0d4828e71993df7","last_reissued_at":"2026-05-18T00:02:07.925087Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:02:07.925087Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Some permutations over ${\\mathbb F}_p$ concerning primitive roots","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Hao Pan, Li-Yuan Wang","submitted_at":"2018-10-27T13:40:22Z","abstract_excerpt":"Let $p$ be an odd prime and let ${\\mathbb F}_p$ denote the finite field with $p$ elements. Suppose that $g$ is a primitive root of ${\\mathbb F}_p$. Define the permutation $\\tau_g:\\,{\\mathcal H}_p\\to{\\mathcal H}_p$ by $$ \\tau_g(b):=\\begin{cases} g^b,&\\text{if }g^b\\in{\\mathcal H}_p,\\\\ -g^b,&\\text{if }g^b\\not\\in{\\mathcal H}_p,\\\\ \\end{cases} $$ for each $b\\in{\\mathcal H}_p$, where ${\\mathcal H}_p=\\{1,2,\\ldots,(p-1)/2\\}$ is viewed as a subset of ${\\mathbb F}_p$. In this paper, we investigate the sign of $\\tau_g$. 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