{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2019:YDP65TLSRBKIBNTMSME6PMHLK2","short_pith_number":"pith:YDP65TLS","canonical_record":{"source":{"id":"1907.01890","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2019-06-25T14:42:21Z","cross_cats_sorted":[],"title_canon_sha256":"24d67a2b246291be31fb227253f5631b4397ce6318034db2144d0dbea0f181e1","abstract_canon_sha256":"d2f65e59afd7850a22b6e03b5dd07b29af2cfae6e4db9791ebe5bcb9cf461990"},"schema_version":"1.0"},"canonical_sha256":"c0dfeecd72885480b66c9309e7b0eb56b098af040811069e36cf8e44e7fc1a97","source":{"kind":"arxiv","id":"1907.01890","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1907.01890","created_at":"2026-05-17T23:41:34Z"},{"alias_kind":"arxiv_version","alias_value":"1907.01890v1","created_at":"2026-05-17T23:41:34Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1907.01890","created_at":"2026-05-17T23:41:34Z"},{"alias_kind":"pith_short_12","alias_value":"YDP65TLSRBKI","created_at":"2026-05-18T12:33:33Z"},{"alias_kind":"pith_short_16","alias_value":"YDP65TLSRBKIBNTM","created_at":"2026-05-18T12:33:33Z"},{"alias_kind":"pith_short_8","alias_value":"YDP65TLS","created_at":"2026-05-18T12:33:33Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2019:YDP65TLSRBKIBNTMSME6PMHLK2","target":"record","payload":{"canonical_record":{"source":{"id":"1907.01890","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2019-06-25T14:42:21Z","cross_cats_sorted":[],"title_canon_sha256":"24d67a2b246291be31fb227253f5631b4397ce6318034db2144d0dbea0f181e1","abstract_canon_sha256":"d2f65e59afd7850a22b6e03b5dd07b29af2cfae6e4db9791ebe5bcb9cf461990"},"schema_version":"1.0"},"canonical_sha256":"c0dfeecd72885480b66c9309e7b0eb56b098af040811069e36cf8e44e7fc1a97","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:41:34.636392Z","signature_b64":"aKvNGL+Urs1slEAXE8bucKxIZW/S+uHZ27Om+V4F+Q0BWC0A34i7LoorjHc/Vi3dkjjTvTkj5Tqw0iygYY64CA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"c0dfeecd72885480b66c9309e7b0eb56b098af040811069e36cf8e44e7fc1a97","last_reissued_at":"2026-05-17T23:41:34.635906Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:41:34.635906Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1907.01890","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-17T23:41:34Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"e7Wh+c/3huucJm5rF4uSXJ2pd95SJfAj6pavTdixhX114Hsr0HBNWGmXloKhv2u6l8g67jB9bUvaqmzU7kQkDQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-28T17:14:56.937096Z"},"content_sha256":"53d41ba90ec20d7fd59335138b70c67e22611e5e5140260ca3b58778ec01a8b5","schema_version":"1.0","event_id":"sha256:53d41ba90ec20d7fd59335138b70c67e22611e5e5140260ca3b58778ec01a8b5"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2019:YDP65TLSRBKIBNTMSME6PMHLK2","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"On permutations derived from integer powers $x^n$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"John S. McCaskill, Peter R.Wills","submitted_at":"2019-06-25T14:42:21Z","abstract_excerpt":"We present a general theorem characterizing the relationship between the prime base $p$ representations of non-negative integers $x$ and their positive integer powers, $x^n$. For any positive integer $l$, the theorem establishes the existence of bijective mappings (permutations) between all $p^l$ members $x$ of each non-zero residue class mod $p$ satisfying $x < p^{l+1}$. These mappings are obtained as the integer part of ${x^p}{p^{-\\alpha}}$ for a particular positive integer $\\alpha$, depending on $n$ and $p$, called the \"coding shift\", for which an explicit formula is given. For relatively p"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1907.01890","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-17T23:41:34Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"dYcTbTI/M2s6hW/gj2l3yYpExUGeARwqCxr8c1D3LzYs1ucJa61cNR3KJKpy5hOAKlYA8Fhht4JS/w5mzMwxAg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-28T17:14:56.937457Z"},"content_sha256":"368b0a57e2d179a959ac10024ca967a8cc2af896354d3efaf9bb9cce9bf7cdd2","schema_version":"1.0","event_id":"sha256:368b0a57e2d179a959ac10024ca967a8cc2af896354d3efaf9bb9cce9bf7cdd2"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/YDP65TLSRBKIBNTMSME6PMHLK2/bundle.json","state_url":"https://pith.science/pith/YDP65TLSRBKIBNTMSME6PMHLK2/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/YDP65TLSRBKIBNTMSME6PMHLK2/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-06-28T17:14:56Z","links":{"resolver":"https://pith.science/pith/YDP65TLSRBKIBNTMSME6PMHLK2","bundle":"https://pith.science/pith/YDP65TLSRBKIBNTMSME6PMHLK2/bundle.json","state":"https://pith.science/pith/YDP65TLSRBKIBNTMSME6PMHLK2/state.json","well_known_bundle":"https://pith.science/.well-known/pith/YDP65TLSRBKIBNTMSME6PMHLK2/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2019:YDP65TLSRBKIBNTMSME6PMHLK2","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"d2f65e59afd7850a22b6e03b5dd07b29af2cfae6e4db9791ebe5bcb9cf461990","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2019-06-25T14:42:21Z","title_canon_sha256":"24d67a2b246291be31fb227253f5631b4397ce6318034db2144d0dbea0f181e1"},"schema_version":"1.0","source":{"id":"1907.01890","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1907.01890","created_at":"2026-05-17T23:41:34Z"},{"alias_kind":"arxiv_version","alias_value":"1907.01890v1","created_at":"2026-05-17T23:41:34Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1907.01890","created_at":"2026-05-17T23:41:34Z"},{"alias_kind":"pith_short_12","alias_value":"YDP65TLSRBKI","created_at":"2026-05-18T12:33:33Z"},{"alias_kind":"pith_short_16","alias_value":"YDP65TLSRBKIBNTM","created_at":"2026-05-18T12:33:33Z"},{"alias_kind":"pith_short_8","alias_value":"YDP65TLS","created_at":"2026-05-18T12:33:33Z"}],"graph_snapshots":[{"event_id":"sha256:368b0a57e2d179a959ac10024ca967a8cc2af896354d3efaf9bb9cce9bf7cdd2","target":"graph","created_at":"2026-05-17T23:41:34Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"We present a general theorem characterizing the relationship between the prime base $p$ representations of non-negative integers $x$ and their positive integer powers, $x^n$. For any positive integer $l$, the theorem establishes the existence of bijective mappings (permutations) between all $p^l$ members $x$ of each non-zero residue class mod $p$ satisfying $x < p^{l+1}$. These mappings are obtained as the integer part of ${x^p}{p^{-\\alpha}}$ for a particular positive integer $\\alpha$, depending on $n$ and $p$, called the \"coding shift\", for which an explicit formula is given. For relatively p","authors_text":"John S. McCaskill, Peter R.Wills","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2019-06-25T14:42:21Z","title":"On permutations derived from integer powers $x^n$"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1907.01890","kind":"arxiv","version":1},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:53d41ba90ec20d7fd59335138b70c67e22611e5e5140260ca3b58778ec01a8b5","target":"record","created_at":"2026-05-17T23:41:34Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"d2f65e59afd7850a22b6e03b5dd07b29af2cfae6e4db9791ebe5bcb9cf461990","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2019-06-25T14:42:21Z","title_canon_sha256":"24d67a2b246291be31fb227253f5631b4397ce6318034db2144d0dbea0f181e1"},"schema_version":"1.0","source":{"id":"1907.01890","kind":"arxiv","version":1}},"canonical_sha256":"c0dfeecd72885480b66c9309e7b0eb56b098af040811069e36cf8e44e7fc1a97","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"c0dfeecd72885480b66c9309e7b0eb56b098af040811069e36cf8e44e7fc1a97","first_computed_at":"2026-05-17T23:41:34.635906Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-17T23:41:34.635906Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"aKvNGL+Urs1slEAXE8bucKxIZW/S+uHZ27Om+V4F+Q0BWC0A34i7LoorjHc/Vi3dkjjTvTkj5Tqw0iygYY64CA==","signature_status":"signed_v1","signed_at":"2026-05-17T23:41:34.636392Z","signed_message":"canonical_sha256_bytes"},"source_id":"1907.01890","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:53d41ba90ec20d7fd59335138b70c67e22611e5e5140260ca3b58778ec01a8b5","sha256:368b0a57e2d179a959ac10024ca967a8cc2af896354d3efaf9bb9cce9bf7cdd2"],"state_sha256":"5e5a805d5d85bfe648dab9eec8eb37fd12eb3514cae0b68eea6977512654aba9"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"O8bLpKVtSiDfY5ASEOiO7aTMfqNNMV8KyH8TcMVBUKPKPVOI0TlTAXCoae2k2heiFCHGaMYAXGUKrf7CXQPUAg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-28T17:14:56.939381Z","bundle_sha256":"2c2d893f41bf62dd00094b1dd6bc478c39e094ec4bcd00ca122ca41578a5b061"}}