{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2015:JINQTFJHPC4X3EPRICKHJIHSJY","short_pith_number":"pith:JINQTFJH","canonical_record":{"source":{"id":"1508.02464","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2015-08-11T01:24:16Z","cross_cats_sorted":[],"title_canon_sha256":"5d01891d1c69a559d82b7a6d5ebd92782ad709513a13a5970b466d1f5ba80e99","abstract_canon_sha256":"5a2c6294673f682429e2ae233c3e107a5d2be7494627e0bd10fc8eb41753f8f7"},"schema_version":"1.0"},"canonical_sha256":"4a1b09952778b97d91f1409474a0f24e021a36334373e4c28b49031716cb1c9a","source":{"kind":"arxiv","id":"1508.02464","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1508.02464","created_at":"2026-05-18T00:25:35Z"},{"alias_kind":"arxiv_version","alias_value":"1508.02464v2","created_at":"2026-05-18T00:25:35Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1508.02464","created_at":"2026-05-18T00:25:35Z"},{"alias_kind":"pith_short_12","alias_value":"JINQTFJHPC4X","created_at":"2026-05-18T12:29:27Z"},{"alias_kind":"pith_short_16","alias_value":"JINQTFJHPC4X3EPR","created_at":"2026-05-18T12:29:27Z"},{"alias_kind":"pith_short_8","alias_value":"JINQTFJH","created_at":"2026-05-18T12:29:27Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2015:JINQTFJHPC4X3EPRICKHJIHSJY","target":"record","payload":{"canonical_record":{"source":{"id":"1508.02464","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2015-08-11T01:24:16Z","cross_cats_sorted":[],"title_canon_sha256":"5d01891d1c69a559d82b7a6d5ebd92782ad709513a13a5970b466d1f5ba80e99","abstract_canon_sha256":"5a2c6294673f682429e2ae233c3e107a5d2be7494627e0bd10fc8eb41753f8f7"},"schema_version":"1.0"},"canonical_sha256":"4a1b09952778b97d91f1409474a0f24e021a36334373e4c28b49031716cb1c9a","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:25:35.123860Z","signature_b64":"yZWfBzAOt4bJFDn3jZDDvBfcKWmC+5wuJQUeDdAlxWJx9cY9ukKJElKN0Ph60aVZ9B1pJL+g6Eb831QWJ5JeAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"4a1b09952778b97d91f1409474a0f24e021a36334373e4c28b49031716cb1c9a","last_reissued_at":"2026-05-18T00:25:35.123124Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:25:35.123124Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1508.02464","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T00:25:35Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"pgCiYCGY2SZLHaDSQgMx18MSLVTsi0LN5eg5Kstbk6nJOySBlHDfA41Mr0tUiLhiuSR4NBFTGUVMdD81ILCuBA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-09T13:48:30.701154Z"},"content_sha256":"7b413f5b6e401e9042eed9171ef6d369513915cc926490363b70bd71ba493125","schema_version":"1.0","event_id":"sha256:7b413f5b6e401e9042eed9171ef6d369513915cc926490363b70bd71ba493125"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2015:JINQTFJHPC4X3EPRICKHJIHSJY","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"The density of primes dividing a particular non-linear recurrence sequence","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Alexi Block Gorman, Heesu Hwang, Jeremy Rouse, Noam Kantor, Sarah Parsons, Tyler Genao","submitted_at":"2015-08-11T01:24:16Z","abstract_excerpt":"Define the sequence $\\{b_n\\}$ by $b_0=1,b_1=1, b_2=2,b_3=1$, and $$b_n=\\begin{cases} \\frac{b_{n-1}b_{n-3}-b_{n-2}^2}{b_{n-4}}&\\textrm{if}~ n\\not\\equiv 0\\pmod 3, \\frac{b_{n-1}b_{n-3}-3b_{n-2}^2}{b_{n-4}}&\\textrm{if}~ n\\equiv 0\\pmod 3. We relate this sequence $\\{b_n\\}$ to the coordinates of points on the elliptic curve $E:y^2+y=x^3-3x+4$. We use Galois representations attached to $E$ to prove that the density of primes dividing a term in this sequence is equal to $\\frac{179}{336}$. Furthermore, we describe an infinite family of elliptic curves whose Galois images match that of $E$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1508.02464","kind":"arxiv","version":2},"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":"2026-05-18T00:25:35Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"HAIX0ZivoG4/Rf0SmbbdCc3Cbq31IyHRGQ/QTvpPTNCKDWELK2EUDxrSUxNtaWGipzL2GkWOW7Rz+x4CRFg+Ag==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-09T13:48:30.701563Z"},"content_sha256":"ba61d6bf749ecd236b572536509cf89d9bd56c91bbc395d20f63d20d3b164dda","schema_version":"1.0","event_id":"sha256:ba61d6bf749ecd236b572536509cf89d9bd56c91bbc395d20f63d20d3b164dda"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/JINQTFJHPC4X3EPRICKHJIHSJY/bundle.json","state_url":"https://pith.science/pith/JINQTFJHPC4X3EPRICKHJIHSJY/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/JINQTFJHPC4X3EPRICKHJIHSJY/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":"2026-06-09T13:48:30Z","links":{"resolver":"https://pith.science/pith/JINQTFJHPC4X3EPRICKHJIHSJY","bundle":"https://pith.science/pith/JINQTFJHPC4X3EPRICKHJIHSJY/bundle.json","state":"https://pith.science/pith/JINQTFJHPC4X3EPRICKHJIHSJY/state.json","well_known_bundle":"https://pith.science/.well-known/pith/JINQTFJHPC4X3EPRICKHJIHSJY/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:JINQTFJHPC4X3EPRICKHJIHSJY","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":"5a2c6294673f682429e2ae233c3e107a5d2be7494627e0bd10fc8eb41753f8f7","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2015-08-11T01:24:16Z","title_canon_sha256":"5d01891d1c69a559d82b7a6d5ebd92782ad709513a13a5970b466d1f5ba80e99"},"schema_version":"1.0","source":{"id":"1508.02464","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1508.02464","created_at":"2026-05-18T00:25:35Z"},{"alias_kind":"arxiv_version","alias_value":"1508.02464v2","created_at":"2026-05-18T00:25:35Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1508.02464","created_at":"2026-05-18T00:25:35Z"},{"alias_kind":"pith_short_12","alias_value":"JINQTFJHPC4X","created_at":"2026-05-18T12:29:27Z"},{"alias_kind":"pith_short_16","alias_value":"JINQTFJHPC4X3EPR","created_at":"2026-05-18T12:29:27Z"},{"alias_kind":"pith_short_8","alias_value":"JINQTFJH","created_at":"2026-05-18T12:29:27Z"}],"graph_snapshots":[{"event_id":"sha256:ba61d6bf749ecd236b572536509cf89d9bd56c91bbc395d20f63d20d3b164dda","target":"graph","created_at":"2026-05-18T00:25:35Z","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":"Define the sequence $\\{b_n\\}$ by $b_0=1,b_1=1, b_2=2,b_3=1$, and $$b_n=\\begin{cases} \\frac{b_{n-1}b_{n-3}-b_{n-2}^2}{b_{n-4}}&\\textrm{if}~ n\\not\\equiv 0\\pmod 3, \\frac{b_{n-1}b_{n-3}-3b_{n-2}^2}{b_{n-4}}&\\textrm{if}~ n\\equiv 0\\pmod 3. We relate this sequence $\\{b_n\\}$ to the coordinates of points on the elliptic curve $E:y^2+y=x^3-3x+4$. We use Galois representations attached to $E$ to prove that the density of primes dividing a term in this sequence is equal to $\\frac{179}{336}$. Furthermore, we describe an infinite family of elliptic curves whose Galois images match that of $E$.","authors_text":"Alexi Block Gorman, Heesu Hwang, Jeremy Rouse, Noam Kantor, Sarah Parsons, Tyler Genao","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2015-08-11T01:24:16Z","title":"The density of primes dividing a particular non-linear recurrence sequence"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1508.02464","kind":"arxiv","version":2},"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":"sha256:7b413f5b6e401e9042eed9171ef6d369513915cc926490363b70bd71ba493125","target":"record","created_at":"2026-05-18T00:25:35Z","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":"5a2c6294673f682429e2ae233c3e107a5d2be7494627e0bd10fc8eb41753f8f7","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2015-08-11T01:24:16Z","title_canon_sha256":"5d01891d1c69a559d82b7a6d5ebd92782ad709513a13a5970b466d1f5ba80e99"},"schema_version":"1.0","source":{"id":"1508.02464","kind":"arxiv","version":2}},"canonical_sha256":"4a1b09952778b97d91f1409474a0f24e021a36334373e4c28b49031716cb1c9a","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"4a1b09952778b97d91f1409474a0f24e021a36334373e4c28b49031716cb1c9a","first_computed_at":"2026-05-18T00:25:35.123124Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:25:35.123124Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"yZWfBzAOt4bJFDn3jZDDvBfcKWmC+5wuJQUeDdAlxWJx9cY9ukKJElKN0Ph60aVZ9B1pJL+g6Eb831QWJ5JeAg==","signature_status":"signed_v1","signed_at":"2026-05-18T00:25:35.123860Z","signed_message":"canonical_sha256_bytes"},"source_id":"1508.02464","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:7b413f5b6e401e9042eed9171ef6d369513915cc926490363b70bd71ba493125","sha256:ba61d6bf749ecd236b572536509cf89d9bd56c91bbc395d20f63d20d3b164dda"],"state_sha256":"32092d8e85d23da8f6cd976bc069e99612660340eef42f50c87d05dc47faf28f"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"eBPqh0g+BRIjvc5iAUdLaDLXNlp5ZStsUkruT+kobrMrk4B/7lY7aVm5/cUEMDjft4yNwfvLyqQpAeW6Dmx9AQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-09T13:48:30.704017Z","bundle_sha256":"a1c9a78d7d22cb8efb26b0a9d17431bc2c3725cece1376c1a7f716ba118b99ee"}}