{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2016:6ZWVFTNYAPSA57LKU2HVSMSOXF","short_pith_number":"pith:6ZWVFTNY","canonical_record":{"source":{"id":"1606.07334","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2016-06-23T14:44:43Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"a3c178b58ae1b256a0bcb0589fea971b68b352a8005692afa2036f91b2d9ce06","abstract_canon_sha256":"b7bddb6a9f685c114e8d9cfb819fbc3dacba9afb31b7ecdd0c0e01322dca609f"},"schema_version":"1.0"},"canonical_sha256":"f66d52cdb803e40efd6aa68f59324eb95b0c9e0a9709742f8ac89462e52d74fb","source":{"kind":"arxiv","id":"1606.07334","version":3},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1606.07334","created_at":"2026-05-18T00:45:02Z"},{"alias_kind":"arxiv_version","alias_value":"1606.07334v3","created_at":"2026-05-18T00:45:02Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1606.07334","created_at":"2026-05-18T00:45:02Z"},{"alias_kind":"pith_short_12","alias_value":"6ZWVFTNYAPSA","created_at":"2026-05-18T12:30:04Z"},{"alias_kind":"pith_short_16","alias_value":"6ZWVFTNYAPSA57LK","created_at":"2026-05-18T12:30:04Z"},{"alias_kind":"pith_short_8","alias_value":"6ZWVFTNY","created_at":"2026-05-18T12:30:04Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2016:6ZWVFTNYAPSA57LKU2HVSMSOXF","target":"record","payload":{"canonical_record":{"source":{"id":"1606.07334","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2016-06-23T14:44:43Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"a3c178b58ae1b256a0bcb0589fea971b68b352a8005692afa2036f91b2d9ce06","abstract_canon_sha256":"b7bddb6a9f685c114e8d9cfb819fbc3dacba9afb31b7ecdd0c0e01322dca609f"},"schema_version":"1.0"},"canonical_sha256":"f66d52cdb803e40efd6aa68f59324eb95b0c9e0a9709742f8ac89462e52d74fb","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:45:02.697449Z","signature_b64":"vj1muqPWVEkF7D7C2yzbqLrjOAcGYwuBd7rfdO9bZABHfaAWbewt0c7Ok+hF+kF5yig0Q8GggKLlBn3pYEzaCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"f66d52cdb803e40efd6aa68f59324eb95b0c9e0a9709742f8ac89462e52d74fb","last_reissued_at":"2026-05-18T00:45:02.697073Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:45:02.697073Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1606.07334","source_version":3,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T00:45:02Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"pOPTNaGpnY20uzISEJb2jchRqRtI4fjJIiqoXoWtXPTNCJfqrwmn/HbG8FZIayYV13GuezGWsPDDfjIbshP8CA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-24T18:04:42.142964Z"},"content_sha256":"931631a31c48dd0b63c64328abb56295c684fef1b2d2951f9fe16bdf7891d92c","schema_version":"1.0","event_id":"sha256:931631a31c48dd0b63c64328abb56295c684fef1b2d2951f9fe16bdf7891d92c"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2016:6ZWVFTNYAPSA57LKU2HVSMSOXF","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Partition regularity of generalised Fermat equations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.NT","authors_text":"Sofia Lindqvist","submitted_at":"2016-06-23T14:44:43Z","abstract_excerpt":"Let $\\alpha,\\beta,\\gamma\\in\\mathbb{N}$. We prove that given an $r$-colouring of $\\mathbb{F}_p$ with $p$ prime, there are more than $c_{r,\\alpha,\\beta,\\gamma} p^2$ solutions to the equation $x^\\alpha+y^\\beta=z^\\gamma$ with all of $x,y,z$ of the same colour. Here $c_{r,\\alpha,\\beta,\\gamma}>0$ is some constant depending on the number of colours and the exponents in the equation. This is already a new result for $\\alpha=\\beta=1$ and $\\gamma=2$, that is to say for the equation $x+y=z^2$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1606.07334","kind":"arxiv","version":3},"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":"2026-05-18T00:45:02Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"qXMWBddrFBpGxmnQxdYUhr92SZtlij8KoiIPU/JiinrbC7LMvnXotMk+unNcEvh/3nU3IFcbb0cADC/b0e7WCA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-24T18:04:42.143304Z"},"content_sha256":"e06c47a2e106fab8d9af0e25465315b7f0481383fcc56f11a29a4206e52da1ce","schema_version":"1.0","event_id":"sha256:e06c47a2e106fab8d9af0e25465315b7f0481383fcc56f11a29a4206e52da1ce"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/6ZWVFTNYAPSA57LKU2HVSMSOXF/bundle.json","state_url":"https://pith.science/pith/6ZWVFTNYAPSA57LKU2HVSMSOXF/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/6ZWVFTNYAPSA57LKU2HVSMSOXF/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":"2026-06-24T18:04:42Z","links":{"resolver":"https://pith.science/pith/6ZWVFTNYAPSA57LKU2HVSMSOXF","bundle":"https://pith.science/pith/6ZWVFTNYAPSA57LKU2HVSMSOXF/bundle.json","state":"https://pith.science/pith/6ZWVFTNYAPSA57LKU2HVSMSOXF/state.json","well_known_bundle":"https://pith.science/.well-known/pith/6ZWVFTNYAPSA57LKU2HVSMSOXF/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:6ZWVFTNYAPSA57LKU2HVSMSOXF","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":"b7bddb6a9f685c114e8d9cfb819fbc3dacba9afb31b7ecdd0c0e01322dca609f","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2016-06-23T14:44:43Z","title_canon_sha256":"a3c178b58ae1b256a0bcb0589fea971b68b352a8005692afa2036f91b2d9ce06"},"schema_version":"1.0","source":{"id":"1606.07334","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1606.07334","created_at":"2026-05-18T00:45:02Z"},{"alias_kind":"arxiv_version","alias_value":"1606.07334v3","created_at":"2026-05-18T00:45:02Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1606.07334","created_at":"2026-05-18T00:45:02Z"},{"alias_kind":"pith_short_12","alias_value":"6ZWVFTNYAPSA","created_at":"2026-05-18T12:30:04Z"},{"alias_kind":"pith_short_16","alias_value":"6ZWVFTNYAPSA57LK","created_at":"2026-05-18T12:30:04Z"},{"alias_kind":"pith_short_8","alias_value":"6ZWVFTNY","created_at":"2026-05-18T12:30:04Z"}],"graph_snapshots":[{"event_id":"sha256:e06c47a2e106fab8d9af0e25465315b7f0481383fcc56f11a29a4206e52da1ce","target":"graph","created_at":"2026-05-18T00:45:02Z","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":"Let $\\alpha,\\beta,\\gamma\\in\\mathbb{N}$. We prove that given an $r$-colouring of $\\mathbb{F}_p$ with $p$ prime, there are more than $c_{r,\\alpha,\\beta,\\gamma} p^2$ solutions to the equation $x^\\alpha+y^\\beta=z^\\gamma$ with all of $x,y,z$ of the same colour. Here $c_{r,\\alpha,\\beta,\\gamma}>0$ is some constant depending on the number of colours and the exponents in the equation. This is already a new result for $\\alpha=\\beta=1$ and $\\gamma=2$, that is to say for the equation $x+y=z^2$.","authors_text":"Sofia Lindqvist","cross_cats":["math.CO"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2016-06-23T14:44:43Z","title":"Partition regularity of generalised Fermat equations"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1606.07334","kind":"arxiv","version":3},"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":"sha256:931631a31c48dd0b63c64328abb56295c684fef1b2d2951f9fe16bdf7891d92c","target":"record","created_at":"2026-05-18T00:45:02Z","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":"b7bddb6a9f685c114e8d9cfb819fbc3dacba9afb31b7ecdd0c0e01322dca609f","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2016-06-23T14:44:43Z","title_canon_sha256":"a3c178b58ae1b256a0bcb0589fea971b68b352a8005692afa2036f91b2d9ce06"},"schema_version":"1.0","source":{"id":"1606.07334","kind":"arxiv","version":3}},"canonical_sha256":"f66d52cdb803e40efd6aa68f59324eb95b0c9e0a9709742f8ac89462e52d74fb","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"f66d52cdb803e40efd6aa68f59324eb95b0c9e0a9709742f8ac89462e52d74fb","first_computed_at":"2026-05-18T00:45:02.697073Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:45:02.697073Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"vj1muqPWVEkF7D7C2yzbqLrjOAcGYwuBd7rfdO9bZABHfaAWbewt0c7Ok+hF+kF5yig0Q8GggKLlBn3pYEzaCA==","signature_status":"signed_v1","signed_at":"2026-05-18T00:45:02.697449Z","signed_message":"canonical_sha256_bytes"},"source_id":"1606.07334","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:931631a31c48dd0b63c64328abb56295c684fef1b2d2951f9fe16bdf7891d92c","sha256:e06c47a2e106fab8d9af0e25465315b7f0481383fcc56f11a29a4206e52da1ce"],"state_sha256":"ddfd15ec2fb4b73f2c36db1b1774b1f0582e38e495e5e9c1c00d45d21a09acf8"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"qNMPPaFf/24hUdusHGbrhldsQC3H1KH/oUJugfIg3r4XDAnnSeMXcKbOlnS4i6T5sTd7+GFAnvuJfcMFP/57Cg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-24T18:04:42.145252Z","bundle_sha256":"72580d515dca0278b2264594f513315c3d55805d9c8077d741dc2b01b53064ca"}}