{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2016:YNDSOAAIQZMAAMHOG4IIDWKMOU","short_pith_number":"pith:YNDSOAAI","canonical_record":{"source":{"id":"1610.07776","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2016-10-25T08:13:49Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"aeadcd82a977051820e37d71cbb0a6a5f4f77fab2f2a8874bc3a011871ea9295","abstract_canon_sha256":"7cdfa2575d6b390b39c6764b6a174c0e3d7a1998be271f27923a45583b062420"},"schema_version":"1.0"},"canonical_sha256":"c34727000886580030ee371081d94c752c45b2614db5a6efa067486a07eabf28","source":{"kind":"arxiv","id":"1610.07776","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1610.07776","created_at":"2026-05-18T01:01:17Z"},{"alias_kind":"arxiv_version","alias_value":"1610.07776v1","created_at":"2026-05-18T01:01:17Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1610.07776","created_at":"2026-05-18T01:01:17Z"},{"alias_kind":"pith_short_12","alias_value":"YNDSOAAIQZMA","created_at":"2026-05-18T12:30:53Z"},{"alias_kind":"pith_short_16","alias_value":"YNDSOAAIQZMAAMHO","created_at":"2026-05-18T12:30:53Z"},{"alias_kind":"pith_short_8","alias_value":"YNDSOAAI","created_at":"2026-05-18T12:30:53Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2016:YNDSOAAIQZMAAMHOG4IIDWKMOU","target":"record","payload":{"canonical_record":{"source":{"id":"1610.07776","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2016-10-25T08:13:49Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"aeadcd82a977051820e37d71cbb0a6a5f4f77fab2f2a8874bc3a011871ea9295","abstract_canon_sha256":"7cdfa2575d6b390b39c6764b6a174c0e3d7a1998be271f27923a45583b062420"},"schema_version":"1.0"},"canonical_sha256":"c34727000886580030ee371081d94c752c45b2614db5a6efa067486a07eabf28","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:01:17.603800Z","signature_b64":"OevZAA+T7VRG37/iz6kzrnbvqea98yMokScjYEpxiQWEWqOjVbxwvLw4knT1mRpsIMLuGpg1At8845vcRz/hDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"c34727000886580030ee371081d94c752c45b2614db5a6efa067486a07eabf28","last_reissued_at":"2026-05-18T01:01:17.603190Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:01:17.603190Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1610.07776","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":"2026-05-18T01:01:17Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Dmbk3l35xh1lm0f8F7lisLb64UaMcTcw/zsB6vm9nH+WoJOdUWteLWpq6QwkTknd+RX87Bz6T9hFvHj27u3GCQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-27T03:17:39.592852Z"},"content_sha256":"82eedfcc7362b8aed7d3234c30810db190af4e032ac1b85396073dd03abf6076","schema_version":"1.0","event_id":"sha256:82eedfcc7362b8aed7d3234c30810db190af4e032ac1b85396073dd03abf6076"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2016:YNDSOAAIQZMAAMHOG4IIDWKMOU","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"On a restricted linear congruence","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.NT","authors_text":"Bruce M. Kapron, Khodakhast Bibak, Venkatesh Srinivasan","submitted_at":"2016-10-25T08:13:49Z","abstract_excerpt":"Let $b,n\\in \\mathbb{Z}$, $n\\geq 1$, and ${\\cal D}_1, \\ldots, {\\cal D}_{\\tau(n)}$ be all positive divisors of $n$. For $1\\leq l \\leq \\tau(n)$, define ${\\cal C}_l:=\\lbrace 1 \\leqslant x\\leqslant n \\; : \\; (x,n)={\\cal D}_l\\rbrace$. In this paper, by combining ideas from the finite Fourier transform of arithmetic functions and Ramanujan sums, we give a short proof for the following result: the number of solutions of the linear congruence $x_1+\\cdots +x_k\\equiv b \\pmod{n}$, with $\\kappa_{l}=|\\lbrace x_1, \\ldots, x_k \\rbrace \\cap {\\cal C}_l|$, $1\\leq l \\leq \\tau(n)$, is \\begin{align*} \\frac{1}{n}\\ma"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1610.07776","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":"2026-05-18T01:01:17Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"QZF+2FIU7CGAOLjEsBSsVG3Lr/5OUHI6YHlRt7fqD6lqBh5px7gpzMxXrbdVxxIjVtE12YgSusAhKpTFJ796DQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-27T03:17:39.593568Z"},"content_sha256":"51974bbbba2becc581e651e01d3f205c493b80f4e0f4a13a3de8ec76f925ba7b","schema_version":"1.0","event_id":"sha256:51974bbbba2becc581e651e01d3f205c493b80f4e0f4a13a3de8ec76f925ba7b"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/YNDSOAAIQZMAAMHOG4IIDWKMOU/bundle.json","state_url":"https://pith.science/pith/YNDSOAAIQZMAAMHOG4IIDWKMOU/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/YNDSOAAIQZMAAMHOG4IIDWKMOU/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":"2026-05-27T03:17:39Z","links":{"resolver":"https://pith.science/pith/YNDSOAAIQZMAAMHOG4IIDWKMOU","bundle":"https://pith.science/pith/YNDSOAAIQZMAAMHOG4IIDWKMOU/bundle.json","state":"https://pith.science/pith/YNDSOAAIQZMAAMHOG4IIDWKMOU/state.json","well_known_bundle":"https://pith.science/.well-known/pith/YNDSOAAIQZMAAMHOG4IIDWKMOU/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:YNDSOAAIQZMAAMHOG4IIDWKMOU","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":"7cdfa2575d6b390b39c6764b6a174c0e3d7a1998be271f27923a45583b062420","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2016-10-25T08:13:49Z","title_canon_sha256":"aeadcd82a977051820e37d71cbb0a6a5f4f77fab2f2a8874bc3a011871ea9295"},"schema_version":"1.0","source":{"id":"1610.07776","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1610.07776","created_at":"2026-05-18T01:01:17Z"},{"alias_kind":"arxiv_version","alias_value":"1610.07776v1","created_at":"2026-05-18T01:01:17Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1610.07776","created_at":"2026-05-18T01:01:17Z"},{"alias_kind":"pith_short_12","alias_value":"YNDSOAAIQZMA","created_at":"2026-05-18T12:30:53Z"},{"alias_kind":"pith_short_16","alias_value":"YNDSOAAIQZMAAMHO","created_at":"2026-05-18T12:30:53Z"},{"alias_kind":"pith_short_8","alias_value":"YNDSOAAI","created_at":"2026-05-18T12:30:53Z"}],"graph_snapshots":[{"event_id":"sha256:51974bbbba2becc581e651e01d3f205c493b80f4e0f4a13a3de8ec76f925ba7b","target":"graph","created_at":"2026-05-18T01:01:17Z","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 $b,n\\in \\mathbb{Z}$, $n\\geq 1$, and ${\\cal D}_1, \\ldots, {\\cal D}_{\\tau(n)}$ be all positive divisors of $n$. For $1\\leq l \\leq \\tau(n)$, define ${\\cal C}_l:=\\lbrace 1 \\leqslant x\\leqslant n \\; : \\; (x,n)={\\cal D}_l\\rbrace$. In this paper, by combining ideas from the finite Fourier transform of arithmetic functions and Ramanujan sums, we give a short proof for the following result: the number of solutions of the linear congruence $x_1+\\cdots +x_k\\equiv b \\pmod{n}$, with $\\kappa_{l}=|\\lbrace x_1, \\ldots, x_k \\rbrace \\cap {\\cal C}_l|$, $1\\leq l \\leq \\tau(n)$, is \\begin{align*} \\frac{1}{n}\\ma","authors_text":"Bruce M. Kapron, Khodakhast Bibak, Venkatesh Srinivasan","cross_cats":["math.CO"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2016-10-25T08:13:49Z","title":"On a restricted linear congruence"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1610.07776","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":"sha256:82eedfcc7362b8aed7d3234c30810db190af4e032ac1b85396073dd03abf6076","target":"record","created_at":"2026-05-18T01:01:17Z","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":"7cdfa2575d6b390b39c6764b6a174c0e3d7a1998be271f27923a45583b062420","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2016-10-25T08:13:49Z","title_canon_sha256":"aeadcd82a977051820e37d71cbb0a6a5f4f77fab2f2a8874bc3a011871ea9295"},"schema_version":"1.0","source":{"id":"1610.07776","kind":"arxiv","version":1}},"canonical_sha256":"c34727000886580030ee371081d94c752c45b2614db5a6efa067486a07eabf28","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"c34727000886580030ee371081d94c752c45b2614db5a6efa067486a07eabf28","first_computed_at":"2026-05-18T01:01:17.603190Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:01:17.603190Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"OevZAA+T7VRG37/iz6kzrnbvqea98yMokScjYEpxiQWEWqOjVbxwvLw4knT1mRpsIMLuGpg1At8845vcRz/hDw==","signature_status":"signed_v1","signed_at":"2026-05-18T01:01:17.603800Z","signed_message":"canonical_sha256_bytes"},"source_id":"1610.07776","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:82eedfcc7362b8aed7d3234c30810db190af4e032ac1b85396073dd03abf6076","sha256:51974bbbba2becc581e651e01d3f205c493b80f4e0f4a13a3de8ec76f925ba7b"],"state_sha256":"aa4a004fd087c663b3052697f3df7d1f9900e2ecc3f6148f2a480731d0b21e8b"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"w2EEfZo3Utm6rRryfN6uLeW+NP96jVssgiHdBl5hcuAZN/Y+q3ouDCCz+8mXXjs19WPDpO6gr/JHDioJmwnBBA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-27T03:17:39.597729Z","bundle_sha256":"2a82c3c513d1e62fd9eb7e5787439825becefc6dd6e9bf8a41cf3eecb14d02e7"}}