Pith Number

pith:WWJQJF7C

pith:2018:WWJQJF7CC726T3SSXTJXPFPI6I

not attested not anchored not stored refs pending

Congruences for Ap\'ery numbers $\beta_{n}=\sum_{k=0}^{n}\binom{n}{k}^2\binom{n+k}{k}$

Hui-Qin Cao, Yuri Matiyasevich, Zhi-Wei Sun

arxiv:1812.10351 v5 · 2018-12-26 · math.NT

Open paper page JSON Open Graph Bundle Merged state Verified badge What is a Pith Number?

Add to your LaTeX paper

\usepackage{pith}
\pithnumber{WWJQJF7CC726T3SSXTJXPFPI6I}

Prints a linked badge after your title and injects PDF metadata. Compiles on arXiv. Learn more · Embed verified badge

Record completeness

1 Bitcoin timestamp

2 Internet Archive

3 Author claim open · sign in to claim

4 Citations open

5 Replications open

✓ Portable graph bundle live · download bundle · merged state

The bundle contains the canonical record plus signed events. A mirror can host it anywhere and recompute the same current state with the deterministic merge algorithm.

Receipt and verification

First computed	2026-07-05T03:25:08.767793Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

b5930497e217f5e9ee52bcd37795e8f21f22d683c303c7cbbaf819aed9ad7f25

Aliases

arxiv: 1812.10351 · arxiv_version: 1812.10351v5 · doi: 10.48550/arxiv.1812.10351 · pith_short_12: WWJQJF7CC726 · pith_short_16: WWJQJF7CC726T3SS · pith_short_8: WWJQJF7C

Agent API

Resolver JSON Graph JSON Events JSON Schema Signing key

Verify this Pith Number yourself

curl -sH 'Accept: application/ld+json' https://pith.science/pith/WWJQJF7CC726T3SSXTJXPFPI6I \
  | jq -c '.canonical_record' \
  | python3 -c "import sys,json,hashlib; b=json.dumps(json.loads(sys.stdin.read()), sort_keys=True, separators=(',',':'), ensure_ascii=False).encode(); print(hashlib.sha256(b).hexdigest())"
# expect: b5930497e217f5e9ee52bcd37795e8f21f22d683c303c7cbbaf819aed9ad7f25

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "8d6256af1c31e36d6aa04a0b05ddfd7a4e09782e1c64c22a717f2fc621307de1",
    "cross_cats_sorted": [],
    "license": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
    "primary_cat": "math.NT",
    "submitted_at": "2018-12-26T15:58:42Z",
    "title_canon_sha256": "0b683bd0561818938efac6ea76482e90f4c5eba8e192f72b76fd09b6573ff151"
  },
  "schema_version": "1.0",
  "source": {
    "id": "1812.10351",
    "kind": "arxiv",
    "version": 5
  }
}