Pith Number

pith:KEESX37Y

pith:2013:KEESX37YAZVGM4DNPWBIW6FIKD

not attested not anchored not stored refs pending

Pl\"unnecke inequalities for countable abelian groups

Alexander Fish, Michael Bj\"orklund

arxiv:1311.5372 v2 · 2013-11-21 · math.DS · math.GR · 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{KEESX37YAZVGM4DNPWBIW6FIKD}

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	2026-05-18T03:05:57.247532Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

51092beff8066a66706d7d828b78a850d4da75106ac09655c70093dfcd6318de

Aliases

arxiv: 1311.5372 · arxiv_version: 1311.5372v2 · doi: 10.48550/arxiv.1311.5372 · pith_short_12: KEESX37YAZVG · pith_short_16: KEESX37YAZVGM4DN · pith_short_8: KEESX37Y

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/KEESX37YAZVGM4DNPWBIW6FIKD \
  | 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: 51092beff8066a66706d7d828b78a850d4da75106ac09655c70093dfcd6318de

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "0d8bd410344683ddcf1e356571140c4ef3daa34a2962c7e4dfc8e60b96a72195",
    "cross_cats_sorted": [
      "math.GR",
      "math.NT"
    ],
    "license": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
    "primary_cat": "math.DS",
    "submitted_at": "2013-11-21T11:39:20Z",
    "title_canon_sha256": "1d57b7cfe30da810bf2e5a8d9e75e97d9abf5c0cf4e93162d1c5a84fd40ddc58"
  },
  "schema_version": "1.0",
  "source": {
    "id": "1311.5372",
    "kind": "arxiv",
    "version": 2
  }
}