Pith Number

pith:IF4XMGLQ

pith:2026:IF4XMGLQBZAHH2SNKBVNS6H5DG

not attested not anchored not stored refs pending

Solution to the Erdos problem on distinct residues of factorials

Vyacheslav M. Abramov

There is no prime number p > 5 such that the residues of 2!, 3!, …, (p-1)! modulo p are all distinct.

arxiv:2604.26429 v3 · 2026-04-29 · math.NT · math.CO

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Add to your LaTeX paper

\usepackage{pith}
\pithnumber{IF4XMGLQBZAHH2SNKBVNS6H5DG}

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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.

Claims

C1strongest claim

There is no prime number p>5 such that the residues of 2!, 3!,…, (p-1)! modulo p all are distinct.

C2weakest assumption

The elementary proof applies to every prime p>5 with no exceptions arising from special cases or additional modular constraints.

C3one line summary

No prime p>5 exists such that the residues of 2!, 3!, ..., (p-1)! modulo p are all distinct.

Receipt and verification

First computed	2026-05-28T01:04:41.007472Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

41797619700e4073ea4d506ad978fd19b5bf0ca29be00d2c196a3e31e8466e2a

Aliases

arxiv: 2604.26429 · arxiv_version: 2604.26429v3 · doi: 10.48550/arxiv.2604.26429 · pith_short_12: IF4XMGLQBZAH · pith_short_16: IF4XMGLQBZAHH2SN · pith_short_8: IF4XMGLQ

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/IF4XMGLQBZAHH2SNKBVNS6H5DG \
  | 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: 41797619700e4073ea4d506ad978fd19b5bf0ca29be00d2c196a3e31e8466e2a

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "d800140b95b97bde6ed5233df37a02b2e01dda97b264197493ef3d5d370713d7",
    "cross_cats_sorted": [
      "math.CO"
    ],
    "license": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
    "primary_cat": "math.NT",
    "submitted_at": "2026-04-29T08:39:51Z",
    "title_canon_sha256": "ef5ba2200b67d71cda266fb96bebabfd0c7b04ff69b256b851862ee554a67ae4"
  },
  "schema_version": "1.0",
  "source": {
    "id": "2604.26429",
    "kind": "arxiv",
    "version": 3
  }
}