Pith Number

pith:G3TA4ONP

pith:2026:G3TA4ONPJBQAFWVDTMCUHA2RUV

not attested not anchored not stored refs pending

A formal proof of the Ramanujan--Nagell theorem in Lean 4

Barinder S. Banwait

A complete Lean 4 formalization establishes that x squared plus 7 equals 2 to the n only for the five listed solution pairs.

arxiv:2604.09808 v2 · 2026-04-10 · math.NT · cs.LO

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\usepackage{pith}
\pithnumber{G3TA4ONPJBQAFWVDTMCUHA2RUV}

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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

the only integer solutions to the Diophantine equation x^2 + 7 = 2^n are (n,x) in {(3,±1),(4,±3),(5,±5),(7,±11),(15,±181)}

C2weakest assumption

The Lean formalization accurately captures the mathematical content of the textbook proof without omissions or errors in the encoding of algebraic number theory concepts such as the ring of integers, class number, and unit group.

C3one line summary

A complete Lean 4 formalization proves the Ramanujan-Nagell theorem: the only integer solutions to x^2 + 7 = 2^n are the five listed pairs.

Receipt and verification

First computed	2026-06-02T03:05:05.461384Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

36e60e39af486002daa39b05438351a5765f8347ef06da40fdeda0567d1a22ee

Aliases

arxiv: 2604.09808 · arxiv_version: 2604.09808v2 · doi: 10.48550/arxiv.2604.09808 · pith_short_12: G3TA4ONPJBQA · pith_short_16: G3TA4ONPJBQAFWVD · pith_short_8: G3TA4ONP

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/G3TA4ONPJBQAFWVDTMCUHA2RUV \
  | 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: 36e60e39af486002daa39b05438351a5765f8347ef06da40fdeda0567d1a22ee

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "082d1768416fc47a4b8db4b1c7d11f9a900de12120282a0cddbcb1c1cdb3ec9c",
    "cross_cats_sorted": [
      "cs.LO"
    ],
    "license": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
    "primary_cat": "math.NT",
    "submitted_at": "2026-04-10T18:32:19Z",
    "title_canon_sha256": "0329556da7a82069f2eb0c4cd80bdaca627db7081b24722be83a623294e6005d"
  },
  "schema_version": "1.0",
  "source": {
    "id": "2604.09808",
    "kind": "arxiv",
    "version": 2
  }
}