Pith Number

pith:UDPW66L5

pith:2026:UDPW66L53WAAI3MUFJDMK2HW6K

not attested not anchored not stored refs pending

The transcendence of $\mathrm{e}$ via formal power series

Martin Klazar

The transcendence of e follows from algebraic operations on formal power series alone.

arxiv:2601.01019 v8 · 2026-01-03 · math.NT

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4 Citations open

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Claims

C1strongest claim

We give two proofs of the transcendence of e based on FPS. The first of them is a specialization of the 1990 proof by Beukers, Bézivin and Robba of the Lindemann-Weierstrass theorem. The second proof is due to this author and is an adaptation of Hilbert's argument to FPS.

C2weakest assumption

That the algebraic operations on formal power series can fully replicate the key contradiction steps of Hilbert's analytic proof without requiring any convergence or analytic continuation properties.

C3one line summary

Two formal power series proofs establish the transcendence of e: one specializes a 1990 Lindemann-Weierstrass proof, and the other adapts Hilbert's argument algebraically.

Formal links

2 machine-checked theorem links

Cited by

1 paper in Pith

Extending the symbolic method in enumerative combinatorics. I

Receipt and verification

First computed	2026-05-17T23:39:16.741580Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

a0df6f797ddd80046d942a46c568f6f2a6398336f64fda174163a1ceb9f3f6da

Aliases

arxiv: 2601.01019 · arxiv_version: 2601.01019v8 · doi: 10.48550/arxiv.2601.01019 · pith_short_12: UDPW66L53WAA · pith_short_16: UDPW66L53WAAI3MU · pith_short_8: UDPW66L5

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Verify this Pith Number yourself

curl -sH 'Accept: application/ld+json' https://pith.science/pith/UDPW66L53WAAI3MUFJDMK2HW6K \
  | 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: a0df6f797ddd80046d942a46c568f6f2a6398336f64fda174163a1ceb9f3f6da

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "dccbd01f8e79e34db6927b2c5b43dbc1e57577670dc997d97f32aa62b9877006",
    "cross_cats_sorted": [],
    "license": "http://creativecommons.org/licenses/by/4.0/",
    "primary_cat": "math.NT",
    "submitted_at": "2026-01-03T00:59:06Z",
    "title_canon_sha256": "ee1502157baf5f55e99ecc065c9d72082d7cbed3db7d9ddfed5e15b88cf7ba83"
  },
  "schema_version": "1.0",
  "source": {
    "id": "2601.01019",
    "kind": "arxiv",
    "version": 8
  }
}