{"paper":{"title":"The transcendence of $\\mathrm{e}$ via formal power series","license":"http://creativecommons.org/licenses/by/4.0/","headline":"The transcendence of e follows from algebraic operations on formal power series alone.","cross_cats":[],"primary_cat":"math.NT","authors_text":"Martin Klazar","submitted_at":"2026-01-03T00:59:06Z","abstract_excerpt":"We review Hilbert's classical analytical proof of the transcendence of the number $\\mathrm{e}$. Then, we show how this result can be obtained algebraically by means of formal power series (FPS). We give two proofs of the transcendence of $\\mathrm{e}$ based on FPS. The first of them is a specialization of the 1990 proof by Beukers, B\\'ezivin 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."},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"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.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"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.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"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.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"The transcendence of e follows from algebraic operations on formal power series alone.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"920c3ee449bba74dd17a60e69125b2dd26d2d574d225c12bf247599bd9dc0f4b"},"source":{"id":"2601.01019","kind":"arxiv","version":8},"verdict":{"id":"6c36488e-9166-43d0-9159-303ee2c6cb21","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-16T18:32:47.250313Z","strongest_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.","one_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.","pipeline_version":"pith-pipeline@v0.9.0","weakest_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.","pith_extraction_headline":"The transcendence of e follows from algebraic operations on formal power series alone."},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":2,"snapshot_sha256":"0a4b8afb14206842e41bb3181cd7e60a4ecf3ad406bab94eaaea165ddfaa2748"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}