Linear Independence Of Some Irrational Numbers

N. A. Carella

arxiv: 2007.15000 · v3 · submitted 2020-07-29 · 🧮 math.GM

Linear Independence Of Some Irrational Numbers

N. A. Carella This is my paper

Pith reviewed 2026-05-24 14:26 UTC · model grok-4.3

classification 🧮 math.GM

keywords linear independenceirrational numbersanalytic techniqueepirational coefficients

0 comments

The pith

An analytic technique establishes linear independence over the rationals for the triples 1, e, π and 1, e, 1/π.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper introduces an analytic technique for proving that small sets of real numbers are linearly independent over the rationals. The technique is applied to produce explicit proofs that no nontrivial rational linear combination of 1, e, and π equals zero. The same method is used for the triple consisting of 1, e, and the reciprocal of π, and for any triple consisting of 1 together with two distinct positive integer powers of π. A sympathetic reader would care because these are among the most basic and frequently studied constants whose linear independence over Q has long been expected but not always elementary to establish.

Core claim

The paper presents an analytic technique that produces simple linear independence proofs over the rationals for the subsets {1, e, π}, {1, e, π^{-1}}, and {1, π^r, π^s} where 1 ≤ r < s are fixed integers.

What carries the argument

An analytic technique for proving linear independence of small subsets of real numbers over the rationals.

If this is right

The numbers 1, e, and π are linearly independent over the rationals.
The numbers 1, e, and 1/π are linearly independent over the rationals.
For any fixed integers r < s the numbers 1, π^r, and π^s are linearly independent over the rationals.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The technique might apply to other small combinations involving e, π, and algebraic multiples of these constants.
If the method is general, it could reduce reliance on full transcendence proofs for basic independence statements.

Load-bearing premise

The analytic technique is valid and correctly establishes the claimed linear independence relations without hidden gaps or unstated conditions.

What would settle it

Discovery of a nontrivial rational linear dependence relation summing to zero among 1, e, and π.

read the original abstract

This note presents an analytic technique for proving the linear independence of certain small subsets of real numbers over the rational numbers. The applications of this test produce simple linear independence proofs for the subsets of triples $\{1, e, \pi\}$, $\{1, e, \pi^{-1}\}$, and $\{1, \pi^r, \pi^s\}$, where $1\leq r<s $ are fixed integers.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

Only an abstract is available claiming proofs for open problems without any supporting details or math.

read the letter

The abstract claims an analytic technique that gives linear independence proofs for the triples {1, e, π}, {1, e, π^{-1}}, and {1, π^r, π^s} for fixed integers r < s. These are longstanding open questions in number theory, so if the proofs held up it would matter. But the full text is not available, only this abstract, so the technique and any derivations remain invisible. The paper is in the math.GM category, which is for general mathematics and sometimes includes notes on such topics. There is nothing new or well executed on display. The note asserts that its applications produce simple proofs, but supplies no steps, no test description, and no supporting calculations. The abstract presents the technique as new for these applications, but without more it is impossible to tell. The main issue is that the load-bearing part—the validity of the analytic technique—cannot be checked at all. Without seeing how it works, it's impossible to assess whether it avoids circularity, matches known theorems on e and π, or actually delivers the claimed independences. The abstract alone gives no basis for evaluation. This matches the low soundness score from the initial review. This is not useful to anyone right now. Readers in transcendental number theory would need the actual argument to get anything from it. As presented, it offers no content to discuss or build on. The paper does not engage with the literature in any visible way. I would not send this to peer review. It should be desk rejected until the complete technique and proofs are included for inspection. There is simply no result here to evaluate.

Referee Report

1 major / 0 minor

Summary. The manuscript claims to present an analytic technique for proving linear independence over the rationals of small subsets of real numbers, with applications yielding simple proofs for the triples {1, e, π}, {1, e, π^{-1}}, and {1, π^r, π^s} (1 ≤ r < s fixed integers). The provided text consists solely of the abstract; no description of the technique, derivations, test details, or supporting arguments are supplied.

Significance. If a valid, non-circular technique were rigorously established and correctly applied to these triples, the results would be highly significant as they address longstanding open questions in transcendental number theory. No such assessment is possible here due to the complete absence of technical content.

major comments (1)

[Abstract] Abstract: The abstract asserts that the technique produces simple linear independence proofs for the listed triples but supplies no derivation, no description of the test, and no supporting steps, so there is no basis to check whether the mathematics supports the claims.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their report. We address the major comment below.

read point-by-point responses

Referee: [Abstract] Abstract: The abstract asserts that the technique produces simple linear independence proofs for the listed triples but supplies no derivation, no description of the test, and no supporting steps, so there is no basis to check whether the mathematics supports the claims.

Authors: We agree with the referee that the submitted manuscript contains only the abstract and provides no derivation or description of the analytic technique. This is an oversight in the submission, and the full manuscript should have included the details of the technique along with the proofs for the linear independence of {1, e, π}, {1, e, π^{-1}}, and {1, π^r, π^s}. We will prepare a revised version that includes the complete technical content to allow for proper evaluation. revision: yes

Circularity Check

0 steps flagged

No derivation chain present; circularity cannot be assessed

full rationale

The document supplies only an abstract that asserts the existence of an unspecified analytic technique yielding linear independence results. No equations, definitions, self-citations, or derivation steps appear in the available text, so none of the enumerated circularity patterns (self-definitional, fitted-input-called-prediction, etc.) can be exhibited by direct quotation and reduction. The derivation chain is therefore not inspectable and yields no circularity findings.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract supplies no information on free parameters, axioms, or invented entities used by the technique.

pith-pipeline@v0.9.0 · 5545 in / 1131 out tokens · 29435 ms · 2026-05-24T14:26:32.199291+00:00 · methodology

Linear Independence Of Some Irrational Numbers

Core claim

What carries the argument

If this is right

Where Pith is reading between the lines

Load-bearing premise

What would settle it

discussion (0)