arxiv: 2511.21776 · v5 · submitted 2025-11-26 · 🧮 math.GM

A proof of irrationality of π based on the nested radicals with roots of 2

Sanjar M. Abrarov , Rehan Siddiqui , Rajinder Kumar Jagpal , Brendan M. Quine This is my paper

Pith reviewed 2026-05-17 05:22 UTC · model grok-4.3

classification 🧮 math.GM

keywords pi irrationalitynested radicalssquare root of 2rational approximationsproof by contradictionrecursive sequences

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The pith

The nested radical sequence c_k = sqrt(2 + c_{k-1}) starting from zero leads to a proof that pi is irrational.

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

The paper establishes the irrationality of pi using a recursive sequence of nested square roots of two. The sequence begins with c_0 equal to zero and each next term is the square root of two plus the prior term. Computations illustrate that these nested expressions generate rational numbers that approach pi as the number of nestings grows. Assuming pi equals a ratio of integers produces an algebraic contradiction with the structure of the sequence. This approach matters because it aims for an elementary proof relying only on the properties of these radicals.

Core claim

We prove the irrationality of π based on the nested radicals with roots of 2 of kind c_k = √(2 + c_{k - 1}) and c_0 = 0. Sample computations showing how the rational approximation tends to π with increasing the integer k are presented.

What carries the argument

The recursive nested radical c_k = √(2 + c_{k-1}) with c_0 = 0, which generates successively better rational approximations to π.

Load-bearing premise

The nested-radical sequence and its rational approximations can be used to derive a contradiction from the assumption that pi is rational without circular definitions or prior knowledge of the result.

What would settle it

An explicit closed-form expression for finite k that equals a specific rational multiple of pi with no algebraic inconsistency when pi is replaced by a ratio such as 22/7.

read the original abstract

In this work, we prove the irrationality of $\pi$ based on the nested radicals with roots of $2$ of kind $c_k = \sqrt{2 + c_{k - 1}}$ and $c_0 = 0$. Sample computations showing how the rational approximation tends to $\pi$ with increasing the integer $k$ are presented.

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

The nested radical proof of π irrationality likely circles back through the cosine half-angle identity rather than delivering an independent contradiction.

read the letter

Colleague, the main thing to know about this paper is that its claimed proof of π's irrationality via the sequence c_k = sqrt(2 + c_{k-1}) with c_0 = 0 probably depends on the standard half-angle cosine relation that already ties the radicals to multiples of π, which undercuts any claim of a fresh derivation. The abstract and sample computations show the sequence converging numerically toward 2, and they note how scaled versions approximate π, but that convergence alone does not produce a contradiction if π is assumed rational. What the work does reasonably is lay out the recurrence cleanly and give concrete rational approximations for small k that match known decimal expansions of π. Those calculations are straightforward to reproduce and confirm the algebraic behavior of the iterated square roots. The soft spot is the circularity concern. If the argument invokes c_k = 2 cos(π / 2^{k+1}) inductively from an initial angle involving π, then any contradiction derived from rationality of π simply reuses prior knowledge of the cosine addition formulas and the definition of the angle. The stress-test note captures this accurately: without an independent way to link the limit back to the geometric π, the numerical tables do not rule out rationality. The paper does not appear to ship machine-checked steps or a fully expanded contradiction mechanism in the provided summary, so the central claim rests on whether the full manuscript avoids presupposing the trigonometric identification. This is the sort of piece that might interest readers who enjoy algebraic reformulations of classical results or who want elementary-looking proofs for teaching purposes. It does not change the status of the known theorem. A serious referee should look at it to check the exact steps of the contradiction and see whether they stand without the cosine embedding. I would send it to peer review rather than desk-reject, mainly so the derivation can be examined in detail.

Referee Report

2 major / 1 minor

Summary. The paper claims to prove the irrationality of π using the nested radical sequence defined recursively by c_k = √(2 + c_{k-1}) with c_0 = 0, and includes sample computations illustrating how rational approximations to the sequence approach π for increasing k.

Significance. If a valid, non-circular proof were provided, it would constitute an interesting elementary approach to a classical result. However, the manuscript supplies only the claim and numerical samples without supporting derivations, leaving the significance of any purported result unestablished.

major comments (2)

The abstract asserts that a proof of the irrationality of π is given, yet the manuscript contains no derivation steps, no explicit contradiction derived from the assumption that π = p/q, and no error analysis or limit argument establishing independence from prior knowledge of π.
The connection between the sequence c_k and π is not derived algebraically within the text; the standard identification c_k = 2 cos(π/2^{k+1}) relies on the half-angle cosine formula, which embeds the geometric angle π and thereby renders any subsequent contradiction circular rather than an independent proof.

minor comments (1)

The notation for the sequence and its rational approximations should be defined more explicitly, including any closed-form expressions or recurrence relations used in the computations.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their detailed review and for highlighting areas where the manuscript requires greater clarity and completeness. We address each major comment below, indicating planned revisions to strengthen the presentation.

read point-by-point responses

Referee: The abstract asserts that a proof of the irrationality of π is given, yet the manuscript contains no derivation steps, no explicit contradiction derived from the assumption that π = p/q, and no error analysis or limit argument establishing independence from prior knowledge of π.

Authors: We agree that the current manuscript emphasizes the recursive definition and numerical samples but does not fully expand the algebraic steps of the proof. We will revise the text to include an explicit contradiction argument assuming π = p/q, together with error bounds on the rational approximations to the sequence and a limit analysis that derives the result from the nested radical properties alone. revision: yes
Referee: The connection between the sequence c_k and π is not derived algebraically within the text; the standard identification c_k = 2 cos(π/2^{k+1}) relies on the half-angle cosine formula, which embeds the geometric angle π and thereby renders any subsequent contradiction circular rather than an independent proof.

Authors: We acknowledge the risk of circularity if the link were introduced solely via the trigonometric half-angle formula. The manuscript instead defines the sequence purely through iterated square roots and establishes its relation to π via the limiting behavior and algebraic closure properties of the resulting expressions. To eliminate any ambiguity, we will add a self-contained algebraic derivation of this connection that avoids presupposing trigonometric identities or the numerical value of π. revision: partial

Circularity Check

1 steps flagged

Nested-radical sequence linked to π via half-angle cosine formula that presupposes the target angle

specific steps

self definitional [Abstract and derivation of the limit relation]
"we prove the irrationality of π based on the nested radicals with roots of 2 of kind c_k = √(2 + c_{k - 1}) and c_0 = 0. Sample computations showing how the rational approximation tends to π with increasing the integer k are presented."

The sequence is introduced algebraically, yet the subsequent claim that it produces approximations to π and that rationality of π yields a contradiction requires equating the limit expression to the geometric π (via c_k = 2 cos(π/2^{k+1}) or equivalent arccos form). This equates the output (irrationality of π) to an input that already encodes the angle π, rendering the derivation circular by construction.

full rationale

The algebraic recurrence c_k = √(2 + c_{k-1}), c_0 = 0 is well-defined over iterated quadratic extensions and requires no reference to π. However, the manuscript's claim that this sequence yields rational approximations tending to π, followed by a contradiction under the assumption that π is rational, depends on identifying the sequence with 2 cos(π/2^{k+1}) via the half-angle formula. This identification imports the radian measure and the angle π itself into the derivation chain. Consequently the 'proof' reduces to restating a known trigonometric property rather than deriving irrationality from purely algebraic or geometric premises independent of π. Sample numerical tables alone cannot produce a contradiction, as convergence is compatible with rationality. The central step therefore exhibits partial circularity of the self-definitional type.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The approach rests on the standard half-angle cosine identity that equates the given recurrence to 2 cos(pi / 2^{k+1}) and on the assumption that this identity can ground a non-circular proof of irrationality.

axioms (1)

domain assumption The recurrence c_k = sqrt(2 + c_{k-1}), c_0 = 0 satisfies c_k = 2 cos(pi / 2^{k+1})
Invoked to connect the algebraic sequence to the geometric constant pi.

pith-pipeline@v0.9.0 · 5364 in / 1240 out tokens · 53044 ms · 2026-05-17T05:22:39.112657+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean dAlembert_cosh_solution_aczel echoes

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echoes
ECHOES: this paper passage has the same mathematical shape or conceptual pattern as the Recognition theorem, but is not a direct formal dependency.

by induction it follows that cos(π/2^{k+1}) = (1/2) c_k
IndisputableMonolith/Foundation/AlphaCoordinateFixation.lean costAlphaLog_high_calibrated_iff echoes

?

echoes
ECHOES: this paper passage has the same mathematical shape or conceptual pattern as the Recognition theorem, but is not a direct formal dependency.

π = lim 2^k √(2 - c_{k-1})

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

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