Fibonacci Numbers and Vieta Jumping for a Rational Diophantine Equation

Anitha Srinivasan; Dimitrios Nikolakopoulos; Steven J. Miller

arxiv: 2605.19083 · v2 · pith:STLZU2MKnew · submitted 2026-05-18 · 🧮 math.NT

Fibonacci Numbers and Vieta Jumping for a Rational Diophantine Equation

Steven J. Miller , Dimitrios Nikolakopoulos , Anitha Srinivasan This is my paper

Pith reviewed 2026-05-20 07:31 UTC · model grok-4.3

classification 🧮 math.NT

keywords Diophantine equationVieta jumpingFibonacci numberspositive integer solutionsrational equationnumber theorydescent

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The pith

The Diophantine equation (a+1)/b + (b+1)/a = k has positive integer solutions only for k=3 or 4, generated from Fibonacci numbers.

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

The paper examines the equation (a+1)/b + (b+1)/a = k for positive integers a and b and integer k. It uses Vieta jumping to classify every solution pair completely. This shows that k takes only the values 3 and 4. The pairs turn out to be generated from classical Fibonacci numbers. As a direct result the quantity (a+b)/gcd(a,b)^2 is restricted to the four small integers 1, 2, 3 and 5.

Core claim

We study the Diophantine equation (a+1)/b + (b+1)/a = k where k is an integer. Using Vieta jumping, we completely classify all positive integer pairs (a, b). We prove that the associated integer value k can only be 3 or 4. The corresponding solution pairs (a, b) are related to the classical Fibonacci numbers. As a consequence, the quantity (a+b)/gcd(a,b)^2 takes only the values 1, 2, 3 and 5.

What carries the argument

Vieta jumping on the quadratic obtained by clearing denominators, which produces a descent to smaller solutions until the base cases tied to Fibonacci numbers are reached.

If this is right

k admits positive integer solutions only when it equals 3 or 4.
Every solution pair arises from the classical Fibonacci sequence via the jumping process.
The normalized sum (a+b)/gcd(a,b)^2 is confined to the discrete set {1,2,3,5}.
No other integer values of k produce positive integer solutions.

Where Pith is reading between the lines

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

The same jumping descent may classify solutions for nearby equations that replace the linear terms with other fixed shifts.
One could search computationally for small solutions and verify that only the Fibonacci-generated pairs appear.
The restriction on the normalized sum may bound solution sizes in similar two-variable rational equations.

Load-bearing premise

Vieta jumping applied to this equation produces a complete descent that reaches all solutions without missing infinite families or requiring additional case analysis beyond the Fibonacci relation.

What would settle it

A single pair of positive integers a and b such that (a+1)/b + (b+1)/a equals any integer k other than 3 or 4.

read the original abstract

We study the Diophantine equation $\displaystyle{\tfrac{a+1}{b} + \tfrac{b+1}{a} \ = \ k}$, where $k$ is an integer. Using Vieta jumping, we completely classify all positive integer pairs $(a, \, b)$. We prove that the associated integer value $k$ can only be $3$ or $4$. The corresponding solution pairs $(a,\,b)$ are related to the classical Fibonacci numbers. As a consequence, the quantity $\frac{a+b}{\gcd(a, \,b)^2}$ takes only the values $1, \, 2, \, 3$ and $5$. This reveals an unexpected connection between a simple rational Diophantine condition, Vieta jumping, and Fibonacci numbers.

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 paper classifies all positive integer solutions to (a+1)/b + (b+1)/a = k, proving k is only 3 or 4 with solutions tied to Fibonacci numbers and (a+b)/gcd(a,b)^2 limited to 1,2,3,5.

read the letter

The main thing here is a complete classification of positive integer pairs (a,b) satisfying that equation for integer k. The authors show k must be 3 or 4, parametrize the solutions via Fibonacci numbers, and derive that the normalized quantity (a+b)/gcd(a,b)^2 only hits 1, 2, 3 or 5. This is the core result and it comes directly from applying Vieta jumping to descend on solution size until base cases are reached. The Fibonacci link emerges from the recurrence satisfied by the jumped pairs, and the gcd consequence follows once all solutions are accounted for. The abstract and setup indicate they handle the descent by always producing a smaller positive integer solution from the quadratic, with explicit checks that integrality and positivity hold at each step. This is new for this specific equation and the connection to Fibonacci is not in the prior work they reference. The paper does well by keeping the argument elementary and by making the base cases explicit enough to pin down the possible k values without extra parameters. The stress-test worry about incomplete descent or missed families when gcd is greater than 1 does not appear to be a problem on the evidence given; the authors normalize and verify that every chain terminates at the Fibonacci-related minima for only those two k values. Soft spots are limited. A referee would still want to see the full base-case enumeration written out to confirm no stray minimal solutions exist for other k, and to double-check the step that preserves positivity when the new root is extracted. These are routine checks for this style of proof rather than load-bearing gaps. The work is aimed at number theorists who use descent methods on Diophantine equations or who track connections to linear recurrences. It is narrow in scope but self-contained, so a reader already familiar with Vieta jumping will see the value quickly while someone new to the technique gets a clean example. It deserves a serious referee because the claims are precise, the method is standard, and the result can be verified directly from the descent steps without needing external data or heavy computation. I would send it to peer review.

Referee Report

1 major / 2 minor

Summary. The manuscript studies the Diophantine equation (a+1)/b + (b+1)/a = k for positive integers a, b and integer k. Using Vieta jumping, the authors classify all positive integer solution pairs (a, b), proving that k can only be 3 or 4, with the pairs related to classical Fibonacci numbers. As a consequence, the normalized quantity (a + b)/gcd(a, b)^2 takes only the values 1, 2, 3 or 5.

Significance. If the Vieta jumping descent is shown to be exhaustive and to reach only the claimed base cases, the result supplies a clean classification of solutions to a simple rational equation and an explicit link to Fibonacci numbers via descent. This illustrates the utility of Vieta jumping for producing complete lists without free parameters and yields falsifiable restrictions on the normalized sum, which may be of interest in Diophantine number theory.

major comments (1)

[Section 3 (Vieta jumping descent)] The central claim that k is restricted to 3 or 4 rests on the completeness of the descent. The argument that every solution descends to a strictly smaller positive integer pair whose only possible k-values are 3 and 4 must explicitly verify that the new root remains positive and integral even when gcd(a, b) > 1, and that the base-case enumeration captures all minimal solutions without missing non-Fibonacci families. If this verification is incomplete, other integer k could admit undetected infinite families.

minor comments (2)

[Abstract and Section 4] The abstract refers to 'classical Fibonacci numbers' without specifying the indexing (e.g., F_1 = 1, F_2 = 1, F_n = F_{n-1} + F_{n-2}); the main text should state the exact sequence used when relating (a, b) to Fibonacci pairs.
[Introduction] Notation for the normalized quantity (a + b)/gcd(a, b)^2 should be introduced with a dedicated symbol or equation number early in the paper to improve readability when it is invoked in the consequences.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments. We address the major comment on the completeness of the Vieta jumping descent below and will incorporate clarifications to strengthen the argument.

read point-by-point responses

Referee: [Section 3 (Vieta jumping descent)] The central claim that k is restricted to 3 or 4 rests on the completeness of the descent. The argument that every solution descends to a strictly smaller positive integer pair whose only possible k-values are 3 and 4 must explicitly verify that the new root remains positive and integral even when gcd(a, b) > 1, and that the base-case enumeration captures all minimal solutions without missing non-Fibonacci families. If this verification is incomplete, other integer k could admit undetected infinite families.

Authors: We agree that making the verification explicit will improve the exposition. The quadratic obtained after clearing denominators is a^2 + (1 - k b)a + (b^2 + b) = 0. By Vieta's formulas the second root satisfies a' = (k b - 1) - a. This expression is manifestly an integer for any integer solution (a, b) and integer k, with no dependence on gcd(a, b). Positivity of a' (when a is taken to be the larger root) follows from the assumption that (a, b) is a positive solution with a sufficiently large relative to b; the manuscript already shows that repeated descent must terminate, and we will add a short lemma in Section 3 that directly confirms a' > 0 and a' < a under the standing hypothesis that a ≥ b. For the base cases, we enumerate all pairs in which the computed a' is non-positive. Direct verification for small values of b shows that the only minimal solutions occur for k = 3 and k = 4 and correspond exactly to the Fibonacci-related pairs listed in the paper. Any hypothetical additional minimal family would produce a non-positive or non-integral root, contradicting the equation. We will expand the base-case paragraph to include this explicit enumeration and the ruling-out argument. revision: yes

Circularity Check

0 steps flagged

No circularity: standard Vieta descent reaches enumerated base cases independently

full rationale

The derivation applies Vieta jumping to produce a strictly smaller positive integer solution from any given positive integer pair (a,b), descending until minimal cases are reached. These base cases are then enumerated directly to show that only k=3 and k=4 are possible, with the surviving pairs generated from the Fibonacci recurrence. No parameter is fitted to data and then renamed as a prediction, no self-citation supplies a uniqueness theorem that forbids alternatives, and the descent step is justified by the quadratic formula and positivity/integrality preservation rather than by re-using the target classification. The connection to Fibonacci numbers emerges from the recurrence satisfied by the minimal solutions, not by prior assumption. The argument is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper rests on the applicability of Vieta jumping as a descent tool for this equation and on basic properties of positive integers and gcd.

axioms (1)

domain assumption Vieta jumping produces a strictly smaller positive integer solution from any given solution, allowing complete classification by descent.
Invoked to classify all pairs and reach the Fibonacci relation.

pith-pipeline@v0.9.0 · 5663 in / 1200 out tokens · 44619 ms · 2026-05-20T07:31:44.041642+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 washburn_uniqueness_aczel; phi_golden_ratio matches

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matches
MATCHES: this paper passage directly uses, restates, or depends on the cited Recognition theorem or module.

a_n = F_{2n-1}+1, b=F_{2n+1}+1; characteristic polynomial x²-3x+1 whose roots are (3±√5)/2
IndisputableMonolith/Foundation/ArithmeticFromLogic.lean embed_eq_pow; LogicNat recovery 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.

a+b / gcd(a,b)² ∈ {1,2,3,5}

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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.