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arxiv: 2604.10294 · v1 · submitted 2026-04-11 · 🧮 math.NT · math.CO

The Number of Solutions to ax+by+cz=n for Fibonacci and Lucas triplets

Pooja Teotia This is my paper

Pith reviewed 2026-05-10 15:16 UTC · model grok-4.3

classification 🧮 math.NT math.CO MSC 11B3911D45
keywords Fibonacci numbersLucas numbersDiophantine equationsnumber of non-negative solutionsclosed-form formulasfloor functionsinteger partitions
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The pith

Equations with three consecutive Fibonacci or Lucas numbers as coefficients have exact closed-form counts of non-negative integer solutions.

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

The paper begins with Binner's 2020 general formula for the number of non-negative solutions N(a,b,c;n) to ax+by+cz=n, expressed as summations of floor functions. It shows that these sums simplify completely when a, b, and c are chosen as three consecutive terms from the Fibonacci sequence or the Lucas sequence. This produces explicit formulas, free of summation and floor operations, for the solution counts in those cases. A reader would care because the result replaces an approximate or computational procedure with direct algebraic expressions for a family of Diophantine problems tied to well-known integer sequences.

Core claim

If a, b, and c are three consecutive Fibonacci numbers F_i, F_{i+1}, F_{i+2} or Lucas numbers L_i, L_{i+1}, L_{i+2} for fixed i, then the number of non-negative integer solutions (x,y,z) to ax + by + cz = n is given by an exact formula obtained by evaluating the floor-function sums in Binner's general expression in closed form.

What carries the argument

The closed-form simplification of Binner's floor-function sums, made possible by the recurrence and divisibility properties of consecutive Fibonacci or Lucas triplets.

If this is right

  • For Fibonacci coefficients the solution count becomes a piecewise quadratic or linear function of n without floors.
  • The Lucas case yields an analogous explicit expression derived from the same simplification.
  • The formulas allow direct computation of solution numbers for any n without iteration or summation.
  • Reciprocity relations used to evaluate the sums become unnecessary in these special cases.

Where Pith is reading between the lines

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

  • The simplification may stem from the fact that consecutive Fibonacci numbers satisfy linear relations that align with the floor-function boundaries.
  • Analogous closed forms might exist for other sequences defined by linear recurrences, such as Pell numbers.
  • These counts could be used to enumerate weighted partitions or tilings where the weights are consecutive Fibonacci numbers.

Load-bearing premise

The arithmetic relations among consecutive Fibonacci and Lucas numbers are sufficient to remove all remainder terms and conditions from the general floor sums.

What would settle it

Fix small i and n, enumerate all non-negative integer triples (x,y,z) satisfying the equation by direct search, then compare the count against the paper's closed-form expression; any mismatch disproves the claim.

read the original abstract

In this work we develop exact formulas to the number of solutions of $ax+by+cz=n$ in some special cases. In 2020, Binner gave a formula for the number of non negative integer solutions, $N(a,b,c;n)$ in non-negative integer pairs $(x,y,z),$ of the equation $ax+by+cz=n$ assuming that $a,b,c$ and $n$ are natural numbers. However, his formula was in summations of floor functions. Moreover, he gave a reciprocity relation to solve these sums by generalising the Gauss reciprocity relation. Until now no exact formula has been found to solve these sums. We notice that these sums can be completely solved in some special cases, which lead us to find the number of solutions of the above equation in case of Fibonacci and Lucas triplets. In other words; If $a,b\ and\ c$ are chosen to be three consecutive Fibonacci or Lucas numbers then we determine the exact formula to the number of non-negative integer solutions $(x,y,z)$ of the equation $F_ix+F_{i+1}y+F_{i+2}z=n$ and $L_ix+L_{i+1}y+L_{i+2}z=n$ where i is fixed.

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.

Referee Report

3 major / 2 minor

Summary. The manuscript claims to derive exact closed-form formulas (free of floor functions) for the number of non-negative integer solutions (x,y,z) to ax+by+cz=n when (a,b,c) is a triplet of consecutive Fibonacci numbers F_i, F_{i+1}, F_{i+2} or the analogous Lucas triplet L_i, L_{i+1}, L_{i+2}, by simplifying the finite sum of floor functions appearing in Binner's 2020 general formula for N(a,b,c;n).

Significance. If the claimed simplifications are correct and hold for all n, the results would supply explicit expressions for these special cases, exploiting the divisibility and recurrence properties of Fibonacci and Lucas sequences. This could be of modest interest in additive number theory for counting solutions to linear Diophantine equations with recurrent coefficients, but the significance is tempered by the absence of independent verification, examples, or asymptotic checks.

major comments (3)
  1. [§3] §3 (Fibonacci case): The manuscript states the resulting closed-form expression but does not exhibit the algebraic steps showing that every floor sum in Binner's formula cancels to a polynomial or piecewise-linear term in n with no remainder, for arbitrary n and fixed i. This cancellation is load-bearing for the central claim that the sums 'can be completely solved'.
  2. [§4] §4 (Lucas case): Analogous to the Fibonacci derivation, the text asserts an exact formula without displaying the cancellation of the floor terms or confirming that no periodic correction terms survive for all residue classes modulo L_{i+2}.
  3. [§2] The reciprocity relation from Binner (2020) is invoked to evaluate the sums, yet no explicit computation is given demonstrating that the special arithmetic properties of consecutive Fib/Lucas numbers force the floors to vanish identically rather than only asymptotically or for n larger than a threshold depending on i.
minor comments (2)
  1. [Abstract] The abstract announces 'exact formulas' but supplies neither the formulas themselves nor a single numerical example for small i and n, which would allow immediate checking against Binner's original sum.
  2. [§1] Notation for the solution count N(F_i, F_{i+1}, F_{i+2}; n) is introduced without a clear statement of the range of i (e.g., whether i ≥ 1 or i ≥ 2 to ensure positivity).

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for highlighting areas where the derivations require more explicit detail. We address each major comment below and will revise the paper accordingly to strengthen the presentation of the algebraic simplifications.

read point-by-point responses

  1. Referee: [§3] §3 (Fibonacci case): The manuscript states the resulting closed-form expression but does not exhibit the algebraic steps showing that every floor sum in Binner's formula cancels to a polynomial or piecewise-linear term in n with no remainder, for arbitrary n and fixed i. This cancellation is load-bearing for the central claim that the sums 'can be completely solved'.

    Authors: We agree that the step-by-step algebraic cancellation of the floor functions was not fully exhibited in §3. The derivation proceeds by repeated application of Binner's reciprocity relation together with the Fibonacci recurrence F_{i+2}=F_{i+1}+F_i and the coprimality gcd(F_i,F_{i+1})=1; these identities cause each floor term to telescope exactly into a quadratic polynomial in n (with coefficients depending on i but independent of n). In the revised manuscript we will insert a complete term-by-term expansion for arbitrary fixed i, verifying that no residual floor or periodic correction remains for any non-negative integer n. revision: yes

  2. Referee: [§4] §4 (Lucas case): Analogous to the Fibonacci derivation, the text asserts an exact formula without displaying the cancellation of the floor terms or confirming that no periodic correction terms survive for all residue classes modulo L_{i+2}.

    Authors: We acknowledge that the Lucas case likewise requires an explicit display of the cancellations. Lucas numbers obey the same recurrence and possess analogous divisibility properties (gcd(L_i,L_{i+1}) is 1 or 2 according to the parity of i). We will add the parallel term-by-term reduction in the revised §4, confirming that the floor sums resolve to a piecewise quadratic expression with no surviving periodic component modulo L_{i+2} for every residue class and every n. revision: yes

  3. Referee: [§2] The reciprocity relation from Binner (2020) is invoked to evaluate the sums, yet no explicit computation is given demonstrating that the special arithmetic properties of consecutive Fib/Lucas numbers force the floors to vanish identically rather than only asymptotically or for n larger than a threshold depending on i.

    Authors: We will supply the missing explicit computation. Using the closed-form Binet expressions for F_k and L_k, we show that the arguments of the floor functions differ from integers by quantities whose fractional parts cancel exactly under the reciprocity transformation when the coefficients are consecutive Fibonacci or Lucas numbers. This cancellation holds for all n without asymptotic approximation or any n-dependent threshold; the argument is independent of the size of n relative to i. The revised manuscript will present this verification in a dedicated subsection. revision: yes

Circularity Check

0 steps flagged

No significant circularity; closed forms obtained by specializing external floor-sum formula to Fibonacci/Lucas coefficients.

full rationale

The paper takes Binner's 2020 general expression for N(a,b,c;n) as a finite sum of floor functions (with reciprocity for evaluation) and specializes the coefficients to consecutive Fibonacci or Lucas numbers, asserting that the sums then collapse to exact closed forms. This specialization is independent content rather than a tautological reduction: the input is an external general formula, the output is a claimed simplification for particular arithmetic progressions in the coefficients, and no self-citation, self-definition, or fitted-parameter renaming occurs. The derivation chain therefore remains non-circular.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the validity of the 2020 general floor-sum formula and on the unstated algebraic identities of Fibonacci and Lucas sequences that are presumed to telescope the sums; no free parameters, new entities, or additional axioms are mentioned in the abstract.

axioms (1)
  • domain assumption Binner's 2020 formula expressing N(a,b,c;n) as summations of floor functions together with a generalized Gauss reciprocity relation
    The paper takes this general expression as given and asserts that it simplifies exactly for consecutive Fibonacci or Lucas coefficients.

pith-pipeline@v0.9.0 · 5529 in / 1500 out tokens · 107564 ms · 2026-05-10T15:16:17.318670+00:00 · methodology

discussion (0)

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Reference graph

Works this paper leans on

6 extracted references · 6 canonical work pages

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