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

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.

• The paper derives explicit closed-form expressions for counting non-negative solutions of ax+by+cz=n when coefficients are consecutive Fibonacci or Lucas numbers.
• It leverages Cassini’s identity and modular inverse computations to simplify complex floor function summations in three-variable linear Diophantine equations.
• Numerical examples validate the theory, while the study highlights implications for partition theory, coding, and further generalizations in combinatorics.

Exact Enumeration of Non-Negative Solutions to Linear Equations with Fibonacci and Lucas Triplets

Introduction and Motivation

The paper “The Number of Solutions to $ax+by+cz=n$ for Fibonacci and Lucas Triplets” (2604.10294) addresses the classical problem of enumerating the number of non-negative integer solutions to linear equations of the form $ax + by + cz = n$, restricting the coefficients $a, b, c$ to families of three consecutive Fibonacci or Lucas numbers. While the general solution-counting framework for two variables is well understood through the works of Tripathi, the extension to three variables, specifically with such algebraically structured coefficients, presents notable challenges due to the complexity arising from interactions among the variables and floor function sums.

Binner previously provided a general formula for three-variable equations, involving intricate summations over floor functions and a reliance on modular inverses. However, these formulas are not explicit in most instances because closed forms for the arising floor function sums are generally unknown. The present paper advances this line of research by leveraging algebraic identities among Fibonacci and Lucas numbers to obtain fully explicit, closed-form expressions for the number of non-negative integer solutions for all such parameter settings.

Mathematical Background

For two-variable cases, the number of solutions to $ax + by = n$ is characterized via the formula

$N(a, b; n) = \frac{n + a a'(n) + b b'(n)}{ab} - 1,$

where $a'(n)$ and $b'(n)$ are modular inverses given specific congruence constraints. Tripathi’s result is thus explicit and its applicability relies only on $a, b$ being coprime and $n$ divisible by their greatest common divisor.

Binner’s extension to the three-variable equation $ax + by + cz = n$ introduces a substantially more complex structure:

$ax + by + cz = n$0

where $ax + by + cz = n$1 is a quadratic and cubic expression in $ax + by + cz = n$2 and the $ax + by + cz = n$3 are sums involving floor functions whose summation limits and arguments are determined via modular arithmetic involving $ax + by + cz = n$4.

The difficulty lies in the lack of closed formulas for the floor function sums, which is the obstacle this paper overcomes for specific families.

Main Contributions

Explicit Formulae for Fibonacci Triplets

For $ax + by + cz = n$5 being consecutive Fibonacci numbers, the author exploits Cassini’s identity and the divisibility properties of the sequence to compute all necessary modular inverses in closed form. As a consequence, it becomes possible to resolve all floor function sums, as they simplify to zeros or expressions with a parameterizable quadratic closed form.

The final formula for the number of non-negative integer solutions to $ax + by + cz = n$6 is:

$ax + by + cz = n$7

where $ax + by + cz = n$8 and $ax + by + cz = n$9 are explicitly computable from $a, b, c$0 and $a, b, c$1 using previously encoded recurrences and modular inverse relations deduced from Fibonacci properties.

All occurrences of modular inverses simplify because, for Fibonacci numbers, modular inverses of consecutive terms can be expressed using shifted indices and alternating signs, making the resulting floor function sums tractable due to their periodicity and the division properties among these special sequences.

Explicit Formulae for Lucas Triplets

Analogous results are derived for consecutive Lucas numbers $a, b, c$2. Using the Lucas analog of Cassini’s identity, the necessary modular inverses are computed in terms of $a, b, c$3 and explicit divisions by 5, due to the discriminant of the Lucas characteristic equation. The explicit solution count then has the same structural form as for the Fibonacci case but with coefficients and congruences adjusted for the Lucas sequence.

Modular inverses involving 5 are treated with residue computations and depend on $a, b, c$4 due to periodicity modulo 5 in the Lucas sequence, which the author carefully analyzes.

Numerical Results and Examples

The paper provides concrete numerical examples. For instance, in the case $a, b, c$5, $a, b, c$6, $a, b, c$7, and $a, b, c$8, the explicit solution count is calculated to be $a, b, c$9; similarly, for Lucas triplets with $ax + by = n$0, $ax + by = n$1, $ax + by = n$2, and the same $ax + by = n$3, the count is $ax + by = n$4. The method generalizes uniformly for all $ax + by = n$5 and $ax + by = n$6 as long as $ax + by = n$7 (or their Lucas analogs) are coprime, as is always the case with three consecutive terms for $ax + by = n$8.

Theoretical and Practical Implications

These results significantly advance the explicit enumeration of restricted partitions and compositions in the context of structured coefficients, directly tying the computational combinatorics of integer solutions to classical sequences. The explicit forms are enabled by the recurrence and modular properties intrinsic to Fibonacci and Lucas numbers, which are unavailable in generic tuples.

This advancement not only completes the explicit solution landscape for the three-variable case in these settings, but also gives insight into potential further generalizations. For example, one could consider related explicit formulas for other recursive sequences with known modular inverse properties or analyze the asymptotic growth of the number of solutions as $ax + by = n$9 and $N(a, b; n) = \frac{n + a a'(n) + b b'(n)}{ab} - 1,$0 scale.

Moreover, the interplay between the algebraic identities of the coefficient sequences and the enumeration of solutions to partition-type Diophantine equations is of substantial interest both in number theory and its applications in combinatorics and coding theory. It opens avenues for tighter bounds or explicit expressions in similar families (e.g., generalized Fibonacci or Lucas sequences, Tribonacci, etc.) or higher-dimensional versions.

Future Directions

Several research directions naturally follow from these explicit formulas:

• Generalization to Multivariate Cases: Extending the methodology to $N(a, b; n) = \frac{n + a a'(n) + b b'(n)}{ab} - 1,$1-term recurrences or other linear recurrence sequences and analyzing when closed-form expressions are available.
• Algorithmic Implementation: Efficient algorithms, possibly in symbolic computation systems, to evaluate the solution counts for large $N(a, b; n) = \frac{n + a a'(n) + b b'(n)}{ab} - 1,$2 based on the explicit forms derived.
• Connection with Frobenius Numbers: Deepening the link with the Frobenius problem, particularly for more complex semigroups generated by recursive sequences, and exploring associated semigroup invariants.
• Spectral and Asymptotic Analysis: Studying the asymptotic behavior and distribution of solution numbers as functions of $N(a, b; n) = \frac{n + a a'(n) + b b'(n)}{ab} - 1,$3 and $N(a, b; n) = \frac{n + a a'(n) + b b'(n)}{ab} - 1,$4; exploring random walks or probabilistic interpretations for compositions with Fibonacci or Lucas summands.
• Applications to Coding and Cryptography: The structure and enumeration properties may underpin schemes in coding theory, discrete optimization, or cryptographic primitives where solution sparsity or density is crucial.

Conclusion

The paper (2604.10294) successfully derives exact, closed-form expressions for the enumeration of non-negative integer solutions to $N(a, b; n) = \frac{n + a a'(n) + b b'(n)}{ab} - 1,$5 in the highly structured cases where $N(a, b; n) = \frac{n + a a'(n) + b b'(n)}{ab} - 1,$6 are consecutive Fibonacci or Lucas numbers. The result extends general solution-count formulas to explicit expressions, eliminating the need for floor function summations via exploitation of Fibonacci and Lucas recurrence properties and modular inverse identities. These findings enrich the theory of linear Diophantine equations and structured integer partitions and suggest further fruitful exploration both in theory and applications.