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

This paper derives exact formulas for the number of non-negative integer solutions to the linear Diophantine equation $ax+by+cz=n$ when the coefficients $a, b, c$ are three consecutive Fibonacci or Lucas numbers, thereby resolving previously unsolved summations of floor functions found in Binner's earlier reciprocity-based approach.

The Gist

Imagine you are a baker trying to sell boxes of cookies. You have three types of boxes:

• Small boxes that hold $a$ cookies.
• Medium boxes that hold $b$ cookies.
• Large boxes that hold $c$ cookies.

You have a huge pile of $n$ cookies, and you want to know: How many different ways can you pack exactly $n$ cookies into these boxes using only whole numbers of boxes?

This is a classic math puzzle called the "Coin Problem" or "Frobenius Coin Problem." For a long time, mathematicians had a way to solve this, but the answer was like a messy recipe with a huge list of steps (sums of floor functions) that was hard to calculate quickly. It was like trying to count every single grain of sand on a beach by picking them up one by one.

The Big Breakthrough

In 2020, a mathematician named Binner gave us a better recipe, but it still required doing a lot of repetitive counting. The author of this paper, Pooja Teotia, asked a clever question: "What if the sizes of our boxes ($a, b, c$) aren't just random numbers, but follow a special pattern?"

She looked at two famous number patterns:

1. Fibonacci Numbers: 1, 1, 2, 3, 5, 8, 13... (where each number is the sum of the two before it).
2. Lucas Numbers: 2, 1, 3, 4, 7, 11, 18... (a similar pattern starting with different numbers).

Teotia discovered that if your box sizes are three consecutive numbers from these patterns (like 5, 8, 13 or 11, 18, 29), the messy recipe disappears! The complex counting steps cancel each other out, leaving you with a clean, simple formula.

The Magic of the "Special Boxes"

Why do these specific numbers work so well?

Think of the numbers in the Fibonacci and Lucas sequences as having a secret handshake. Because of how they are built, they have a special mathematical relationship (called "modular inverses") that makes them "talk" to each other perfectly.

In the paper, Teotia shows that when you use these special triplets:

• The complicated "counting loops" in the formula turn into zeros.
• One part of the formula simplifies into a neat triangle number (like stacking blocks).
• The rest becomes a straightforward calculation.

It's like finding a shortcut through a dense forest. Instead of hacking your way through the trees (doing hundreds of calculations), you find a hidden path that takes you straight to the destination.

The Result: A Crystal Clear Answer

The paper provides two "magic spells" (formulas):

1. For Fibonacci Triplets: If your boxes hold $F_i, F_{i+1}, F_{i+2}$ cookies, here is the exact number of ways to pack $n$ cookies.
2. For Lucas Triplets: If your boxes hold $L_i, L_{i+1}, L_{i+2}$ cookies, here is the exact number of ways.

Example from the paper:
Imagine you have boxes of size 144, 233, and 377 (which are three consecutive Fibonacci numbers). You have 425,896 cookies.

• Before this paper: A computer would have to run a slow, complex program to count the possibilities.
• With this paper: You can plug the numbers into Teotia's new formula and instantly get the answer: 7,178 different ways to pack the cookies.

Why Does This Matter?

While packing cookies might seem like a simple game, this kind of math is used in:

• Cryptography: Securing your online banking.
• Computer Science: Optimizing how data is stored.
• Physics: Understanding how particles arrange themselves.

By finding these "exact formulas" for special cases, Teotia has given mathematicians and scientists a powerful new tool. She didn't just solve one puzzle; she showed us that when nature follows a pattern (like the Fibonacci sequence), the universe often hides a simpler, more beautiful solution waiting to be found.

In short: The paper takes a messy, hard-to-solve math problem and shows that if you choose your numbers carefully (using Fibonacci or Lucas sequences), the answer becomes as clear and simple as counting your fingers.

Technical Summary

1. Problem Statement

The paper addresses the classic problem in number theory of finding the number of non-negative integer solutions, denoted as $N(a, b, c; n)$, to the linear Diophantine equation:
$$ax + by + cz = n$$
where $a, b, c,$ and $n$ are natural numbers.

While the case for two variables ($ax + by = n$) has well-known exact formulas (e.g., Tripathi, 2000), the case for three variables is significantly more complex. Previous work by Binner (2020) provided a formula for $N(a, b, c; n)$, but it relied on summations of floor functions (specifically $\sum \lfloor \frac{ic'}{a} \rfloor$). These sums generally lack closed-form solutions and require complex reciprocity relations to evaluate, making the computation of exact values difficult for general coefficients.

The Core Objective: The author aims to derive exact, closed-form formulas (without summations) for the number of solutions in specific cases where the coefficients $(a, b, c)$ are three consecutive Fibonacci numbers or three consecutive Lucas numbers.

2. Methodology

The methodology builds upon Binner's 2020 framework but exploits the unique algebraic properties of Fibonacci and Lucas sequences to simplify the general formula.

A. Foundation: Binner's General Formula

The paper utilizes Binner's Theorem 1.2, which expresses the number of solutions as:
$$N(a, b, c; n) = \frac{N_1}{2abc} + \sum_{i=1}^{b'_1-1} \left\lfloor \frac{ic'_1}{a} \right\rfloor + \sum_{i=1}^{c'_2-1} \left\lfloor \frac{ia'_2}{b} \right\rfloor + \sum_{i=1}^{a'_3-1} \left\lfloor \frac{ib'_3}{c} \right\rfloor - 2$$
Here, $N_1$ is a polynomial term involving $n$ and modular inverses ($b'_1, c'_2, a'_3, c'_1, a'_2, b'_3$). The difficulty lies in evaluating the three summation terms.

B. Specialization to Fibonacci and Lucas Triplets

The author sets $a, b, c$ to be consecutive terms of the sequences:

• Fibonacci: $F_i, F_{i+1}, F_{i+2}$
• Lucas: $L_i, L_{i+1}, L_{i+2}$

The key to the derivation is the application of Cassini's Identity (and its Lucas variant) to simplify the modular inverses required in Binner's formula.

1. Simplification of Modular Inverses:

   • For Fibonacci numbers, the identity $F_{k+1}F_{k-1} - F_k^2 = (-1)^k$ allows the author to explicitly calculate inverses like $F_{i+1}^{-1} \pmod{F_i}$ without complex algorithms.
   • For Lucas numbers, the identity $L_k^2 - L_{k-1}L_{k+1} = 5(-1)^k$ is used. This introduces a factor of 5, requiring the calculation of $5^{-1} \pmod{L_k}$, which depends on $k \pmod 4$.

2. Evaluation of Summations:
   By substituting the specific Fibonacci/Lucas values into the definitions of $c'_1, a'_2,$ and $b'_3$, the author proves that:

   • $c'_1 = 1$ and $a'_2 = 1$.
   • $b'_3 = c - 1$ (where $c$ is the largest coefficient).

   These specific values cause the first two summation terms in Binner's formula to vanish (become 0) because $\lfloor \frac{i}{a} \rfloor = 0$ for $i < a$. The third summation simplifies to a standard arithmetic series $\sum (j-1)$, which yields a closed quadratic form.

3. Key Contributions and Results

A. Exact Formula for Fibonacci Triplets

Theorem 2.1: For fixed natural numbers $i$ and $n$, the number of non-negative integer solutions to $F_i x + F_{i+1} y + F_{i+2} z = n$ is:
$$N(F_i, F_{i+1}, F_{i+2}; n) = \frac{N_2}{2F_i F_{i+1} F_{i+2}} + \frac{(A'_3 - 1)(A'_3 - 2)}{2} - 2$$
Where:

• $A'_3 \equiv (-1)^i n F_{i+1} \pmod{F_{i+2}}$
• $N_2$ is a specific polynomial expression involving $n$ and the modular residues $B'_1, C'_2, A'_3$.
• Significance: This removes the need for floor function summations, providing a direct calculation.

B. Exact Formula for Lucas Triplets

Theorem 3.1: Similarly, for $L_i x + L_{i+1} y + L_{i+2} z = n$, the number of solutions is:
$$N(L_i, L_{i+1}, L_{i+2}; n) = \frac{N_3}{2L_i L_{i+1} L_{i+2}} + \frac{(A''_3 - 1)(A''_3 - 2)}{2} - 2$$
Where the modular residues ($B''_1, C''_2, A''_3$) involve the term $5^{-1} \pmod L_k$, handled via case analysis based on $k \pmod 4$.

C. Numerical Verification

The paper provides concrete examples to validate the formulas:

• Fibonacci Case: For $i=12$ ($F_{12}=144, F_{13}=233, F_{14}=377$) and $n=425896$, the formula yields exactly 7,178 solutions.
• Lucas Case: For $i=10$ ($L_{10}=123, L_{11}=199, L_{12}=322$) and $n=394072$, the formula yields exactly 9,532 solutions.

4. Significance

1. Resolution of Open Sums: The paper demonstrates that while Binner's general formula involves unsolvable summations for arbitrary coefficients, these sums collapse into simple closed forms for Fibonacci and Lucas triplets. This is a rare instance of finding an exact formula for the 3-variable Frobenius coin problem without summation.
2. Algebraic Insight: It highlights the deep connection between the arithmetic properties of linear Diophantine equations and the recursive identities of number sequences (Cassini's identity).
3. Computational Efficiency: The derived formulas allow for the immediate calculation of solution counts for large $n$ without iterative algorithms or complex reciprocity relations, which is valuable for computational number theory and combinatorics.

Conclusion

Pooja Teotia successfully extends previous work on the 3-variable linear Diophantine equation by identifying a special class of coefficients (consecutive Fibonacci and Lucas numbers) where the number of solutions can be determined via an exact, non-summation formula. This work bridges the gap between general theoretical bounds and practical, exact computation for specific, mathematically significant sequences.