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On a Diophantine Equation with Jacobsthal and Fibonacci Numbers

This paper proves that the only Jacobsthal number expressible as the product of three distinct Fibonacci numbers $F_k, F_l, F_m$ with $2 < k < l < m$ is $J_{12} = F_5 F_7 F_8$ , a result established using linear forms in logarithms.

Original authors: Daeyeoul Kim, Zekiye Pinar Cihan, Zeynep Demirkol Ozkaya, Ilker Inam

Published 2026-07-24

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CC BY 4.0

Original authors: Daeyeoul Kim, Zekiye Pinar Cihan, Zeynep Demirkol Ozkaya, Ilker Inam

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

The Great Number Hunt: When Sequences Collide

Imagine a world where numbers aren't just tools for counting your allowance or calculating the score in a video game, but characters in their own right, each with a unique personality and a strict set of rules for how they grow. In the vast, quiet library of mathematics, there are two famous families of numbers that love to follow a simple recipe to create the next number in their line. The first family is the Fibonacci numbers. They start with 0 and 1, and every new member is just the sum of the two parents before it (0, 1, 1, 2, 3, 5, 8, 13...). They are the golden ratio's favorite children, showing up in sunflowers, seashells, and spiral galaxies. The second family is the Jacobsthal numbers. They are a bit more energetic; they also start with 0 and 1, but their recipe is different: they take the previous number, double it, and add the one before that (0, 1, 1, 3, 5, 11, 21...).

Mathematicians have spent centuries studying these families, asking questions like, "Can a Fibonacci number ever be a perfect square?" or "Do these two families ever meet?" Usually, these number families march to the beat of their own drums, rarely overlapping in surprising ways. But sometimes, they collide. The question this paper tackles is a bit like a culinary challenge: If you take three members from the Fibonacci family and multiply them together, can the result ever be exactly equal to a single member from the Jacobsthal family? It's a search for a very specific, very rare coincidence in the infinite ocean of numbers.

The One-in-a-Million Match

In this paper, the authors, DaeYeoul Kim and their team, set out to solve this specific puzzle. They wanted to find every possible instance where the product of three distinct Fibonacci numbers equals a Jacobsthal number. To do this, they didn't just guess and check; they used a powerful mathematical toolkit called "linear forms in logarithms." You can think of this as a super-precise ruler that measures the distance between numbers so accurately that it can tell you if two numbers are ever close enough to be the same, even if they are astronomically large.

The team started by setting up a strict rule: the three Fibonacci numbers had to be different and increasing in size (let's call them $F_k$ , $F_l$ , and $F_m$ where $k < l < m$ ). They then used their mathematical ruler to prove that the numbers in this equation couldn't be just any size. They showed that if a solution existed, the numbers involved couldn't be infinitely large; they had to be smaller than a specific, albeit huge, limit (around $10^{42}$ ). This was like narrowing down a search for a lost coin from "anywhere in the universe" to "somewhere in this specific city."

Once they had this massive but finite range, they used computer algorithms to shrink the search area even further, down to a manageable size where they could check every possibility by hand (or rather, by computer). After all the crunching, they found something remarkable: there is exactly one solution that fits the rules.

The only time this magical equation works is when you take the 5th, 7th, and 8th Fibonacci numbers and multiply them together.

The 5th Fibonacci number is 5.
The 7th Fibonacci number is 13.
The 8th Fibonacci number is 21.

When you multiply them ( $5 \times 13 \times 21$ ), you get 1365. And guess what? The 12th Jacobsthal number is also 1365.

The paper proves with absolute certainty that this is the only time this happens for three distinct Fibonacci numbers in increasing order. There are no other hidden matches waiting to be found. The authors have effectively closed the book on this specific question, showing that while the Fibonacci and Jacobsthal families are vast and full of patterns, this particular three-way handshake happens only once in the history of numbers.

Technical Summary: On a Diophantine Equation with Jacobsthal and Fibonacci Numbers

Problem Statement
This paper investigates the Diophantine equation $F_k F_l F_m = J_n$ , where $F_n$ denotes the $n$ -th Fibonacci number and $J_n$ denotes the $n$ -th Jacobsthal number. The authors seek to identify all positive integer solutions $(k, l, m, n)$ satisfying the condition $2 < k < l < m$ . The study focuses on the multiplicative interplay between two distinct recursive sequences: the Fibonacci sequence, defined by $F_{n+2} = F_{n+1} + F_n$ , and the Jacobsthal sequence, defined by $J_{n+2} = 2J_{n+1} + J_n$ .

Methodology
The proof employs techniques involving linear forms in logarithms of algebraic numbers, specifically utilizing Matveev's theorem to establish upper bounds on the variables. The methodology proceeds through the following stages:

Binet Formulas and Inequalities: The authors express the Fibonacci and Jacobsthal numbers using their respective Binet formulas. By substituting these into the main equation, they derive inequalities relating the sum of the indices $k+l+m$ to the index $n$ . Specifically, they establish that $n$ is approximately proportional to $(k+l+m) \frac{\log \alpha}{\log 2}$ , where $\alpha = \frac{1+\sqrt{5}}{2}$ .
Linear Forms Construction: The equation is rearranged to isolate a linear form in logarithms, denoted as $\Lambda$ . The authors construct three distinct linear forms ( $\Lambda_1, \Lambda_2, \Lambda_3$ ) by fixing different subsets of the variables ( $k$ , then $k$ and $l$ , then all three) and dividing the rearranged equation by dominant terms.
Application of Matveev's Theorem: For each linear form, the authors verify that the form is non-zero (using Galois automorphisms and rational/irrational part comparisons in $\mathbb{Q}(\sqrt{5})$ ). They then apply Matveev's lower bound theorem to obtain an inequality of the form $\log |\Lambda| > -C(1 + \log B)$ , where $B$ is a bound on the exponents.
Upper Bound Derivation: By comparing the lower bound from Matveev's theorem with the upper bound derived from the original equation's structure (which involves terms like $\alpha^{-2k}$ ), the authors derive a massive initial upper bound for the variable $m$ , specifically $m < 5.93 \times 10^{42}$ .
Reduction of Bounds: Recognizing that the initial bound is computationally intractable, the authors apply Lemma 3.3 (a result related to continued fractions and the distance to the nearest integer) to reduce the bounds. By analyzing the continued fraction expansion of $\tau = \frac{\log 2}{\log \alpha}$ and using computational tools (SageMath and Mathematica), they iteratively reduce the bounds for $k$ , $l$ , and $m$ to manageable sizes ( $k \le 110$ , $l \le 114$ , $m \le 236$ ).

Key Results
The paper establishes the following main theorem:

There is exactly one non-trivial solution to the Diophantine equation $F_k F_l F_m = J_n$ under the constraint $2 < k < l < m$ .
The unique solution is $(k, l, m, n) = (5, 7, 8, 12)$ .
This solution corresponds to the identity: $F_5 F_7 F_8 = 5 \cdot 13 \cdot 21 = 1365 = J_{12}$ .

Significance and Context
The authors position this work within the context of recent studies on multiplicative relations between recursive sequences. They cite previous works that identified Tribonacci numbers as products of two Fibonacci numbers and Fibonacci numbers as products of two Jacobsthal numbers. The significance of this paper lies in extending these investigations to a more complex equation involving three Fibonacci factors. The authors state that understanding such multiplicative relations provides insight into the "arithmetic rigidity and coincidence structure" of distinct recursive sequences. The paper does not propose new applications or future experimental directions but focuses strictly on the complete resolution of the specific Diophantine equation presented.

The Great Number Hunt: When Sequences Collide

The One-in-a-Million Match

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