An invitation to Fibonacci digits

The purpose of this short note is to show the interplay between math outreach and conducting original research, in particular how each can build off the other.

💡 Research Summary

This paper, “An invitation to Fibonacci digits,” presents a compelling case study on the symbiotic relationship between mathematical outreach and original research. It begins by detailing a successful outreach activity conducted by the authors in elementary schools within the Mount Greylock Regional School District. The central activity is the “I Love Rectangles” game, where students are given an infinite supply of squares of sizes 1x1, 2x2, 3x3, and so on. The challenge is to place as many squares as possible to form a connected rectangle. Students quickly discover that starting with two 1x1 squares is necessary, and the sequence of square sizes required to continue building the rectangle—2, 3, 5, 8, 13…—reveals the Fibonacci sequence. This hands-on discovery leads to discussions on patterns (odd, odd, even), recursive definitions, and even geometric proofs of Fibonacci identities. The authors emphasize the pedagogical importance of finding the “right perspective,” using an analogy with a non-standard Rubik’s cube that, when viewed from an angle, resembles a standard one.

The paper then pivots to demonstrate how such outreach can directly inspire new research questions. During a summer research program (SMALL at Williams College), a student observed that the first few Fibonacci numbers seem to always contain one of the digits 1, 2, 3, 5, or 8 (the single-digit Fibonacci numbers). This sparked a series of investigative questions: For a given base, what subsets of digits must appear in every Fibonacci number? Does this hold for all sufficiently large Fibonacci numbers if we allow finitely many exceptions? What would a probabilistic model predict about digit appearance?

To tackle the deterministic version of this problem, the authors employ a blend of computational experimentation and theoretical number theory. A key tool is Jarden’s Theorem, which states that the last n digits of the Fibonacci sequence repeat with a period of 15·10^(n-1). This periodicity allows the infinite problem to be reduced to checking a finite set of numbers. The authors complement this with a proof of a more foundational lemma: for any modulus m, the Fibonacci sequence modulo m is periodic with a period (the Pisano period) of at most m²+1, proven elegantly using the Pigeonhole Principle.

Through computer searches leveraging this periodicity, they obtain partial results. They prove that every Fibonacci number contains a digit from the set {0,1,2,3,4,5,7,8,9} (i.e., all digits except 6). Similar results hold for sets excluding 4 or 2 (with the trivial exception of F₂=2 itself). However, the original conjecture about the set {1,2,3,5,8} remains open. They identify specific Fibonacci numbers, like F₂₁=10946 and F₃₀₀, where none of these digits appear in the last several positions, demonstrating that a simple check of the last few digits is insufficient to prove or disprove the statement for all numbers. This highlights the complexity and allure of the problem.

The paper concludes by suggesting avenues for further research, including deeper studies of Pisano periods and probabilistic analyses of digit distribution. Overall, it masterfully illustrates a complete cycle: an engaging outreach activity fosters curiosity; that curiosity is formalized into a precise mathematical question; and standard research techniques are deployed to explore it, yielding partial answers and opening new doors. It serves as an exemplary model of how community engagement and academic inquiry can fuel and enrich each other.