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A supercongruence fantasy on Fibonacci and Lucas (after Guillera)

Motivated by Guillera's observations, this paper generalizes Ramanujan-type supercongruences to a deeper level where Fibonacci, Lucas, and Apéry numbers naturally emerge.

Original authors: Wadim Zudilin

Published 2026-02-09

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Wadim Zudilin

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

Imagine you are looking at a set of magical recipes that mathematicians use to calculate the value of Pi ( $\pi$ ). For over a century, we've known some of these recipes, written down by the legendary Indian mathematician Srinivasa Ramanujan. These recipes are like infinite sums: you add up an endless list of numbers, and if you do it right, the total gets closer and closer to a specific value involving $\pi$ .

Recently, a mathematician named Jesús Guillera noticed something fascinating. He took one of Ramanujan's "irrational" recipes (one involving square roots) and combined it with a "conjugate" version of itself. When he mixed them, the messy square roots canceled out, leaving behind two brand-new, "rational" recipes. But here's the twist: these new recipes didn't just use simple numbers; they used Fibonacci and Lucas numbers (the famous sequence where each number is the sum of the two before it: 1, 1, 2, 3, 5, 8...).

The "Fantasy" of Supercongruences

The author of this paper, Wadim Zudilin, is essentially saying: "Let's see if we can find a hidden pattern in these new recipes."

In the world of numbers, there is a concept called a congruence. Think of it like a clock. If you look at the time 13:00, it's the same as 1:00 on a 12-hour clock. In math, we say $13 \equiv 1 \pmod{12}$ .

A supercongruence is like a super-accurate clock. It's a rule that says two huge, complicated sums of numbers are equal to each other not just on a normal clock, but on a clock with a very high number of hours (specifically, powers of prime numbers).

Zudilin is proposing a "fantasy" (a strong mathematical guess) that these new Fibonacci-based recipes follow a very specific, rigid pattern.

The Core Idea: The "Echo" Effect

The paper suggests that if you take a partial sum of these infinite recipes (stopping after $N$ terms instead of going to infinity), the result behaves in a predictable way when you change the stopping point.

Imagine you are stacking blocks to build a tower.

The Normal Rule: If you build a tower of height $p$ (where $p$ is a prime number), the top block has a certain property.
The Supercongruence Rule: Zudilin guesses that if you build a tower of height $p^2$ , $p^3$ , or even $p^s$ , the top block isn't just random. It "echoes" the structure of the smaller tower ( $p^{s-1}$ ), multiplied by a specific factor related to the prime number.

The paper claims that this "echo" pattern holds true for:

Fibonacci and Lucas numbers mixed with standard binomial coefficients (combinations).
Apéry numbers (another famous sequence used to prove that certain numbers are irrational) mixed with similar structures.

The "Rationalization" Trick

The paper describes a process called "rationalization."

Step 1: Start with a complex formula involving square roots (like $\sqrt{5}$ ).
Step 2: Create a "twin" formula where the sign of the square root is flipped.
Step 3: Add or subtract these twins. The square roots disappear, and you are left with a clean formula involving only whole numbers and the Fibonacci/Lucas sequences.

Zudilin applies this trick to several known formulas, creating new series for $1/\pi$ and $1/\pi^2$ . He then checks the "truncated" versions of these series (the partial sums) to see if they obey the supercongruence rules.

What the Paper Actually Claims

Observation: The author has run computer calculations and observed that these new Fibonacci-based sums do seem to follow the supercongruence rules. For example, the sum up to a certain point is congruent to a multiple of the sum up to a smaller point, modulo a high power of a prime.
The "Fantasy": The title calls it a "fantasy" because, while the computer data is strong, a formal mathematical proof for these specific Fibonacci/Lucas cases is not yet fully established in the same way it is for simpler cases. The author is proposing that these patterns are real and universal for this class of numbers.
The Difference: The paper notes a key difference between these new formulas and the old ones. The old formulas often relied on simple powers of numbers (like $z^n$ ), which follow a simple one-step rule. The new Fibonacci-based formulas rely on sequences that follow more complex, multi-step rules. This suggests a deeper, more complicated layer of arithmetic structure.

In Summary

This paper is a mathematical "what if" story. It takes known recipes for calculating $\pi$ , mixes in the famous Fibonacci and Lucas number sequences, and proposes that these new combinations obey a hidden, high-precision symmetry (supercongruence) when you look at them through the lens of prime numbers. The author has verified this pattern numerically and is calling for a unified theory to explain why these Fibonacci numbers behave so elegantly in these deep mathematical structures.

Note: The paper is purely theoretical mathematics. It does not discuss medical applications, engineering uses, or future technologies. It is strictly about the beauty and patterns of numbers.

Technical Summary: "A Supercongruence Fantasy on Fibonacci and Lucas"

Problem Statement
The paper addresses the extension of Ramanujan-type supercongruences, which traditionally relate truncated hypergeometric series to values of $\pi$ and its powers modulo powers of primes. While previous work by Guillera and others established "rational" Ramanujan-like series by combining irrational formulas with their algebraic conjugates, these rational series typically involve standard binomial coefficients. The central problem explored here is whether such "rationalization" can be extended to series where the coefficients involve Fibonacci ( $F_n$ ) and Lucas ( $L_n$ ) numbers, as well as Apéry numbers. The author investigates whether these sequences, which satisfy higher-order linear recurrences, can generate new families of series for $1/\pi$ and $1/\pi^2$ that exhibit analogous supercongruence properties modulo $p^k$ .

Methodology
The methodology is primarily observational and numerical, driven by the "spirit" of previous discoveries by Guillera. The approach involves:

Rationalization of Irrational Series: Starting with known irrational Ramanujan-type series (e.g., those involving $\sqrt{5}$ or $\sqrt{6}$ ), the author combines them with their algebraic conjugates. This process eliminates irrational terms to produce series with rational coefficients.
Introduction of Fibonacci/Lucas Terms: In the resulting rational series, the coefficients are expressed in terms of $F_n$ and $L_n$ (or generalized sequences $U_n, V_n$ derived from $\sqrt{6}$ ).
Numerical Verification of Supercongruences: The author defines truncated sums ( $S_1(N)$ through $S_6(N)$ ) corresponding to these new series. Using computational evidence, the paper tests whether these sums satisfy supercongruences of the form:
$S(p^s) \equiv \chi(p) p^k S(p^{s-1}) \pmod{p^{m s}}$
where $\chi(p)$ is a Legendre symbol, $p$ is a prime, and $s \geq 1$ .
Generating Function Analysis: To facilitate numerical checks, the paper derives generating functions for the "rarefied" sequences (e.g., $F_{8n}, L_{8n}, F_{15n}$ ) involved in the sums, demonstrating that they satisfy linear recurrences with constant rational coefficients of order greater than 1.

Key Contributions and Results
The paper presents several new conjectures regarding supercongruences involving Fibonacci and Lucas numbers:

Level 1 (Modulo $p^3$ and $p^5$ ):
- By combining Guillera's series for $1/\pi$ (involving $\sqrt{5}$ ), the author derives two new series (Equations 3 and 4) containing $F_{8n}$ and $L_{8n}$ .
- Conjecture: The truncated sums $S_1(N)$ and $S_2(N)$ satisfy supercongruences modulo $p^3$ (and $p^3s$ for higher powers). Specifically:
  $S_1(p^s) \equiv \left(\frac{-5}{p}\right) p S_1(p^{s-1}) \pmod{p^{3s}}$
  $S_2(p^s) \equiv \left(\frac{-1}{p}\right) p S_2(p^{s-1}) \pmod{p^{3s}}$
- Similar congruences are proposed for shorter truncations (at $(p^s+1)/2$ ).
Level 2 (Modulo $p^5$ ):
- Applying the same logic to a series for $1/\pi^2$ (involving $\sqrt{5}$ and higher powers of binomial coefficients), the author constructs series $S_3(N)$ and $S_4(N)$ involving $F_{15n}$ and $L_{15n}$ .
- Conjecture: These sums satisfy stronger supercongruences modulo $p^5$ :
  $S_3(p^s) \equiv \left(\frac{5}{p}\right) p^2 S_3(p^{s-1}) \pmod{p^{5s}}$
  $S_4(p^s) \equiv p^2 S_4(p^{s-1}) \pmod{p^{5s}}$
Apéry-Type Generalization:
- The author extends the framework to include Apéry numbers ( $A_n$ ) and sequences $U_n, V_n$ related to $\sqrt{6}$ .
- Conjecture: Sums $S_5(N)$ and $S_6(N)$ satisfy supercongruences modulo $p^3$ , with specific Legendre symbols $\left(\frac{-3}{p}\right)$ and $\left(\frac{-2}{p}\right)$ , respectively. A single exception ( $S_6(31)$ ) is noted.

Significance and Claims
The paper modestly frames itself as a "fantasy" or a collection of numerical observations rather than a proof of these congruences. Its significance lies in:

Generalization: It proposes that the phenomenon of "rationalizing" Ramanujan-type series to include Fibonacci and Lucas numbers is a generalizable arithmetic pattern, not an isolated curiosity.
Structural Difference: The author highlights a principal difference between classical Ramanujan formulas (where coefficients often come from sequences satisfying order-1 recursions, like pure powers) and these new "Fibonacci-like" versions. The latter involve sequences satisfying higher-order linear recurrences, suggesting a deeper, yet unexplained, arithmetic structure.
Call for Theory: The paper concludes by noting that while there is a substantial body of work on "pure hypergeometric and rational" supercongruences, there is currently no unifying conceptual framework for these Fibonacci- and Apéry-based instances. The primary contribution is the identification of these patterns and the invitation for the mathematical community to develop a theoretical framework to explain them.

The work relies heavily on the "numerical observation" of these congruences, leaving the rigorous proof as an open problem for future research.

Technical Summary: "A Supercongruence Fantasy on Fibonacci and Lucas"

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