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The Quadratic and Cubic Characters of 2

This paper explores the solvability of the cubic congruence $x^3 \equiv 2 \pmod{p}$ by employing Eisenstein integers, Gauss and Jacobi sums, and the law of cubic reciprocity, while providing historical context on the development of higher reciprocity laws and the quadratic character of 2 through the contributions of Fermat, Euler, Legendre, Jacobi, and Eisenstein.

Original authors: Matias C. Relyea

Published 2026-02-03

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Matias C. Relyea

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 a detective trying to solve a mystery about numbers. Specifically, you are looking for a hidden pattern that tells you whether a specific math puzzle can be solved. The puzzle is simple to state: Can you find a number that, when you multiply it by itself three times, leaves a remainder of 2 when divided by a specific prime number?

For example, if your prime number is 7, can you find a number $x$ where $x \times x \times x = 2$ (plus some multiple of 7)? Sometimes the answer is "yes," and sometimes it's "no." This paper is about figuring out exactly when the answer is "yes."

Here is the story of how the author, Matias C. Relyea, solves this mystery, explained in everyday terms.

The Setup: The "Golden Theorem" and Its Cousins

The paper starts by looking at a famous, older mystery called the Quadratic Character of 2. This is the same puzzle, but instead of multiplying a number by itself three times ( $x^3$ ), you only multiply it twice ( $x^2$ ).

The Old Mystery: Can you find $x$ such that $x^2 \equiv 2 \pmod p$ ?
The Solution: Mathematicians like Gauss figured out a simple rule: If your prime number $p$ looks like $8n+1$ or $8n+7$ , the answer is "yes." If it looks like $8n+3$ or $8n+5$ , the answer is "no."

The author uses this old, solved mystery as a training ground. He wants to apply similar logic to the harder, newer mystery: The Cubic Character of 2 (the $x^3$ version).

The Problem: The Puzzle Gets Harder

In the old days, mathematicians like Euler and Fermat guessed that there was a similar simple rule for the cubic case ( $x^3 \equiv 2$ ). They noticed that if a prime number $p$ can be written in a very specific shape, the puzzle is solvable.

The Big Guess (Euler's Conjecture):
If you can write a prime number $p$ in the form $C^2 + 27D^2$ (where $C$ and $D$ are whole numbers), then the puzzle $x^3 \equiv 2$ is solvable. If you cannot write it in that shape, it's not solvable.

Euler made this guess based on pure intuition, but he couldn't prove it. Gauss later found notes suggesting he knew the proof, but it wasn't fully fleshed out until much later. This paper aims to walk through the history and provide a clear, step-by-step proof of this specific rule.

The Toolkit: Building a New World

To solve the cubic puzzle, the author explains that we can't just use regular whole numbers (like 1, 2, 3). We have to build a new "world" of numbers.

The Eisenstein Integers (The New Neighborhood):
Imagine regular numbers live on a straight line. To solve cubic problems, we need to move into a 2D plane. We introduce a special number called $\omega$ (omega), which is a "cube root of unity." Think of $\omega$ as a magical key that, when you turn it three times, brings you back to where you started.
In this new neighborhood, numbers look like $a + b\omega$ . The author calls these Eisenstein Integers. This new world has its own rules for multiplication and division, but it's structured enough that we can still do math on it.
The Magic Sums (Gauss and Jacobi Sums):
To find patterns in this new neighborhood, the author uses tools called Gauss Sums and Jacobi Sums.
- Analogy: Imagine you have a giant bag of colored marbles (numbers). You want to know if a specific pattern exists. Instead of counting them one by one, you shake the bag and listen to the sound they make. If the sound is a specific pitch, you know the pattern exists. These "sums" are like that special pitch—they are complex calculations that reveal hidden properties of the numbers without you having to check every single one.
The "Primary" Filter:
In this new world, numbers can look different but act the same (like how $1$ and $-1$ are related). To avoid confusion, the author uses a filter called Primary Numbers. This is like sorting your mail into specific bins so you only look at one representative from each group. This makes the math much cleaner.

The Solution: Connecting the Dots

The paper walks through a logical journey to prove Euler's guess:

Step 1: The Simple Case. First, the author shows that if you pick a prime number that is "weird" (specifically, if it leaves a remainder of 2 when divided by 3), the puzzle is always solvable. It's like a free pass.
Step 2: The Complex Case. The real challenge is when the prime number is "normal" (leaves a remainder of 1 when divided by 3). Here, the author uses the Eisenstein Integers and the Magic Sums to show that the solvability of the puzzle depends entirely on the shape of the prime number.
Step 3: The Shape Shift. Using the properties of the new number world, the author proves that if the puzzle is solvable, the prime number $p$ $p$ must be able to be rearranged into the shape $C^2 + 27D^2$ .
- The Analogy: It's like saying, "If you can unlock this door, your key must be made of gold." The author proves that if the door opens (the equation has a solution), the key (the prime number) must have the specific shape of $C^2 + 27D^2$ .
Step 4: The Reverse. He also proves the opposite: If you have a prime number that is shaped like $C^2 + 27D^2$ , then the door will open.

The Conclusion

The paper concludes by confirming Euler's centuries-old guess. It tells us that the solvability of the cubic equation $x^3 \equiv 2 \pmod p$ is not random. It is strictly determined by whether the prime number $p$ can be written as the sum of a square and 27 times another square.

In short:

The Mystery: Can $x^3 = 2$ be solved for a given prime?
The Clue: Look at the prime number's shape.
The Answer: If the prime fits the formula $C^2 + 27D^2$ , the answer is Yes. If it doesn't, the answer is No.

The author achieves this by borrowing tools from the past (Gauss's induction), building a new mathematical neighborhood (Eisenstein Integers), and using complex "magic sums" to reveal the hidden geometry of numbers. It's a story of how mathematicians built a bridge from simple arithmetic to complex algebra to solve a puzzle that stumped the greatest minds for hundreds of years.

Technical Summary: The Quadratic and Cubic Characters of 2

Problem Statement
The paper addresses the solvability of the cubic congruence $x^3 \equiv 2 \pmod p$ for primes $p \equiv 1 \pmod 3$ . This solvability is defined as the "cubic character of 2." While the quadratic character of 2 (the solvability of $x^2 \equiv 2 \pmod p$ ) is a standard result in elementary number theory, the cubic case requires tools from algebraic number theory, specifically the theory of Eisenstein integers and higher reciprocity laws. The paper aims to provide a proof for Euler's conjecture regarding the cubic character of 2, which states that $x^3 \equiv 2 \pmod p$ is solvable if and only if $p$ can be represented in the form $p = C^2 + 27D^2$ for integers $C$ and $D$ .

Methodology
The paper employs a historical and constructive approach, utilizing the following mathematical frameworks:

Historical Context and Induction: The author begins by reviewing Gauss's inductive proof of the quadratic character of 2 (Theorem 2) from Disquisitiones Arithmeticae. This serves as a motivational foundation, illustrating how elementary methods can establish results often reserved for more advanced treatments.
Eisenstein Integers ( $\mathbb{Z}[\omega]$ ): The core of the methodology shifts to the ring of Eisenstein integers, where $\omega = e^{2\pi i/3}$ . The paper establishes that $\mathbb{Z}[\omega]$ is a Unique Factorization Domain (UFD) and a Euclidean domain. It defines the norm $N(a+b\omega) = a^2 - ab + b^2$ and introduces "primary" numbers (elements $\pi \equiv \pm 1 \pmod 3$ ) to handle unique factorization issues.
Gauss and Jacobi Sums: The paper defines multiplicative characters on finite fields $\mathbb{F}_p$ and constructs Gauss sums ( $g(\chi)$ ) and Jacobi sums ( $J(\chi, \lambda)$ ). It proves fundamental properties, including the relationship $J(\chi, \lambda) = g(\chi)g(\lambda)/g(\chi\lambda)$ and the magnitude $|J(\chi, \chi)| = \sqrt{p}$ for non-trivial characters of order 3.
Representation of Primes: Using the properties of Jacobi sums, the paper demonstrates that for $p \equiv 1 \pmod 3$ , there exist integers $a, b$ such that $p = a^2 - ab + b^2$ . It further refines this to show the existence of unique integers $A, B$ such that $4p = A^2 + 27B^2$ (Theorem 15).
Cubic Reciprocity: The Law of Cubic Reciprocity (Theorem 3) is stated for primary elements in $\mathbb{Z}[\omega]$ . The paper utilizes the cubic residue character $(\alpha/\pi)_3$ and the property that $(\alpha/\pi)_3 = \alpha^{(N\pi-1)/3} \pmod \pi$ .

Key Contributions and Results
The paper synthesizes historical conjectures with rigorous proofs to establish the following results:

Theorem 4 (Main Result): For a prime $p \equiv 1 \pmod 3$ $p \equiv 1 (mod 3)$ , the congruence $x^3 \equiv 2 \pmod p$ $x^{3} \equiv 2 (mod p)$ is solvable if and only if there exist integers $C$ $C$ and $D$ $D$ such that $p = C^2 + 27D^2$ $p = C^{2} + 27 D^{2}$ .
- Proof Strategy: The proof connects the solvability of $x^3 \equiv 2 \pmod \pi$ (where $\pi$ is a prime in $\mathbb{Z}[\omega]$ with norm $p$ ) to the parity of $\pi$ . By Lemma 19, solvability requires $\pi \equiv 1 \pmod 2$ . This parity constraint, combined with the representation $p = a^2 - ab + b^2$ and the condition $b \equiv 0 \pmod 3$ (derived from properties of Jacobi sums), forces the coefficients to align such that $p$ takes the form $C^2 + 27D^2$ .
Proposition 18: Establishes that if $q \equiv 2 \pmod 3$ is a rational prime, every integer is a cubic residue modulo $q$ . Consequently, $x^3 \equiv 2 \pmod q$ is always solvable in this case.
Theorem 15: Proves that if $p \equiv 1 \pmod 3$ , there exist unique integers $A$ and $B$ (up to sign) such that $4p = A^2 + 27B^2$ . This uniqueness is critical for the final proof of Theorem 4.
Historical Clarification: The paper clarifies the timeline of contributions, noting that while Euler conjectured the result based on the form $a^2 + 3b^2$ , and Gauss provided proofs in posthumous notes, the paper aims to illuminate the specific historical nuances and the "elementary" nature of the proof using Gauss and Jacobi sums.

Significance and Claims
The paper claims significance in bridging the gap between elementary number theory and the more complex machinery of algebraic number theory.

Accessibility: It argues that while higher reciprocity laws (cubic, biquadratic) are often viewed as requiring advanced algebraic number theory, their special cases (like the character of 2) can be approached with "a few additional ideas from algebra," specifically Gauss and Jacobi sums.
Historical Narrative: The paper emphasizes the rich history of reciprocity laws, tracing the development from Fermat and Euler through Gauss, Jacobi, and Eisenstein. It highlights that Gauss's insights into cubic residues relied on the introduction of imaginary quantities (Eisenstein integers), a concept that was not fully realized until Eisenstein's work in 1844.
Pedagogical Value: The author notes that the proof of the quadratic character of 2 (Theorem 2) is rarely covered in elementary texts, and similarly, the cubic case is often omitted. This paper provides a self-contained derivation of the cubic character of 2, motivated by the historical development of the field.

The paper does not propose new applications or future research directions but focuses on providing a rigorous, historically grounded proof of a classical conjecture, demonstrating how fundamental properties of Gauss and Jacobi sums can resolve specific questions regarding the solvability of cubic congruences.