🔢 mathematics

Solving equations of signature $(p,p,2)$ with coefficients over number fields

This paper employs the modular method to establish both asymptotic results for the Diophantine equation $Aa^p+Bb^p=Cc^2$ over general number fields and effective, explicit bounds on the exponent $p$ that guarantee the non-existence of non-trivial solutions for specific quadratic fields.

Original authors: Begum Gulsah Cakti, Erman Isik, Yasemin Kara, Ekin Ozman

Published 2026-02-24

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Begum Gulsah Cakti, Erman Isik, Yasemin Kara, Ekin Ozman

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 very tricky puzzle involving numbers. This puzzle is a variation of Fermat's Last Theorem, the famous math problem that took 350 years to solve. The original puzzle asked: Can you find three whole numbers ( $a, b, c$ ) such that $a^n + b^n = c^n$ for any power $n$ greater than 2? The answer is "No."

This paper tackles a slightly different, but equally stubborn, version of that puzzle. Instead of just adding powers, the equation looks like this:
$A \cdot a^p + B \cdot b^p = C \cdot c^2$
Here, $p$ is a very large prime number (like 17, 19, 23, etc.), and $A, B, C$ are specific coefficients. The authors are asking: "If we make the power $p$ huge enough, can we prove that no 'interesting' solutions exist?"

Here is how the authors solve this, explained through simple analogies.

1. The Detective's Toolkit: The "Modular Method"

To solve this, the authors use a powerful tool called the Modular Method. Think of this as a special pair of X-ray glasses.

The Problem: You have a Diophantine equation (a number puzzle). It's hard to look at the numbers directly.
The Trick: If a solution to the puzzle did exist, you could use it to build a very specific, strange object called a Frey Curve (an elliptic curve).
The X-Ray: The authors then put this Frey Curve under the "Modular Method" X-ray. This tool checks if the curve behaves like a "modular form" (a highly structured, rhythmic pattern in mathematics).
The Contradiction: The authors prove that if a solution to the puzzle existed, the resulting Frey Curve would have to be a "modular form" that doesn't actually exist. It's like finding a footprint that proves a unicorn was in the room, but then realizing unicorns aren't real. Therefore, the solution to the puzzle must never have existed in the first place.

2. The Two Main Strategies

The paper uses two different approaches depending on how much information they have about the numbers involved.

Strategy A: The "Asymptotic" Approach (The Long-Range Telescope)

The Goal: Prove that for any sufficiently large prime number $p$ , there are no solutions.
The Analogy: Imagine you are looking at a forest. You can't count every single leaf, but you know that if the trees get tall enough, the forest will definitely look a certain way.

The authors prove that if you pick a prime number $p$ that is "big enough" (beyond a certain threshold), the mathematical "forest" of possible solutions simply disappears.
The Catch: To do this for complex number fields (which are like multi-dimensional versions of our number line), they have to assume a few "standard conjectures" (like assuming the weather will behave normally). They verify these assumptions for specific types of number fields (like real quadratic fields).

Strategy B: The "Effective" Approach (The Precision Scanner)

The Goal: Find a specific, calculable number (a bound) for $p$ . For example, "If $p > 17$ , there are no solutions."
The Analogy: Instead of just saying "it's too big," they build a metal detector that beeps if a solution exists. They tune the detector so that for any $p$ larger than a specific number (like 17 or 19), the detector stays silent.

They tested this for specific "neighborhoods" of numbers (specific quadratic fields like $\mathbb{Q}(\sqrt{3})$ or $\mathbb{Q}(\sqrt{-11})$ ).
They found that for these specific neighborhoods, if the power $p$ is greater than 17 (or 19, depending on the neighborhood), the equation has no non-trivial solutions.

3. The "S-Unit" Condition: The Gatekeeper

To make their proof work, the authors had to check a condition involving S-units.

Analogy: Imagine a club with a strict bouncer. The "S-unit condition" is the bouncer's rule. The authors had to prove that for certain number fields, the "bouncer" allows only a very limited number of people (solutions) to enter.
They showed that for an infinite family of real quadratic fields, this bouncer is very strict, effectively blocking any "large" solutions from getting in.

4. The Results: What Did They Find?

The authors successfully applied their "X-ray glasses" and "metal detectors" to a list of specific number fields.

Imaginary Quadratic Fields: For fields like $\mathbb{Q}(\sqrt{-3})$ , $\mathbb{Q}(\sqrt{-11})$ , etc., they proved that if $p > 17$ (or 19), the equation $x^p + d y^p = z^2$ has no solutions.
Real Quadratic Fields: For fields like $\mathbb{Q}(\sqrt{3})$ , $\mathbb{Q}(\sqrt{5})$ , etc., they proved the same thing, often without needing to assume the extra conjectures.

5. Why Does This Matter?

Think of Fermat's Last Theorem as the "King" of number puzzles. This paper is exploring the "Knights" and "Jesters" that surround the King.

By solving these specific variations (signatures like $p, p, 2$ ), they are expanding our understanding of how numbers interact in complex, multi-dimensional spaces.
They are showing that the "Modular Method" isn't just for the original Fermat equation; it's a universal key that can unlock many other difficult doors in number theory.

Summary in One Sentence

The authors used a high-tech mathematical "X-ray" to prove that for a wide variety of complex number systems, if you raise numbers to a sufficiently high power, a specific type of number puzzle simply has no solutions, effectively closing the door on those possibilities forever.

1. Problem Statement

The paper investigates the generalized Fermat equation with signature $(p, p, 2)$ and coefficients over number fields $K$ :
$A x^p + B y^p = C z^2$
where $A, B, C$ are pairwise coprime odd elements in the ring of integers $\mathcal{O}_K$ , and $p$ is a prime exponent. The authors focus on finding non-trivial primitive solutions (where $xyz \neq 0$ and $x, y, z$ are pairwise coprime).

The study addresses two main objectives:

Asymptotic Results: Proving that for sufficiently large primes $p$ , no non-trivial solutions exist for general number fields satisfying specific $S$ -unit conditions.
Effective Results: Establishing explicit bounds $C_K$ for specific families of quadratic fields (real and imaginary) such that no solutions exist when $p > C_K$ .

2. Methodology: The Modular Method

The authors employ the modular method, a technique pioneered by Wiles and Taylor-Wiles for Fermat's Last Theorem, which connects Diophantine equations to the theory of elliptic curves and modular forms. The core strategy involves:

Frey Curve Construction:
For a hypothetical solution $(a, b, c)$ , the authors associate a specific Frey elliptic curve $E$ (or $E_1$ for effective cases).
- For the general case ( $Ax^p + By^p = Cz^2$ ), the curve is $Y^2 = X^3 + 2CcX^2 + BCb^pX$ .
- For the effective case ( $x^p + dy^p = z^2$ ), a different model $E_1: Y^2 + XY = X^3 + \frac{c-1}{2^2}X^2 + \frac{db^p}{2^6}X$ is used to facilitate explicit computations.
Galois Representations:
The authors analyze the residual Galois representation $\rho_{E,p}: G_K \to \text{GL}_2(\mathbb{F}_p)$ attached to the $p$ -torsion points of the Frey curve. Key properties proven include:
- Irreducibility: Proving $\rho_{E,p}$ is irreducible (and absolutely irreducible for imaginary fields) for large $p$ .
- Finite Flatness: Establishing that the representation is finite flat at primes above $p$ .
- Oddness: Verifying the representation is odd (determinant of complex conjugation is $-1$).
Modularity and Level Lowering:
- Modularity: They assume or prove that the Frey curve is modular (associated with a Hilbert or Bianchi eigenform). For totally real fields, modularity is a theorem; for general fields, it relies on Conjecture 3.7 (Serre's modularity conjecture).
- Level Lowering: Using theorems by Freitas, Siksek, and others, they show that if a solution exists, the representation $\rho_{E,p}$ must be isomorphic to the representation of a new elliptic curve $E'$ (or a modular form) with a conductor independent of the solution (specifically, supported only on primes dividing $2ABC$).
Contradiction via Form Elimination:
The final step involves showing that no such modular form (or elliptic curve) exists with the required properties.
- Asymptotic Case: They reduce the problem to the solvability of an $S$ -unit equation ( $\alpha + \beta = \gamma^2$ ). By analyzing the valuations of solutions to this equation, they derive a contradiction regarding the reduction type of the curve.
- Effective Case: They explicitly compute all newforms of the predicted levels using the computer algebra system Magma. They then use trace of Frobenius comparisons (Lemma 6.12) to show that no modular form matches the trace of the Frey curve for $p$ exceeding a calculated bound.

3. Key Contributions and Results

A. Asymptotic Results (General Number Fields)

The paper generalizes previous results (which were mostly restricted to totally real fields) to arbitrary number fields.

Theorem 1.1: Establishes that if $K$ $K$ satisfies a specific $S$ $S$ -unit condition (related to the solvability of $\alpha + \beta = \gamma^2$ $α + β = γ^{2}$ ) and the 2-torsion of a specific class group is trivial, then for sufficiently large $p$ $p$ , no solutions exist.
- Note: For non-totally real fields, this relies on Conjecture 3.7 (Serre's conjecture) and Conjecture 4.7 (Langlands-type conjecture regarding eigenforms and elliptic curves/fake elliptic curves).
Theorem 1.3: Applies the asymptotic result to real quadratic fields $K = \mathbb{Q}(\sqrt{q})$ with specific congruence conditions, proving the non-existence of solutions for $x^p + \ell y^p = z^2$ .

B. Effective Results (Quadratic Fields)

The authors provide explicit bounds $C_K$ for specific quadratic fields where the $S$ -unit equation can be solved effectively and modular forms can be eliminated computationally.

Imaginary Quadratic Fields ( $K = \mathbb{Q}(\sqrt{-d})$ ):
- For $d \in \{3, 11, 19, 43\}$ , the equation $x^p + dy^p = z^2$ has no non-trivial primitive solutions with $P|b$ (where $P|2$ ) when $p > C_K$ .
- Bounds: $C_K = 17$ for $d \in \{3, 11, 43\}$ and $C_K = 19$ for $d=19$ .
- Constraint: Relies on Conjecture 3.7.
Real Quadratic Fields ( $K = \mathbb{Q}(\sqrt{d})$ ):
- For $d \in \{3, 5, 11, 13, 19, 29\}$ , the equation $x^p + dy^p = z^2$ has no non-trivial primitive solutions with $P|b$ when $p > C_K$ .
- Bounds: $C_K = 17$ for $d \in \{3, 5, 11, 13, 29\}$ and $C_K = 19$ for $d=19$ .
- Advantage: Does not rely on Conjecture 3.7 because modularity is proven for real quadratic fields.

C. Appendix: Signature (p, p, 3)

The authors extend their effective methods to the equation $x^p + dy^p = z^3$ over real quadratic fields for $d \in \{2, 5, 14\}$ , proving non-existence of solutions for $p > 17$ .

4. Significance and Impact

Generalization to Non-Totally Real Fields: The paper breaks the barrier of working exclusively with totally real fields. By assuming standard conjectures (Serre's and Langlands), it extends the modular method to complex quadratic fields and general number fields.
Bridging Asymptotic and Effective: It successfully combines asymptotic arguments (relying on finiteness of $S$ -unit solutions) with effective computational elimination (using Magma to rule out specific modular forms).
Handling Coefficients: While much work exists on $x^p + y^p = z^2$ , this paper systematically handles equations with coefficients ( $A, B, C$ ), which introduces significant complexity in the conductor calculation and the associated modular forms.
Computational Verification: The paper provides a rigorous framework for using computational tools (Magma) to eliminate potential counterexamples, offering explicit bounds that were previously unknown for these specific families of equations.

5. Limitations

Conjectural Nature: The results for general number fields and imaginary quadratic fields depend on Conjecture 3.7 (Serre's modularity conjecture) and Conjecture 4.7. While widely believed, these are not yet proven theorems in full generality.
Scope of Fields: The effective results are currently limited to specific quadratic fields where the class number and unit group structures allow for the complete elimination of modular forms. The authors note that fields like $\mathbb{Q}(\sqrt{7})$ and $\mathbb{Q}(\sqrt{23})$ were excluded because they could not eliminate all Hilbert newforms in those cases.

In conclusion, this paper represents a significant advancement in the application of the modular method to generalized Fermat equations over number fields, providing both theoretical asymptotic guarantees and concrete effective bounds for quadratic fields.