On an elementary method for solving $Ax^4-By^2=1$

This paper investigates a new elementary method by Luo and Lin for solving quartic equations, successfully applying it to provide a straightforward solution for Bumby's equation $3X^4-2Y^2=1$ and proposing a conjecture that could extend this approach to an infinite family of similar equations.

The Gist

Imagine you are a detective trying to solve a very specific, stubborn riddle written in the language of numbers. The riddle is: "Find all whole numbers $x$ and $y$ that make the equation $Ax^4 - By^2 = 1$ true."

For a long time, mathematicians have been trying to crack this code. Some of the best detectives in history (like Ljunggren and Cohn) have solved specific versions of it, but the general case remains a mystery.

This paper, written by P.G. Walsh, is like a new detective stepping up to the case. He's testing a brand-new, "elementary" (meaning it doesn't require a PhD in advanced physics to understand) method that was recently discovered by two other researchers, Lin and Luo.

Here is the story of the paper, broken down into simple analogies:

1. The "Bumby" Mystery

The paper starts by looking at a famous specific case solved by a detective named Richard Bumby. The equation was $3x^4 - 2y^2 = 1$.

• The Solution: Bumby found that there are only two answers: $(1, 1)$ and $(3, 11)$.
• The Old Way: Bumby's proof was like using a sledgehammer. It required very complex, high-level math tools (working in a special "ring" of numbers involving $\sqrt{-2}$). It was brilliant, but hard to copy for other cases.

2. The New "Sieve" Method

Lin and Luo found a new way to solve a similar equation ($2x^4 - y^2 = 1$). Their method is like a giant sieve (a kitchen strainer used to separate pasta from water).

• Step 1: The Big Sieve (The Factor Base): Imagine you have a huge list of potential number candidates. You pour them through a sieve made of specific prime numbers (like 11, 13, 29, etc.). Most candidates fall through the holes and are eliminated because they can't possibly be the answer.
• Step 2: The Tiny Sieve (The Jacobi Symbol): After the big sieve, a few candidates might still be stuck. The method then uses a very clever mathematical trick (involving something called a "Jacobi symbol," which acts like a truth-telling test) to check these remaining few. If they fail the test, they are thrown out.

Walsh's paper asks: "Can we use this new sieve method to solve Bumby's equation ($3x^4 - 2y^2 = 1$)?"

3. The Investigation (Section 2)

Walsh takes the new method and applies it to Bumby's equation.

• The Setup: He builds a custom sieve with a "modulus" (a cycle length) of 1680. He uses a list of 27 prime numbers as his "witnesses."
• The Result: The sieve works beautifully! It filters out almost every possible number. It leaves only a tiny handful of candidates (specifically, numbers that fit into certain patterns like $1, 3, 837, 839$, etc., when divided by 1680).
• The Final Blow: For the remaining candidates, Walsh uses the "truth-telling test" (the Jacobi symbol). He constructs a specific mathematical scenario where the test always says "No" (the result is always -1). This proves that no other solutions exist.

The Victory: Walsh successfully proves Bumby's result using this new, simpler method. He shows that the "sledgehammer" isn't needed; a clever sieve works just as well.

4. The Big Question: Can We Scale Up? (Section 3)

Now that the method works for one case, can it solve all equations of this type?

• The Experiment: Walsh runs computer simulations for hundreds of different equations (up to $t = 1000$).
• The Discovery: The method is incredibly picky. It works like a master key for a very specific type of lock, but it jams for almost everything else.
• The Pattern: The method only seems to work when the numbers in the equation follow a very specific pattern: $t = du^2 - 1$, where $d$ is 2, 3, 4, or 6.

5. The Conjecture (The Cliffhanger)

Walsh ends with a bold guess (a conjecture). He believes that if we can prove a specific mathematical formula works for the "pickiest" cases (where $d=3$), then this simple sieve method could solve an infinite family of these equations.

The Takeaway

Think of this paper as a mechanic testing a new, simple tool.

1. He successfully uses the new tool to fix a famous, difficult car engine (Bumby's equation).
2. He tries the tool on 1,000 other engines.
3. He finds that the tool only works on a very specific brand of engine.
4. He leaves a note for future mechanics: "If you can figure out why this tool works on this specific brand, you might be able to fix an infinite number of engines with it."

It's a story of taking a complex problem, finding a simpler way to solve a specific instance, and then wondering if that simplicity can be the key to unlocking a whole new world of math.

Technical Summary

Here is a detailed technical summary of the paper "On an Elementary Method for Solving $Ax^4 - By^2 = 1$" by P.G. Walsh.

1. Problem Statement

The paper addresses the Diophantine equation of the form:
$$Ax^4 - By^2 = 1$$
where $A$ and $B$ are positive integers. The author focuses specifically on the subfamily of equations reducible to the form:
$$(t+1)X^4 - tY^2 = 1$$
This reduction is based on arguments involving Jacobi symbols established in prior literature. The primary goal is to determine all positive integer solutions $(X, Y)$ for specific values of $t$.

Context:

• Historically, Ljunggren proved fundamental results on these equations.
• Richard Bumby (1967) famously solved the case $3x^4 - 2y^2 = 1$(where$t=2$), proving the only positive integer solutions are$(1,1)$and$(3,11)$. Bumby's proof utilized the ring$\mathbb{Z}[\sqrt{-2}]$.
• Recent work by Lin and Luo (2026) provided an "elementary" method (avoiding complex algebraic number theory) to solve the case $t=1$ ($2x^4 - y^2 = 1$).
• The Gap: It was unclear if Lin and Luo's elementary method could be generalized to other values of $t$, particularly $t=2$ (Bumby's equation) and beyond.

2. Methodology

Walsh investigates the applicability of the Lin-Luo method to the general case. The methodology consists of two distinct phases:

Phase 1: The Factor Base and Congruence Elimination

The problem is reduced to finding indices $n$ such that a term in a specific recurrence sequence, $P_n$, is a perfect square.

1. Sequence Definition: For a given $t$, define $\alpha = \sqrt{t+1} + \sqrt{t}$. The sequences $\{P_k\}$ and $\{Q_k\}$ are defined via the expansion of $\alpha^k$.
   • For odd $k$: $P_k\sqrt{t+1} + Q_k\sqrt{t} = \alpha^k$.
2. Modulus Selection: A primary modulus $m$ (typically 840) and a working modulus $M$ are chosen.
3. Factor Base Construction: A set of primes (the factor base) is selected such that the order of the sequence $\{P_k\}$ modulo each prime divides $M$.
4. Elimination: For a large set of congruence classes modulo $M$, the author identifies a prime $p$ in the factor base where the Legendre symbol $(\frac{P_k}{p}) = -1$. This proves $P_k$ cannot be a square for those indices.
5. Result: This reduces the potential solutions to a very small set of congruence classes (typically $k \equiv \pm 1, \pm 3 \pmod{840}$).

Phase 2: The Main Case and Jacobi Symbol Reduction

The remaining congruence classes (specifically $n \equiv 1 \pmod{840}$) are handled via a constructive argument involving divisibility and Jacobi symbols.

1. Construction of Index $b$: For $n = 1 + 840t$, a specific integer $b$ is constructed based on the properties of $t$ (specifically the 3-adic valuation and modulo 24 properties).
2. Divisibility Properties: The author utilizes identities relating $P_n$ and $Q_n$. Specifically, it is shown that $P_n \equiv P_{8b-1} \pmod{Q_{6b}}$.
3. Polynomial Reduction: Using the identity $Q_{6k}/Q_{2k} = 3(4P_k^2-1)(12P_k^2-1)$, the term $P_{8b-1}$ is reduced modulo a divisor of $Q_{6b}$ (specifically $2P_b + 1$).
4. Jacobi Symbol Evaluation: The core of the proof involves showing that the Jacobi symbol $(\frac{P_n}{2P_b+1}) = -1$. This is achieved by expanding the polynomial expressions of $P_n$ and $Q_n$ in terms of $P_b$ and $Q_b$, eventually reducing the expression to a form that evaluates to $-1$.
5. Conclusion: If the Jacobi symbol is $-1$, $P_n$ cannot be a perfect square, thereby eliminating the infinite family of candidates in that congruence class.

3. Key Contributions and Results

A. Solution to Bumby's Equation ($t=2$)

The paper successfully applies the Lin-Luo method to solve $3x^4 - 2y^2 = 1$.

• Modulus: $M = 1680$ with a factor base of 27 primes.
• Reduction: The method proves that if $P_n = x^2$, then $n \equiv \pm 1, \pm 3 \pmod{840}$.
• Elimination:
  • The case $n \equiv 1 \pmod{840}$ is eliminated using the Jacobi symbol construction detailed in Section 2.2.
  • The cases $n \equiv 3, -1, -3$ are handled via descent arguments and symmetry ($P_n = P_{-n}$).
• Outcome: This provides a completely elementary proof (relying only on integer arithmetic and Jacobi symbols, not algebraic number theory) that the only positive integer solutions are $(1,1)$ and $(3,11)$.

B. Computational Investigation for General $t$

The author extends the method computationally to $t \le 1000$.

• Success Rate: The factor base elimination (Phase 1) works reliably for almost all $t$, reducing the search space to the expected congruence classes.
• Limitation (Phase 2): The second phase (Jacobi symbol reduction) is highly sensitive to the specific form of $t$. The method only succeeds in producing a consistent $-1$ Jacobi symbol for a very thin set of $t$ values.

C. The Conjecture

Based on computational evidence, the author proposes Conjecture 3.1:
The elementary method works for an infinite family of equations where $t = du^2 - 1$ with $d \in \{2, 3, 4, 6\}$.

• Specifically, for $d=3$ ($t = 3i^2 - 1$), the author conjectures that specific polynomials $p(x)$ and inputs $b$ will always yield $(\frac{P_n}{p(b)}) = -1$.
• Significance: If proven, this would extend the elementary solution to an infinite family of quartic Diophantine equations, generalizing the results of Lin-Luo and Bumby.

4. Significance

• Elementary Proof: The paper demonstrates that deep results in Diophantine equations, previously requiring advanced tools (like hypergeometric methods or algebraic number theory), can potentially be solved using "elementary" number theory (congruences, factor bases, and Jacobi symbols).
• Generalization: It moves beyond the isolated case of $t=1$ (Lin-Luo) and $t=2$ (Bumby) to propose a framework for solving a broader class of equations.
• New Toolkit: The method adds a new, computationally accessible tool to the arsenal for solving $Ax^4 - By^2 = 1$, complementing existing software like Magma and Pari.
• Open Problem: The paper highlights the difficulty of proving the polynomial identities required for the Jacobi symbol reduction for $t > 2$, leaving a clear path for future research to resolve Conjecture 3.1.

In summary, Walsh provides a rigorous elementary proof for Bumby's equation and establishes a computational framework suggesting that this method is applicable to a specific, infinite family of similar Diophantine equations, contingent on the resolution of a specific conjecture regarding polynomial Jacobi symbols.