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Integers representable as a difference of two rational fourth powers

Motivated by Cohen's investigation of sums of rational fourth powers, this paper determines the complete list of positive integers up to 10,000 that can be expressed as the difference of two nonzero rational fourth powers.

Original authors: Ashleigh Ratcliffe, Tho Nguyen Xuan

Published 2026-04-28

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Ashleigh Ratcliffe, Tho Nguyen Xuan

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 have a giant box of building blocks. Some blocks are perfect squares (like $2^2 = 4$ ), and some are perfect cubes ( $2^3 = 8$ ). In this paper, the authors are playing with a very specific, slightly more difficult type of block: fourth powers. A fourth power is just a number multiplied by itself four times (like $2^4 = 16$ ).

The big question the authors are asking is: Can you build any whole number (like 1, 2, 3... up to 10,000) by taking one of these fourth-power blocks and subtracting another one?

For example, can you find two fractions (rational numbers) such that when you raise them to the fourth power and subtract them, you get exactly the number 5?
$x^4 - y^4 = 5$

The Big Challenge

The authors aren't just looking for whole number blocks; they are allowed to use fractional blocks (like $3/2$ or $7/5$ ). This makes the puzzle much harder because there are infinitely many fractions to check. You can't just try them one by one; you need a map.

The paper is essentially a massive "Yes/No" list for every number from 1 to 10,000.

Yes: "Here is a specific pair of fractions that works."
No: "It is mathematically impossible to build this number this way."

How They Solved It: The Detective's Toolkit

To solve this, the authors used a mix of high-tech math tools and clever detective work. Think of it like a three-step investigation:

1. The "Easy Search" (Looking for Small Clues)
First, they used a computer to look for "small" solutions. Imagine searching a room for a lost key. If the key is on the floor, you'll find it quickly. The computer scanned for simple fraction combinations. If it found a match, the number was marked as "Solvable."

2. The "Shape Shifter" (Turning the Problem into a Curve)
If the computer couldn't find a simple answer, the authors had to prove that no answer exists. They used a mathematical trick to turn the equation $x^4 - y^4 = n$ into a different shape: an Elliptic Curve.

Analogy: Imagine the original equation is a tangled knot. The authors found a way to untie it and lay it flat as a smooth, looping curve.
If this curve has a "rank" of zero, it means the curve is too small or broken to hold any solutions. It's like a road that ends in a cliff; you can't drive anywhere. This proved many numbers were impossible.

3. The "Mordell-Weil Sieve" (The Ultimate Filter)
For the stubborn numbers where the curve looked promising (it had a "positive rank," meaning it looked like it could have solutions), they used a sophisticated filter called the Mordell-Weil sieve.

Analogy: Imagine you are trying to find a specific person in a massive crowd. You know they have a red hat. You ask everyone in the crowd, "Do you have a red hat?" Then you ask, "Do you have a blue shirt?"
The sieve checks the numbers against different "moduli" (like checking remainders when divided by 5, 7, or 11). If the math says, "For a solution to exist, the numbers must be divisible by 5," but another rule says, "They must not be divisible by 5," you have a contradiction. The person (the solution) cannot exist. The sieve found these contradictions for the remaining difficult numbers.

4. The "Pythagorean Trick" (Using Old Friends)
For some specific cases, they used a method involving Pythagorean triples (the famous $a^2 + b^2 = c^2$ triangles). They realized that if a solution existed, it would have to fit into the pattern of these ancient triangles. By checking if the numbers fit the triangle rules, they could rule out more possibilities.

The Final Result

After running these tests on every number up to 10,000, the authors produced Table 3.

This table lists the "lucky" numbers that can be written as the difference of two rational fourth powers.
For these lucky numbers, they even provided the actual fractions (the "keys") that make the equation work.
For all the other numbers, they proved mathematically that no such fractions exist.

Why This Matters (According to the Paper)

The paper doesn't claim this will help build bridges or cure diseases. Instead, it completes a puzzle that mathematicians have been working on for a long time.

It finishes the job started by other mathematicians (like Cohen, Grechuk, and Tho) who solved similar problems for sums of powers or smaller ranges.
It provides a definitive "map" for the difference of fourth powers up to 10,000, filling in the gaps that previous methods couldn't solve.

In short, the authors acted as master cartographers, drawing the complete boundary line between the numbers that can be built from fourth-power fractions and those that cannot.

1. Problem Statement

The paper addresses the solubility of the Diophantine equation:
$n = x^4 - y^4$
where $n$ is a positive integer and $x, y$ are nonzero rational numbers.

This problem is motivated by Fermat's Last Theorem for the exponent $n=4$ , which implies that $1$ cannot be written as a difference of two nonzero rational fourth powers. While the case of the sum of two rational fourth powers ( $x^4 + y^4 = n$ ) was largely resolved by Cohen, Bremner, and Tho for $n \le 10,000$ , the difference case remained incomplete for certain values. The authors aim to provide a complete list of all positive integers $n \le 10,000$ that can be represented in this form.

Mathematically, the problem is equivalent to finding integer solutions $(x, y, z)$ with $xyz \neq 0$ to the homogeneous equation:
$x^4 - y^4 = nz^4$
Due to the properties of fourth powers, it suffices to analyze $n$ that are fourth-power free.

2. Methodology

The authors employ a multi-layered computational and theoretical approach, combining methods from elliptic curves, local obstructions, and factorization in Gaussian integers. The workflow is divided into three main stages:

A. Easy Reductions and Elliptic Curves (Sections 2.1 & 2.2)

The first step involves reducing the problem to finding rational points on specific elliptic curves.

Transformation: Any nontrivial integer solution to $x^4 - y^4 = nz^4$ $x^{4} - y^{4} = n z^{4}$ implies the existence of rational points on one of three associated elliptic curves:
1. $Y^2 = X^3 + nX$
2. $Y^2 = X^3 - nX$
3. $Y^2 = X^3 - n^2X$
Rank 0 Check: If any of these curves has rank 0, the set of rational points is finite (torsion points only). The authors use the computer algebra system Magma to compute the rank and torsion points. If the torsion points do not yield a valid solution (where $xyz \neq 0$ ), the equation is proven insoluble.
Reduction to Octic Equations: For cases where the curves have positive rank, the problem is reduced (via Theorem 2) to solving equations of the form:
$a^2u^8 + b^2v^8 = cw^4$
where $a, b, c$ are pairwise coprime integers such that $abc \in \{n/8, 2n\}$ .
Local Obstructions & Mordell-Weil Sieve:
- Local Obstructions: The authors check for solutions modulo $p^k$ . If no coprime solutions exist modulo a prime, the global equation has no solution.
- Mordell-Weil Sieve: For equations where local obstructions fail to rule out solutions, the authors map the problem to finding rational points on elliptic curves derived from the octic equation. They use the Mordell-Weil sieve to check if the generated group of points modulo a prime $p$ is disjoint from the set of possible local solutions, thereby proving insolubility.

B. Pythagorean Triple Method (Section 2.3)

For specific cases where the coefficient $c$ in the octic equation $a^2u^8 + b^2v^8 = cw^4$ is a perfect square ( $c=k^2$ ), the equation can be transformed into a system involving Pythagorean triples:
$x^2 + y^2 = z^2$
Using the standard parametrization of primitive Pythagorean triples, the authors reduce the problem to solving systems of equations involving fourth powers (e.g., $d^2U^8 - e^2V^8 = bv^4$ ). This method successfully resolves many cases that the sieve method could not.

C. Factorization in $\mathbb{Z}[i]$ (Section 3 & Appendix B)

For the remaining stubborn cases (listed in Table 2 of the paper), the authors utilize the method of factorization in the ring of Gaussian integers $\mathbb{Z}[i]$ .

Factorization: The equation $a^2u^8 + b^2v^8 = cw^4$ is rewritten as:
$(au^4 + bv^4i)(au^4 - bv^4i) = cw^4$
Gaussian Integers: By analyzing the gcd in $\mathbb{Z}[i]$ and the prime factorization of $c$ , the authors deduce that:
$au^4 + bv^4i = \alpha(s + ti)^4$
where $\alpha$ is a divisor of $c$ in $\mathbb{Z}[i]$ .
Local Insolubility of Schemes: Equating real and imaginary parts yields a system of two homogeneous quartic equations in variables $s, t, u, v$ . This defines a scheme in projective space $\mathbb{P}^3(\mathbb{Q})$ . The authors use Magma to check if this scheme is locally insoluble at a specific prime (usually 2, 3, 5, or 17). If the scheme has no solutions modulo $p$ , the original Diophantine equation has no integer solutions.

3. Key Contributions

Complete Classification: The paper provides the complete list of all positive integers $n \le 10,000$ representable as a difference of two nonzero rational fourth powers.
Resolution of Open Cases: The authors resolve the specific cases left open by previous work (Grechuk and Tho), including $n = 219, 463, 514$ , and others up to $n=10,000$ .
Algorithmic Framework: The paper details a robust computational pipeline combining elliptic curve ranks, the Mordell-Weil sieve, Pythagorean parametrization, and Gaussian integer factorization. This framework is applicable to similar Diophantine problems involving higher powers.
OEIS Sequences: The results are cataloged in the On-Line Encyclopedia of Integer Sequences (OEIS) as sequences A393616 (nonzero rational fourth powers) and A393617 (allowing zero).

4. Results

Soluble Integers: Table 3 in the paper lists 136 integers $n \le 10,000$ that are representable as $x^4 - y^4$ . For each, a specific rational representation is provided (e.g., $5 = (3/2)^4 - (1/2)^4$ ).
Insoluble Integers: The paper proves that the remaining integers in the range $1 \le n \le 10,000$ have no nonzero rational solutions.
Specific Cases: The appendices provide exhaustive proofs for the 33 "hard" cases that required the factorization method in $\mathbb{Z}[i]$ . For example, for $n=463$ , the equation $214369u^8 + v^8 = 2w^4$ is shown to have no solutions by proving the associated scheme is locally insoluble at $p=2$ and $p=3$ .

5. Significance

Completing the Survey: This work parallels Cohen's survey on the sum of fourth powers, completing the picture for the difference case up to $10,000$.
Methodological Advancement: It demonstrates the power of combining classical number theory (Gaussian integers, Pythagorean triples) with modern computational tools (Magma, Mordell-Weil sieve) to solve high-degree Diophantine equations.
Foundation for Future Research: The detailed appendices and the specific techniques used for the "hard" cases provide a blueprint for tackling similar problems, such as sums or differences of sixth powers or higher exponents.

In conclusion, Ratcliffe and Tho have successfully closed the gap in the literature regarding the difference of rational fourth powers, providing a definitive list of representable integers and rigorous proofs for the non-representable ones within the specified range.