Integers representable as a difference of two rational fourth powers

Abstract: In Section 6.6 of the book {\it Number Theory, Volume I: Tools and Diophantine Equations, Graduate Texts in Mathematics, Volume 239, Springer (2007)}, Cohen investigated the solubility of the equation $n=x<sup>4+y<sup>4$ in the rational numbers $x,y$ for all positive integers $n \leq 10000$. Motivated by this, we investigate the equation $n=x<sup>4-y<sup>4$ and obtain the complete list of positive integers $n\leq 10000$ that can be represented in this form for some nonzero rational numbers $x$ and $y$.

• The paper establishes a complete classification for all integers up to 10,000 that can be expressed as a difference of two nonzero rational fourth powers.
• It combines explicit algebraic computations, elliptic curve theory, and descent methods to analyze local and global solubility.
• The computational implementation in Magma confirms nonrepresentability or provides explicit representations, refining earlier studies in rational fourth power equations.

Summary of "Integers representable as a difference of two rational fourth powers" (2604.15832)

Introduction and Motivation

This paper addresses a long-standing problem in the theory of Diophantine equations: characterizing all positive integers $n$ (specifically $1 \leq n \leq 10000$) for which there exist nonzero rational numbers $x, y$ such that $n = x^4 - y^4$. The investigation parallels earlier work (e.g., Cohen, Demjanenko, Serre) focusing on sums of fourth powers, but here the main focus is the difference of rational fourth powers—a case intricately tied to the arithmetic of elliptic curves and Fermat's equation in degree four.

The impossibility of representing $1$ as a sum or difference of two rational fourth powers (a reformulation of the $n=4$ case of Fermat's Last Theorem) sets a natural context: which other integers $n$ are so representable?

Main Result

The central result is a complete classification:

Theorem: The only integers $1 \leq n \leq 10000$ representable as a difference of two nonzero rational fourth powers are those listed explicitly in the paper.

This complements and completes prior computational surveys on sums, and extends previous works (notably, Cohen, Grechuk, Tho) by providing the full answer for the difference case in the large range $n \leq 10000$.

Methodology

The characterization leverages a layered approach, combining explicit algebraic computations, techniques from the arithmetic of elliptic curves, and descent methods. The essential methods can be summarized as follows:

Connection to Elliptic Curves

Any integer $n>0$ is representable as a difference of rational fourth powers iff the equation

$1 \leq n \leq 10000$0

has an integer solution $1 \leq n \leq 10000$1 with all variables nonzero. This transformation enables the application of elliptic curve theory via several key facts:

• Rational solutions to $1 \leq n \leq 10000$2 correspond to rational points on associated quadratic twists of the modular curve $1 \leq n \leq 10000$3 or $1 \leq n \leq 10000$4.
• If any of these curves has rank zero, all rational points are torsion and correspond to trivial (zero) solutions.

Reduction to Ternary Equations

Descent arguments further reduce the problem to finding integer solutions to equations of the form

$1 \leq n \leq 10000$5

with coprimality conditions among the variables. For each $1 \leq n \leq 10000$6, all possible factorizations into such triple $1 \leq n \leq 10000$7 (with $1 \leq n \leq 10000$8 in certain related sets) are enumerated and checked.

Algorithmic Sieve: Local Obstructions and Mordell-Weil Sieve

• For each $1 \leq n \leq 10000$9, the associated quartic-to-elliptic curve reductions are subjected to computational analysis for local solubility (i.e., solvability modulo small primes).
• When local solubility does not rule out global solutions, a Mordell-Weil sieve is applied: using knowledge of the finitely generated structure of elliptic curve rational points, congruence conditions modulo primes are extracted and used to rule out possible solutions, unless certain exceptional cases remain.
• Explicit elementary analysis via parametrizations of Pythagorean triples is used in particular configurations (e.g., when coefficients reduce the problem to norm equations or quadratic forms admitting explicit parametrizations).

Final Exceptional Cases and Factorization over $x, y$0

For the finitely many $x, y$1 not eliminated by the preceding methods, the authors employ a descent using factorization in the Gaussian integers $x, y$2. This entails expressing suitable values as norms or as products where local solubility at primes can be effectively checked, often by computer algebra (Magma). The process always leads to local obstructions, confirming the non-solubility globally.

Computational Implementation and Results

The authors implemented these procedures in Magma, with code designed to automate:

• Generation and checking of rational points on the relevant curves up to the prescribed bound ($x, y$3),
• Analysis of local solvability and reduction to explicit finite computational questions using the Mordell-Weil sieve,
• Handling all resulting exceptional cases, which include a small list of $x, y$4 for which more intricate factorization or descent is required.

A technical note is the focus on $x, y$5 that are fourth-power free, since higher powers can be absorbed via scaling in the variables.

The conclusion is exhaustive for the stated range: for every $x, y$6, either an explicit rational representation is found, or nonexistence is proven.

Implications and Comparison with Prior Work

• The result constitutes the first unconditional and effective complete classification for the difference case up to $x, y$7. In terms of methodology, it demonstrates the critical importance of bringing together advanced arithmetic geometry methods (curves of rank computation, sieving) with explicit descent arguments in classical settings.
• By tracing which $x, y$8 do and do not have rational representations as such differences, the work informs further investigations in rational points on higher-degree curves, the minimal sizes of solutions, and the statistical distributions of such representations.
• The contrast with the "sum of fourth powers" case is instructive: distribution and density questions, as well as potential analogues in higher degree or other forms (e.g., cubes, higher powers), can be formulated and investigated.

Future Directions

• Algorithmic Generalization: The approach can, in principle, be extended to even larger ranges, or to similar equations involving higher powers (e.g., differences of rational $x, y$9-th powers for $n = x^4 - y^4$0), though computational and theoretical complexity grow rapidly.
• Statistical and Asymptotic Behavior: With a complete list for a large range, one can approach the question of density or frequency: how often is an integer representable as such a difference, or what is the distribution of such $n = x^4 - y^4$1 within a given interval?
• Refinement of Sieve Techniques: The interplay of the Mordell-Weil sieve, explicit descent, and local obstruction analysis may spur algorithmic improvements or new heuristics for related integral or rational representation problems.
• Connections to Rational Points and the BSD Conjecture: The practical computation of curve ranks—and the identification of cases where all twists have positive rank but still no solution—is tightly linked to foundational conjectures and to the practical toolkit for arithmetic geometry.

Conclusion

This paper achieves a comprehensive resolution to the problem of determining which positive integers up to $n = x^4 - y^4$2 can be expressed as a difference of two nonzero rational fourth powers. The work is notable for the thorough integration of computational methods with deep arithmetic theory, and for definitively closing this chapter in explicit Diophantine analysis. Its tools and conclusions will inform further work both in classical number theory and in the arithmetic of curves.