[Paper Review] Diophantine Equation $X^4+Y^4=2(U^4+V^4)$

Farzali Izadi, Kamran Nabardi|arXiv (Cornell University)|Jan 23, 2015

Algebraic Geometry and Number Theory3 references3 citations

TL;DR

This paper uses the theory of elliptic curves to generate infinitely many integer solutions to the Diophantine equation $X^4 + Y^4 = 2(U^4 + V^4)$, leveraging the congruent number curve $y^2 = x^3 - 36x$. The key contribution is a parametric family of solutions derived from the Mordell-Weil group of the curve, with rank 1 and generator $(-3, 9)$, yielding primitive solutions via recursive point multiplication and normalization.

ABSTRACT

In this paper, the theory of elliptic curves is used for finding the solutions of the quartic Diophantine equation $X^4+Y^4=2(U^4+V^4)$ Keywords: Diophantine equation, Elliptic curve, Congruent number

Motivation & Objective

• To find infinitely many integer solutions to the quartic Diophantine equation $X^4 + Y^4 = 2(U^4 + V^4)$.
• To apply the theory of elliptic curves, particularly congruent number curves, to parametrize solutions of this equation.
• To generate primitive solutions by normalizing integer solutions derived from rational points on the elliptic curve $E_6: y^2 = x^3 - 36x$.
• To demonstrate that the equation has infinitely many solutions by exploiting the Mordell-Weil group structure of the associated elliptic curve.

Proposed method

• Transform the equation $X^4 + Y^4 = 2(U^4 + V^4)$ via substitution $X = U + t$, $Y = U - t$, leading to $6t^2U^2 + t^4 = V^4$.
• Use the identity $6Z^2 = V^4 - t^4$ with $Z = tU$, linking the equation to the generalized congruent number problem.
• Apply Proposition 1.1, which connects solutions of $X^4 - Y^4 = cZ^2$ to rational points on $E_c: y^2 = x^3 - c^2x$, with $c = 6$.
• Use the generator $P = (-3, 9)$ of $E_6(bQ)$ with rank 1 to generate an infinite sequence of rational points via scalar multiplication $nP$.
• Express solutions in terms of division polynomials $\psi_n$, $\phi_n$, and $\omega_n$ evaluated at $(-3, 9)$, yielding parametric formulas for $X_n, Y_n, U_n, V_n$.
• Normalize each solution by dividing by the GCD $d_n = \gcd(X_n, Y_n, U_n, V_n)$ to obtain primitive solutions $(A_n, B_n, C_n, D_n)$.

Experimental results

Research questions

• RQ1 Does the Diophantine equation $X^4 + Y^4 = 2(U^4 + V^4)$ admit infinitely many integer solutions?
• RQ2 Can the theory of elliptic curves be used to parametrize solutions of this quartic equation?
• RQ3 What is the role of congruent number curves in solving $X^4 + Y^4 = 2(U^4 + V^4)$?
• RQ4 How can rational points on $E_6: y^2 = x^3 - 36x$ be used to generate integer solutions to the equation?
• RQ5 What is the structure of the solution set, and how can primitive solutions be systematically extracted?

Key findings

• The elliptic curve $E_6: y^2 = x^3 - 36x$ has rank 1 and is generated by the point $(-3, 9)$, enabling the construction of infinitely many rational points.
• The smallest known solution $(X, Y, U, V) = (21, 19, 20, 7)$ arises from the point $2P = (-3, 9)$, scaled by a factor of 9.
• For $n = 2$, the primitive solution is $(A_2, B_2, C_2, D_2) = (988521, -1661081, -336280, -1437599)$.
• For $n = 3$, the primitive solution is $(A_3, B_3, C_3, D_3) = (-22394369951939, -59719152671941, -41056761311940, 43690772126393)$.
• For $n = 4$, the solution reaches values on the order of $10^{21}$, such as $A_4 = 5009010521962601088594641$.
• For $n = 5$, the solution components exceed $10^{28}$, with $A_5 = 385103462588108468740542460457075040101$, confirming rapid growth of the solution sequence.