[Paper Review] Newer sums of three cubes

Sander G. Huisman|arXiv (Cornell University)|Apr 26, 2016

Analytic Number Theory Research7 references3 citations

TL;DR

This paper presents a computational search for integer solutions to the Diophantine equation $x^3 + y^3 + z^3 = k$ with bounds up to $10^{15}$, extending previous work. It reports the first solution for $k = 74$ and discovers 966 new solutions, leaving only 13 values of $k < 1000$ unresolved, including $k = 33$ and $k = 42$. The method leverages lattice reduction and rational point approximation on cubic curves to efficiently explore solution space near $X^3 + Y^3 = 1$.

ABSTRACT

The search of solutions of the Diophantine equation $x^3 + y^3 + z^3 = k$ for $k&lt;1000$ has been extended with bounds of $|x|$, $|y|$ and $|z|$ up to $10^{15}$. The first solution for $k=74$ is reported. This only leaves $k=33$ and $k=42$ for $k&lt;100$ for which no solution has yet been found. A total of 966 new solutions were found.

Motivation & Objective

• Address the long-standing open problem of determining which natural numbers $k$ can be expressed as the sum of three integer cubes.
• Extend the search bounds for $x$, $y$, and $z$ beyond previous limits to $10^{15}$ to uncover previously unknown solutions.
• Provide computational evidence supporting the conjecture that all $k \not\equiv \pm 4 \pmod{9}$ and $k \neq 0$ can be written as the sum of three cubes.
• Identify new solutions for $k$ values with few or no known solutions, including multiple solutions for certain $k$.
• Contribute to the understanding of solution density and distribution in the context of cubic Diophantine equations.

Proposed method

• The search employs a method based on Elkies’ algorithm, reformulating the equation as $X^3 + Y^3 = 1 + (k/z)^3$ with $X = x/z$, $Y = y/z$ to locate rational points near the curve $X^3 + Y^3 = 1$.
• Lattice basis reduction is applied to transform the search space into a geometry more amenable to efficient iteration over integer lattice points within a pyramid-shaped region.
• Solution candidates are identified by covering the cubic curve with small parallelogram-shaped 'flagstones' that tile the neighborhood of the curve, ensuring no solutions are missed.
• Flagstones are sized according to the search bound $B$ and the maximum $k$, enabling systematic coverage of potential solutions within the bounds.
• The algorithm uses symmetry to restrict the search to $0 < X < \sqrt[3]{1/2}$, reducing computational load by half.
• High-performance computing on the PSMN cluster at École Normale Supérieure de Lyon enabled the execution of over $10^5$ hours of computation to achieve the results.

Experimental results

Research questions

• RQ1 Can the Diophantine equation $x^3 + y^3 + z^3 = k$ be solved for $k = 74$, which had no known solution prior to this work?
• RQ2 Are there new solutions for $k < 1000$ that were previously unknown, particularly for values with only one or two known solutions?
• RQ3 Does extending the search bound to $10^{15}$ significantly increase the number of known solutions, and how does this compare to theoretical expectations?
• RQ4 Can the computational method efficiently uncover solutions with very large absolute values of $x$, $y$, and $z$?
• RQ5 Which values of $k < 1000$ remain unsolved after this extended search, and what does this imply for the broader conjecture on the solvability of the equation?

Key findings

• The first solution for $k = 74$ was discovered: $74 = (-284650292555885)^3 + 66229832190556^3 + 283450105697727^3$.
• A total of 966 new solutions were found for $k < 1000$, significantly expanding the known solution set.
• Two new solutions were found for $k = 606$, $k = 830$, and $k = 966$, indicating multiple solutions exist for these values.
• Nine values of $k$ that previously had only two known solutions now have additional solutions, including $k = 30, 87, 327, 402, 483, 735, 758, 786, 831$.
• Despite the extended search bound of $10^{15}$, 13 values of $k < 1000$ remain unsolved: $33, 42, 114, 165, 390, 579, 627, 633, 732, 795, 906, 921, 975$.
• The solution density aligns with the expected $\log(B)$ scaling, supporting the conjecture that solutions exist for all $k \not\equiv \pm 4 \pmod{9}$, $k \neq 0$. The search results are consistent with the Brauer–Manin obstruction being trivial for these equations.