Sums of three cubes¶
Integer representability by three signed cubes, subject to a modulo-nine obstruction.
Core Idea¶
The sum-of-three-cubes question fixes an integer k and asks whether three signed integer cubes add to it. A solution triple is an exact witness. Elementary modular arithmetic excludes k≡4 or 5 (mod 9), since each cube is 0 or ±1 modulo 9. For other residue classes, the modular filter alone neither produces a solution nor proves one must exist. The general characterization remains open.
Small targets can have obvious witnesses, while others demand enormous signed integers with cancellation. Booker's solution for 33 is a published computational discovery, not merely a reported approximation. Booker and Sutherland later found a solution for 42 by large-scale search. Each solved target adds knowledge about that integer while leaving the cross-target sufficiency question unresolved.
Scope of Application¶
The name denotes a precise integer existence question, not merely any use of cubic terms.
- Number theory. Analyze a Diophantine equation over integers.
- Modular reasoning. Exclude two residue classes before search.
- Computational mathematics. Find and verify large signed witnesses.
- Open-problem tracking. Distinguish known solutions from the unresolved universal question.
Clarity¶
For a chosen integer k, find signed integers x,y,z with x³+y³+z³=k. If k is 4 or 5 modulo 9, this is impossible. Passing that filter is not a guarantee: the general sufficiency question remains open. A published triple for k=33 proves only that instance.
Manages Complexity¶
A tiny target may require enormous positive and negative cubes that almost cancel. Search strategies exploit arithmetic structure; exact integer verification is essential. A failed search within a bound is not an impossibility theorem, and the modular obstruction must not be promoted from a necessary condition into an unproved sufficiency claim.
Abstract Reasoning¶
Fix k; apply the mod-nine obstruction; if eligible, construct or search for signed integers; verify equality exactly; separate one-target evidence from a universal theorem.
Knowledge Transfer¶
Modular filtering and witness verification recur in Diophantine problems, but the exact three signed cubes and residues ⅘ mod 9 define this specific problem. Replacing cubes or the number of variables changes the mathematical identity.
Relationships to Other Abstractions¶
Current abstraction Sums of three cubes Domain-specific
Parents (1) — more general patterns this builds on
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Sums of three cubes is a kind of Diophantine Equation Domain-specific
Sums of three cubes asks for integer solutions to x3+y3+z^3=k, which is by definition a polynomial equation with integer coefficients requiring integer solutions.
Hierarchy path (1) — routes to 1 parentless root
- Sums of three cubes → Diophantine Equation → Constraint
Neighborhood in Abstraction Space¶
Sums of three cubes sits in a moderately populated region (47th percentile for distinctiveness): it has near-neighbors but no dense thicket of look-alikes.
Family — Cryptographic & Combinatorial Hardness Problems (5 abstractions)
Nearest neighbors
- Numerical Method — 0.89
- Diffie–Hellman problem — 0.88
- Short Integer Solution Problem — 0.88
- Primitive Semiperfect Number — 0.86
- Minimal Axioms for Boolean Algebra — 0.85
Computed from structural-signature embeddings · 2026-10-08