Sums of two cubes: is the representation count $1_A*1_A(n)\ll(\log n)^{O(1)}$? (Erdős #829)
Statement
Let $A=\{1,8,27,64,\ldots\}\subset\mathbb{N}$ be the set of positive perfect cubes, and for $n\in\mathbb{N}$ let $1_A\ast 1_A(n)=\#\{(a,b):a,b\in A,\ a+b=n\}$ be the number of representations of $n$ as an ordered sum of two positive cubes. Is it true that $$1_A\ast 1_A(n)\ll (\log n)^{O(1)}?$$ That is, is the number of such representations bounded by a fixed power of $\log n$?
Acceptance. FULLY RESOLVES: a complete proof (Lean/Coq preferred, otherwise a full written proof) that $1_A\ast 1_A(n)\ll(\log n)^{O(1)}$ — i.e. there exist constants $C,K$ with $1_A\ast 1_A(n)\leq K(\log n)^{C}$ for all $n\geq 2$ — OR a disproof exhibiting a mechanism that forces $1_A\ast 1_A(n)$ above every fixed power of $\log n$ infinitely often. ADVANCES: a proven improvement of the infinitely-often lower bound beyond Stewart's $(\log n)^{11/13}$ (state $11/13$ as the bar; require a strictly larger proved exponent), with full proof; or a proven partial upper bound, e.g. $1_A\ast 1_A(n)\ll n^{o(1)}$ with an explicit decay rate, or a rigorous average/normal-order bound on the representation count. Deliver the proof (formal or written) with the explicit exponent or bound.
Background
Posed by Erdős [Er83]; listed as open on erdosproblems.com/829 (fetched 2026-07-21, status 'open'). The known frontier concerns how large the representation count can be infinitely often (the opposite direction to the conjectured upper bound). Mordell proved $\limsup_{n\to\infty}1_A\ast 1_A(n)=\infty$; Mahler [Ma35b] proved $1_A\ast 1_A(n)\gg(\log n)^{1/4}$ for infinitely many $n$; and Stewart [St08] improved this to $1_A\ast 1_A(n)\gg(\log n)^{11/13}$ for infinitely many $n$. The conjectured upper bound $(\log n)^{O(1)}$ would show the count never exceeds polylogarithmic size — an enormous gap remains, as no polylogarithmic (indeed no $n^{o(1)}$ with explicit rate) upper bound is currently known. A Lean formalisation exists in the DeepMind formal-conjectures repository. Attacker's tool: bounds on the number of integer points on the surface $a^3+b^3=n$ via Thue-equation / elliptic-curve rank estimates, together with computational enumeration of 'generalised taxicab' numbers ($n$ admitting many representations as a sum of two cubes) to extend the empirical lower-bound records.
References
| Ref | Source | Type |
|---|---|---|
| REF-01 | Erdős Problem #829 (T. F. Bloom) | website |
| REF-02 | Lean formalisation of Erdős #829 (DeepMind formal-conjectures) | website |
Investigations · 0
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