Sums of a triangular number and three weighted cubes

Conjecture (i): Any positive integer can be written as w(w+1)/2 + x^3 + by^3 + cz^3 with w > 0 and x, y, z ≥ 0, provided that (b, c) ∈ (1,2), (1,3), (1,4), (1,6), (2,2), (2,3), (2,4), (2,5), (2,6), (2,7), (2,20), (2,21), (2,34), (3,3), (3,4), (3,5), (3,6), (4,10).

From the catalogue. Imported from The Formal Conjectures Authors (Google DeepMind and contributors) (Apache-2.0) — original. Nobody has started on it here yet: tasks are created as soon as someone asks for one or submits a claim.

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Cite

@misc{cairn-oeis-262880,
  title        = {Sums of a triangular number and three weighted cubes},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/oeis-262880}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-11-11}
}

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Claims

Verified

Disputed

Refuted

On the literature board

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What a result looks like: C-16, a newcomer working with an AI model found C(85,24,3) ≤ 76 (previous best 77), and a computer verified it within minutes. A documented dead end counts too.

The problem

The question

conjecture_1. Conjecture (i): Any positive integer can be written as $\frac{w ( w + 1 )}{2} + x^{3} + b y^{3} + c z^{3}$ with $w > 0$ and $x, y, z \geq 0$ , provided that $(b, c) \in \{(1,2), (1,3), (1,4), (1,6), (2,2), (2,3), (2,4), (2,5), (2,6), (2,7), (2,20), (2,21), (2,34), (3,3), (3,4), (3,5), (3,6), (4,10)\}$.

conjecture_2. Conjecture (ii): For $(b, c) \in {(3, 4), (3, 6), (4, 8)}$ , every nonnegative integer $n$ is of the form $\frac{w ( w + 1 )}{2} + 2 x^{3} + b y^{3} + c z^{3}$ for some nonnegative integers $w, x, y, z$ .

Number of ordered ways to write $n$ as $\frac{w ( w + 1 )}{2} + x^{3} + y^{3} + 2 z^{3}$ with $w > 0$ , $0 \leq x \leq y$ , and $z \geq 0$ .

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.OEIS.«262880» (2 statements).

theorem conjecture_1 (p : ℕ × ℕ) (hp : p ∈ pairs1) (n : ℕ) (hn : 0 < n) :
    ∃ w x y z : ℕ, 0 < w ∧ n = Nat.triangular w + x ^ 3 + p.1 * y ^ 3 + p.2 * z ^ 3

theorem conjecture_2 (p : ℕ × ℕ) (hp : p ∈ pairs2) (n : ℕ) :
    ∃ w x y z : ℕ, n = Nat.triangular w + 2 * x ^ 3 + p.1 * y ^ 3 + p.2 * z ^ 3

What counts as progress

A Lean proof of one of the statements above, pinned as the claim's formal statement.
Partial results: special cases, weaker bounds, reductions — as verified claims.
Computations and numerical evidence with published code (reproducible).
Literature: the problem may have been solved or partly solved already. Report it as a literature claim.
A precise flaw in the formal statement (a misformalisation) — report it upstream too.

References

A262880

Source and licence

Imported from Formal Conjectures (OEIS), commit 2f91d251c07f. Statements and descriptions © The Formal Conjectures Authors, Apache License 2.0; reformatted for this page.