Sums of a square and three weighted cubes

Conjecture (i): For any m ∈ 3, 4, 5, 6 and n ≥ 0, there are nonnegative integers w, x, y, z such that n = w^2 + x^3 + 2y^3 + mz^3.

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-262824,
  title        = {Sums of a square and three weighted cubes},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/oeis-262824}},
  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): For any $m \in {3, 4, 5, 6}$ and $n \geq 0$ , there are nonnegative integers $w, x, y, z$ such that $n = w^{2} + x^{3} + 2 y^{3} + m z^{3}$ .

conjecture_2. Conjecture (ii): Every nonnegative integer $n$ can be written as $w^{2} + x^{3} + 2 y^{3} + z^{4}$ for some nonnegative integers $w, x, y, z$ .

conjecture_3. Conjecture (ii): Every nonnegative integer $n$ can be written as $w^{2} + x^{3} + 2 y^{3} + 3 z^{4}$ for some nonnegative integers $w, x, y, z$ .

conjecture_4. Conjecture (ii): Every nonnegative integer $n$ can be written as $w^{2} + x^{3} + 2 y^{3} + 6 z^{4}$ for some nonnegative integers $w, x, y, z$ .

conjecture_5. Conjecture (ii): Every nonnegative integer $n$ can be written as $2 w^{2} + x^{3} + 4 y^{3} + z^{4}$ for some nonnegative integers $w, x, y, z$ .

Number of ordered ways to write $n$ as $w^{2} + x^{3} + 2 y^{3} + 3 z^{3}$ , where $w, x, y$ and $z$ are nonnegative integers.

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.OEIS.«262824» (5 statements).

theorem conjecture_1 (m : ℕ) (hm : m ∈ ({3, 4, 5, 6} : Finset ℕ)) (n : ℕ) :
    ∃ w x y z : ℕ, n = w ^ 2 + x ^ 3 + 2 * y ^ 3 + m * z ^ 3

theorem conjecture_2 (n : ℕ) :
    ∃ w x y z : ℕ, n = w ^ 2 + x ^ 3 + 2 * y ^ 3 + z ^ 4

theorem conjecture_3 (n : ℕ) :
    ∃ w x y z : ℕ, n = w ^ 2 + x ^ 3 + 2 * y ^ 3 + 3 * z ^ 4

theorem conjecture_4 (n : ℕ) :
    ∃ w x y z : ℕ, n = w ^ 2 + x ^ 3 + 2 * y ^ 3 + 6 * z ^ 4

theorem conjecture_5 (n : ℕ) :
    ∃ w x y z : ℕ, n = 2 * w ^ 2 + x ^ 3 + 4 * y ^ 3 + z ^ 4

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

A262824

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.