Taxicab numbers — open problem in number theory

Taxicab numbers

Taxicab number for k=5, m=2, and n=2 is not known. Whether such a number exists is also not known.

Current state (tracked automatically)

All tracked facts and sources

Fact	Value	Source
Formal Conjectures: research statements in the file	2 open, 0 solved	Formal Conjectures

Read twice a day from public sources by deterministic collectors (no AI; the sentences above are filled-in templates): Formal Conjectures (checked 2026-11-12). How it works.

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-taxicab,
  title        = {Taxicab numbers},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/taxicab}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-11-12}
}

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Claims

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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

taxicab_for_5_2_2. Taxicab number for $k = 5$ , $m = 2$ , and $n = 2$ is not known. Whether such a number exists is also not known.

taxicab_for_5_2_n. Taxicab number for $k = 5$ and $m = 2$ is not-known for any $n \geq 2$ . Whether such a number exists is also not known.

A taxicab number for natural numbers $k, m, n$ is the smallest number $x$ that can be expressed as a sum of $m$ positive $k$ -th powers in at least $n$ distinct ways. The most famous taxicab number is $1729 = 1^{3} + 1 2^{3} = 9^{3} + 1 0^{3},$ also known as the Hardy–Ramanujan number.

However, a taxicab number is not known for $k = 5$ , $m = 2$ , and any $n \geq 2$ : No positive integer is known that can be written as the sum of two 5th powers in more than one way, and it is not known whether such a number exists.

In particular, it is not known whether there exists a taxicab number for $k = 5$ , $m = 2$ , and $n = 2$ .

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.Wikipedia.Taxicab (2 statements).

theorem taxicab_for_5_2_2 : ∃ x : ℕ, IsTaxicabFor 5 2 2 x

theorem taxicab_for_5_2_n : ∃ n : ℕ, n ≥ 2 ∧ (∃ x : ℕ, IsTaxicabFor 5 2 n x)

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).
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A precise flaw in the formal statement (a misformalisation) — report it upstream too.

References

Source and licence

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