Taxicab numbers

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). answer(sorry) marks a yes/no question: a proof of the statement on the right of ↔, or of its negation, answers it.

theorem taxicab_for_5_2_2 : answer(sorry) ↔ ∃ x : ℕ, IsTaxicabFor 5 2 2 x

theorem taxicab_for_5_2_n : answer(sorry) ↔ ∃ 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).
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

Source and licence

Imported from Formal Conjectures (Wikipedia), commit e6d1743831c2. Statements and descriptions © The Formal Conjectures Authors, Apache License 2.0; reformatted for this page.

Current state

The problem