Taxicab numbers

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import FormalConjecturesUtil

/-!
# Taxicab numbers

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 = 1³ + 12³ = 9³ + 10³, $
also known as the Hardy–Ramanujan number.

However, a taxicab number is not known for $k=5$, $m=2$, and any $n ≥ 2$:
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 $n=2$.

*References:*
 - [Wikipedia](https://en.wikipedia.org/wiki/Taxicab_number)
 - [Generalized taxicab number](https://en.wikipedia.org/wiki/Generalized_taxicab_number)
 - [OEIS taxicab cubes](https://oeis.org/A001235)
 - [OEIS taxicab 4th powers](https://oeis.org/A018786)
 - [OEIS taxicab conjecture](https://oeis.org/A088703)
-/

namespace Taxicab

/-- $x$ is a candidate for being a taxicab number for $k, m, n$
if there exists a (finite) set of at least $n$ distinct,
pairwise disjoint, non-empty, non-zero lists of length
$m$, such that the sum of the $k$-th powers of the
elements of each list is $x$. The disjointness condition
ensures that the representations do not share any common terms.
-/
def IsTaxicabFor' (k m n x : ℕ) : Prop :=
  ∃ (S : Finset (List ℕ)),
  S.card ≥ n ∧
  ∀ L ∈ S, ∀ M ∈ S, L ≠ M → List.Disjoint L M ∧
  (∀ L ∈  S, L.length = m ∧ L ≠ [] ∧ 0 ∉ L ∧ (L.map (· ^ k)).sum = x)

/-- $1729$ is a possible taxicab number for $k=3, m=2, n=2$.
-/
@[category test, AMS 11]
theorem taxicab_1729 : IsTaxicabFor' 3 2 2 1729 := by
  use {{1, 12}, {9, 10}}
  simp [List.Disjoint]
  simp +decide

/-- $x$ is a taxicab number if it is the smallest number
that can be expressed as a sum of $m$ positive $k$-th
powers in at least $n$ distinct ways.
-/
def IsTaxicabFor (k m n : ℕ) (x : ℕ) : Prop :=
  IsLeast { x : ℕ | IsTaxicabFor' k m n x } x

@[category test, AMS 11]
theorem taxicab_4' : IsTaxicabFor' 1 2 2 4 := by
  use {[1, 3], [2, 2]}
  simp [List.Disjoint]

/-- Using Aristotle (Harmonic) we get a compact proof that
4 is the taxicab number for $k=1, m=2, n=2$. -/
@[category test, AMS 11]
theorem taxicab_4 : IsTaxicabFor 1 2 2 4 := by
  constructor
  · exact taxicab_4'
  · rintro x ⟨ S, hS₁, hS₂ ⟩
    obtain ⟨ s, hs, t, ht, hst ⟩ := Finset.one_lt_card.mp hS₁
    have := hS₂ s hs t ht hst
    rcases this with ⟨ h₁, h₂ ⟩
    specialize h₂ s hs
    rcases s with ( _ | ⟨ a, _ | ⟨ b, _ | s ⟩ ⟩ ) <;> simp_all +arith +decide
    rcases t with ( _ | ⟨ c, _ | ⟨ d, _ | t ⟩ ⟩ ) <;> simp_all +arith +decide
    · grind
    · have := hS₂ _ ht _ hs
      simp_all +decide
      grind
    · have := hS₂ _ hs _ ht
      simp_all +decide [ List.Disjoint ]
      have := this.2 _ hs
      have := this.2.2
      simp_all +arith +decide
      grind
    · have := hS₂ _ hs _ ht
      simp_all +decide [ List.Disjoint ]
      grind +ring

/-- Taxicab number for $k=5$, $m=2$, and $n=2$ is not known.
Whether such a number exists is also not known. -/
@[category research open, AMS 11]
theorem taxicab_for_5_2_2 : answer(sorry) ↔ ∃ x : ℕ, IsTaxicabFor 5 2 2 x := by
  sorry

/-- Taxicab number for $k=5$ and $m=2$ is not-known for any $n ≥ 2$.
Whether such a number exists is also not known. -/
@[category research open, AMS 11]
theorem taxicab_for_5_2_n : answer(sorry) ↔ ∃ n : ℕ, n ≥ 2 ∧ (∃ x : ℕ, IsTaxicabFor 5 2 n x)
  := by sorry

end Taxicab

• Formalization source ↗

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