FormalConjectures.Wikipedia.Taxicab

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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

Generalized taxicab number

OEIS taxicab cubes

OEIS taxicab 4th powers

OEIS taxicab conjecture


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⊢ IsTaxicabFor' 3 2 2 1729
  use {{1, 12}, {9, 10}}h⊢ {{1, 12}, {9, 10}}.card ≥ 2 ∧
  ∀ L ∈ {{1, 12}, {9, 10}},
    ∀ M ∈ {{1, 12}, {9, 10}},
      L ≠ M →
        L.Disjoint M ∧
          ∀ L ∈ {{1, 12}, {9, 10}}, L.length = 2 ∧ L ≠ [] ∧ 0 ∉ L ∧ (List.map (fun x => x ^ 3) L).sum = 1729
  simp [List.Disjoint]h⊢ 2 ≤ {{1, 12}, {9, 10}}.card ∧
  (¬{1, 12} = {9, 10} →
      (∀ ⦃a : ℕ⦄, a ∈ {1, 12} → a ∉ {9, 10}) ∧
        ({1, 12}.length = 2 ∧ ¬{1, 12} = [] ∧ 0 ∉ {1, 12} ∧ (List.map (fun x => x ^ 3) {1, 12}).sum = 1729) ∧
          {9, 10}.length = 2 ∧ ¬{9, 10} = [] ∧ 0 ∉ {9, 10} ∧ (List.map (fun x => x ^ 3) {9, 10}).sum = 1729) ∧
    (¬{9, 10} = {1, 12} →
      (∀ ⦃a : ℕ⦄, a ∈ {9, 10} → a ∉ {1, 12}) ∧
        ({1, 12}.length = 2 ∧ ¬{1, 12} = [] ∧ 0 ∉ {1, 12} ∧ (List.map (fun x => x ^ 3) {1, 12}).sum = 1729) ∧
          {9, 10}.length = 2 ∧ ¬{9, 10} = [] ∧ 0 ∉ {9, 10} ∧ (List.map (fun x => x ^ 3) {9, 10}).sum = 1729)
  simp +decideAll goals completed! 🐙

$x$ 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⊢ IsTaxicabFor' 1 2 2 4
  use {[1, 3], [2, 2]}h⊢ {[1, 3], [2, 2]}.card ≥ 2 ∧
  ∀ L ∈ {[1, 3], [2, 2]},
    ∀ M ∈ {[1, 3], [2, 2]},
      L ≠ M →
        L.Disjoint M ∧ ∀ L ∈ {[1, 3], [2, 2]}, L.length = 2 ∧ L ≠ [] ∧ 0 ∉ L ∧ (List.map (fun x => x ^ 1) L).sum = 4
  simp [List.Disjoint]All goals completed! 🐙

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⊢ IsTaxicabFor 1 2 2 4
  constructorleft⊢ 4 ∈ {x | IsTaxicabFor' 1 2 2 x}right⊢ 4 ∈ lowerBounds {x | IsTaxicabFor' 1 2 2 x}
  ·left⊢ 4 ∈ {x | IsTaxicabFor' 1 2 2 x} exact taxicab_4'All goals completed! 🐙
  ·right⊢ 4 ∈ lowerBounds {x | IsTaxicabFor' 1 2 2 x} rintro x ⟨ S, hS₁, hS₂ ⟩rightx:ℕS:Finset (List ℕ)hS₁:S.card ≥ 2hS₂:∀ L ∈ S, ∀ M ∈ S, L ≠ M → L.Disjoint M ∧ ∀ L ∈ S, L.length = 2 ∧ L ≠ [] ∧ 0 ∉ L ∧ (List.map (fun x => x ^ 1) L).sum = x⊢ 4 ≤ x
    obtain ⟨ s, hs, t, ht, hst ⟩ := Finset.one_lt_card.mp hS₁rightx:ℕS:Finset (List ℕ)hS₁:S.card ≥ 2hS₂:∀ L ∈ S, ∀ M ∈ S, L ≠ M → L.Disjoint M ∧ ∀ L ∈ S, L.length = 2 ∧ L ≠ [] ∧ 0 ∉ L ∧ (List.map (fun x => x ^ 1) L).sum = xs:List ℕhs:s ∈ St:List ℕht:t ∈ Shst:s ≠ t⊢ 4 ≤ x
    have := hS₂ s hs t ht hstrightx:ℕS:Finset (List ℕ)hS₁:S.card ≥ 2hS₂:∀ L ∈ S, ∀ M ∈ S, L ≠ M → L.Disjoint M ∧ ∀ L ∈ S, L.length = 2 ∧ L ≠ [] ∧ 0 ∉ L ∧ (List.map (fun x => x ^ 1) L).sum = xs:List ℕhs:s ∈ St:List ℕht:t ∈ Shst:s ≠ tthis:s.Disjoint t ∧ ∀ L ∈ S, L.length = 2 ∧ L ≠ [] ∧ 0 ∉ L ∧ (List.map (fun x => x ^ 1) L).sum = x := hS₂ s hs t ht hst⊢ 4 ≤ x
    rcases this with ⟨ h₁, h₂ ⟩rightx:ℕS:Finset (List ℕ)hS₁:S.card ≥ 2hS₂:∀ L ∈ S, ∀ M ∈ S, L ≠ M → L.Disjoint M ∧ ∀ L ∈ S, L.length = 2 ∧ L ≠ [] ∧ 0 ∉ L ∧ (List.map (fun x => x ^ 1) L).sum = xs:List ℕhs:s ∈ St:List ℕht:t ∈ Shst:s ≠ th₁:s.Disjoint th₂:∀ L ∈ S, L.length = 2 ∧ L ≠ [] ∧ 0 ∉ L ∧ (List.map (fun x => x ^ 1) L).sum = x⊢ 4 ≤ x
    specialize h₂ s hsrightx:ℕS:Finset (List ℕ)hS₁:S.card ≥ 2hS₂:∀ L ∈ S, ∀ M ∈ S, L ≠ M → L.Disjoint M ∧ ∀ L ∈ S, L.length = 2 ∧ L ≠ [] ∧ 0 ∉ L ∧ (List.map (fun x => x ^ 1) L).sum = xs:List ℕhs:s ∈ St:List ℕht:t ∈ Shst:s ≠ th₁:s.Disjoint th₂:s.length = 2 ∧ s ≠ [] ∧ 0 ∉ s ∧ (List.map (fun x => x ^ 1) s).sum = x⊢ 4 ≤ x
    rcases s with ( _ | ⟨ a, _ | ⟨ b, _ | s ⟩ ⟩ )right.nilx:ℕS:Finset (List ℕ)hS₁:S.card ≥ 2hS₂:∀ L ∈ S, ∀ M ∈ S, L ≠ M → L.Disjoint M ∧ ∀ L ∈ S, L.length = 2 ∧ L ≠ [] ∧ 0 ∉ L ∧ (List.map (fun x => x ^ 1) L).sum = xt:List ℕht:t ∈ Shs:[] ∈ Shst:[] ≠ th₁:[].Disjoint th₂:[].length = 2 ∧ [] ≠ [] ∧ 0 ∉ [] ∧ (List.map (fun x => x ^ 1) []).sum = x⊢ 4 ≤ xright.cons.nilx:ℕS:Finset (List ℕ)hS₁:S.card ≥ 2hS₂:∀ L ∈ S, ∀ M ∈ S, L ≠ M → L.Disjoint M ∧ ∀ L ∈ S, L.length = 2 ∧ L ≠ [] ∧ 0 ∉ L ∧ (List.map (fun x => x ^ 1) L).sum = xt:List ℕht:t ∈ Sa:ℕhs:[a] ∈ Shst:[a] ≠ th₁:[a].Disjoint th₂:[a].length = 2 ∧ [a] ≠ [] ∧ 0 ∉ [a] ∧ (List.map (fun x => x ^ 1) [a]).sum = x⊢ 4 ≤ xright.cons.cons.nilx:ℕS:Finset (List ℕ)hS₁:S.card ≥ 2hS₂:∀ L ∈ S, ∀ M ∈ S, L ≠ M → L.Disjoint M ∧ ∀ L ∈ S, L.length = 2 ∧ L ≠ [] ∧ 0 ∉ L ∧ (List.map (fun x => x ^ 1) L).sum = xt:List ℕht:t ∈ Sa:ℕb:ℕhs:[a, b] ∈ Shst:[a, b] ≠ th₁:[a, b].Disjoint th₂:[a, b].length = 2 ∧ [a, b] ≠ [] ∧ 0 ∉ [a, b] ∧ (List.map (fun x => x ^ 1) [a, b]).sum = x⊢ 4 ≤ xright.cons.cons.consx:ℕS:Finset (List ℕ)hS₁:S.card ≥ 2hS₂:∀ L ∈ S, ∀ M ∈ S, L ≠ M → L.Disjoint M ∧ ∀ L ∈ S, L.length = 2 ∧ L ≠ [] ∧ 0 ∉ L ∧ (List.map (fun x => x ^ 1) L).sum = xt:List ℕht:t ∈ Sa:ℕb:ℕs:ℕtail✝:List ℕhs:a :: b :: s :: tail✝ ∈ Shst:a :: b :: s :: tail✝ ≠ th₁:(a :: b :: s :: tail✝).Disjoint th₂:(a :: b :: s :: tail✝).length = 2 ∧
  a :: b :: s :: tail✝ ≠ [] ∧ 0 ∉ a :: b :: s :: tail✝ ∧ (List.map (fun x => x ^ 1) (a :: b :: s :: tail✝)).sum = x⊢ 4 ≤ x <;>right.nilx:ℕS:Finset (List ℕ)hS₁:S.card ≥ 2hS₂:∀ L ∈ S, ∀ M ∈ S, L ≠ M → L.Disjoint M ∧ ∀ L ∈ S, L.length = 2 ∧ L ≠ [] ∧ 0 ∉ L ∧ (List.map (fun x => x ^ 1) L).sum = xt:List ℕht:t ∈ Shs:[] ∈ Shst:[] ≠ th₁:[].Disjoint th₂:[].length = 2 ∧ [] ≠ [] ∧ 0 ∉ [] ∧ (List.map (fun x => x ^ 1) []).sum = x⊢ 4 ≤ xright.cons.nilx:ℕS:Finset (List ℕ)hS₁:S.card ≥ 2hS₂:∀ L ∈ S, ∀ M ∈ S, L ≠ M → L.Disjoint M ∧ ∀ L ∈ S, L.length = 2 ∧ L ≠ [] ∧ 0 ∉ L ∧ (List.map (fun x => x ^ 1) L).sum = xt:List ℕht:t ∈ Sa:ℕhs:[a] ∈ Shst:[a] ≠ th₁:[a].Disjoint th₂:[a].length = 2 ∧ [a] ≠ [] ∧ 0 ∉ [a] ∧ (List.map (fun x => x ^ 1) [a]).sum = x⊢ 4 ≤ xright.cons.cons.nilx:ℕS:Finset (List ℕ)hS₁:S.card ≥ 2hS₂:∀ L ∈ S, ∀ M ∈ S, L ≠ M → L.Disjoint M ∧ ∀ L ∈ S, L.length = 2 ∧ L ≠ [] ∧ 0 ∉ L ∧ (List.map (fun x => x ^ 1) L).sum = xt:List ℕht:t ∈ Sa:ℕb:ℕhs:[a, b] ∈ Shst:[a, b] ≠ th₁:[a, b].Disjoint th₂:[a, b].length = 2 ∧ [a, b] ≠ [] ∧ 0 ∉ [a, b] ∧ (List.map (fun x => x ^ 1) [a, b]).sum = x⊢ 4 ≤ xright.cons.cons.consx:ℕS:Finset (List ℕ)hS₁:S.card ≥ 2hS₂:∀ L ∈ S, ∀ M ∈ S, L ≠ M → L.Disjoint M ∧ ∀ L ∈ S, L.length = 2 ∧ L ≠ [] ∧ 0 ∉ L ∧ (List.map (fun x => x ^ 1) L).sum = xt:List ℕht:t ∈ Sa:ℕb:ℕs:ℕtail✝:List ℕhs:a :: b :: s :: tail✝ ∈ Shst:a :: b :: s :: tail✝ ≠ th₁:(a :: b :: s :: tail✝).Disjoint th₂:(a :: b :: s :: tail✝).length = 2 ∧
  a :: b :: s :: tail✝ ≠ [] ∧ 0 ∉ a :: b :: s :: tail✝ ∧ (List.map (fun x => x ^ 1) (a :: b :: s :: tail✝)).sum = x⊢ 4 ≤ x simp_all +arith +decideAll goals completed! 🐙
    rcases t with ( _ | ⟨ c, _ | ⟨ d, _ | t ⟩ ⟩ )right.cons.cons.nil.nilx:ℕS:Finset (List ℕ)a:ℕb:ℕhS₁:2 ≤ S.cardhS₂:∀ L ∈ S, ∀ M ∈ S, ¬L = M → L.Disjoint M ∧ ∀ L ∈ S, L.length = 2 ∧ ¬L = [] ∧ 0 ∉ L ∧ L.sum = xhs:[a, b] ∈ Sh₂:(¬0 = a ∧ ¬0 = b) ∧ a + b = xht:[] ∈ Shst:¬[a, b] = []⊢ 4 ≤ xright.cons.cons.nil.cons.nilx:ℕS:Finset (List ℕ)a:ℕb:ℕhS₁:2 ≤ S.cardhS₂:∀ L ∈ S, ∀ M ∈ S, ¬L = M → L.Disjoint M ∧ ∀ L ∈ S, L.length = 2 ∧ ¬L = [] ∧ 0 ∉ L ∧ L.sum = xhs:[a, b] ∈ Sh₂:(¬0 = a ∧ ¬0 = b) ∧ a + b = xc:ℕht:[c] ∈ Shst:¬[a, b] = [c]⊢ 4 ≤ xright.cons.cons.nil.cons.cons.nilx:ℕS:Finset (List ℕ)a:ℕb:ℕhS₁:2 ≤ S.cardhS₂:∀ L ∈ S, ∀ M ∈ S, ¬L = M → L.Disjoint M ∧ ∀ L ∈ S, L.length = 2 ∧ ¬L = [] ∧ 0 ∉ L ∧ L.sum = xhs:[a, b] ∈ Sh₂:(¬0 = a ∧ ¬0 = b) ∧ a + b = xc:ℕd:ℕht:[c, d] ∈ Shst:¬[a, b] = [c, d]⊢ 4 ≤ xright.cons.cons.nil.cons.cons.consx:ℕS:Finset (List ℕ)a:ℕb:ℕhS₁:2 ≤ S.cardhS₂:∀ L ∈ S, ∀ M ∈ S, ¬L = M → L.Disjoint M ∧ ∀ L ∈ S, L.length = 2 ∧ ¬L = [] ∧ 0 ∉ L ∧ L.sum = xhs:[a, b] ∈ Sh₂:(¬0 = a ∧ ¬0 = b) ∧ a + b = xc:ℕd:ℕt:ℕtail✝:List ℕht:c :: d :: t :: tail✝ ∈ Shst:¬[a, b] = c :: d :: t :: tail✝⊢ 4 ≤ x <;>right.cons.cons.nil.nilx:ℕS:Finset (List ℕ)a:ℕb:ℕhS₁:2 ≤ S.cardhS₂:∀ L ∈ S, ∀ M ∈ S, ¬L = M → L.Disjoint M ∧ ∀ L ∈ S, L.length = 2 ∧ ¬L = [] ∧ 0 ∉ L ∧ L.sum = xhs:[a, b] ∈ Sh₂:(¬0 = a ∧ ¬0 = b) ∧ a + b = xht:[] ∈ Shst:¬[a, b] = []⊢ 4 ≤ xright.cons.cons.nil.cons.nilx:ℕS:Finset (List ℕ)a:ℕb:ℕhS₁:2 ≤ S.cardhS₂:∀ L ∈ S, ∀ M ∈ S, ¬L = M → L.Disjoint M ∧ ∀ L ∈ S, L.length = 2 ∧ ¬L = [] ∧ 0 ∉ L ∧ L.sum = xhs:[a, b] ∈ Sh₂:(¬0 = a ∧ ¬0 = b) ∧ a + b = xc:ℕht:[c] ∈ Shst:¬[a, b] = [c]⊢ 4 ≤ xright.cons.cons.nil.cons.cons.nilx:ℕS:Finset (List ℕ)a:ℕb:ℕhS₁:2 ≤ S.cardhS₂:∀ L ∈ S, ∀ M ∈ S, ¬L = M → L.Disjoint M ∧ ∀ L ∈ S, L.length = 2 ∧ ¬L = [] ∧ 0 ∉ L ∧ L.sum = xhs:[a, b] ∈ Sh₂:(¬0 = a ∧ ¬0 = b) ∧ a + b = xc:ℕd:ℕht:[c, d] ∈ Shst:¬[a, b] = [c, d]⊢ 4 ≤ xright.cons.cons.nil.cons.cons.consx:ℕS:Finset (List ℕ)a:ℕb:ℕhS₁:2 ≤ S.cardhS₂:∀ L ∈ S, ∀ M ∈ S, ¬L = M → L.Disjoint M ∧ ∀ L ∈ S, L.length = 2 ∧ ¬L = [] ∧ 0 ∉ L ∧ L.sum = xhs:[a, b] ∈ Sh₂:(¬0 = a ∧ ¬0 = b) ∧ a + b = xc:ℕd:ℕt:ℕtail✝:List ℕht:c :: d :: t :: tail✝ ∈ Shst:¬[a, b] = c :: d :: t :: tail✝⊢ 4 ≤ x simp_all +arith +decideright.cons.cons.nil.cons.cons.consx:ℕS:Finset (List ℕ)a:ℕb:ℕhS₁:2 ≤ S.cardhS₂:∀ L ∈ S, ∀ M ∈ S, ¬L = M → L.Disjoint M ∧ ∀ L ∈ S, L.length = 2 ∧ ¬L = [] ∧ 0 ∉ L ∧ L.sum = xhs:[a, b] ∈ Sh₂:(¬0 = a ∧ ¬0 = b) ∧ a + b = xc:ℕd:ℕt:ℕtail✝:List ℕht:c :: d :: t :: tail✝ ∈ S⊢ 4 ≤ x
    ·right.cons.cons.nil.nilx:ℕS:Finset (List ℕ)a:ℕb:ℕhS₁:2 ≤ S.cardhS₂:∀ L ∈ S, ∀ M ∈ S, ¬L = M → L.Disjoint M ∧ ∀ L ∈ S, L.length = 2 ∧ ¬L = [] ∧ 0 ∉ L ∧ L.sum = xhs:[a, b] ∈ Sh₂:(¬0 = a ∧ ¬0 = b) ∧ a + b = xht:[] ∈ S⊢ 4 ≤ x grindAll goals completed! 🐙
    ·right.cons.cons.nil.cons.nilx:ℕS:Finset (List ℕ)a:ℕb:ℕhS₁:2 ≤ S.cardhS₂:∀ L ∈ S, ∀ M ∈ S, ¬L = M → L.Disjoint M ∧ ∀ L ∈ S, L.length = 2 ∧ ¬L = [] ∧ 0 ∉ L ∧ L.sum = xhs:[a, b] ∈ Sh₂:(¬0 = a ∧ ¬0 = b) ∧ a + b = xc:ℕht:[c] ∈ S⊢ 4 ≤ x have := hS₂ _ ht _ hsright.cons.cons.nil.cons.nilx:ℕS:Finset (List ℕ)a:ℕb:ℕhS₁:2 ≤ S.cardhS₂:∀ L ∈ S, ∀ M ∈ S, ¬L = M → L.Disjoint M ∧ ∀ L ∈ S, L.length = 2 ∧ ¬L = [] ∧ 0 ∉ L ∧ L.sum = xhs:[a, b] ∈ Sh₂:(¬0 = a ∧ ¬0 = b) ∧ a + b = xc:ℕht:[c] ∈ Sthis:¬[c] = [a, b] → [c].Disjoint [a, b] ∧ ∀ L ∈ S, L.length = 2 ∧ ¬L = [] ∧ 0 ∉ L ∧ L.sum = x := hS₂ [c] ht [a, b] hs⊢ 4 ≤ x
      simp_all +decideright.cons.cons.nil.cons.nilx:ℕS:Finset (List ℕ)a:ℕb:ℕhS₁:2 ≤ S.cardc:ℕhS₂:∀ L ∈ S, ∀ M ∈ S, ¬L = M → L.Disjoint Mhs:[a, b] ∈ Sh₂:(¬0 = a ∧ ¬0 = b) ∧ a + b = xht:[c] ∈ Sthis:(¬c = a ∧ ¬c = b) ∧ ∀ L ∈ S, L.length = 2 ∧ ¬L = [] ∧ 0 ∉ L ∧ L.sum = x⊢ 4 ≤ x
      grindAll goals completed! 🐙
    ·right.cons.cons.nil.cons.cons.nilx:ℕS:Finset (List ℕ)a:ℕb:ℕhS₁:2 ≤ S.cardhS₂:∀ L ∈ S, ∀ M ∈ S, ¬L = M → L.Disjoint M ∧ ∀ L ∈ S, L.length = 2 ∧ ¬L = [] ∧ 0 ∉ L ∧ L.sum = xhs:[a, b] ∈ Sh₂:(¬0 = a ∧ ¬0 = b) ∧ a + b = xc:ℕd:ℕht:[c, d] ∈ Shst:a = c → ¬b = d⊢ 4 ≤ x have := hS₂ _ hs _ htright.cons.cons.nil.cons.cons.nilx:ℕS:Finset (List ℕ)a:ℕb:ℕhS₁:2 ≤ S.cardhS₂:∀ L ∈ S, ∀ M ∈ S, ¬L = M → L.Disjoint M ∧ ∀ L ∈ S, L.length = 2 ∧ ¬L = [] ∧ 0 ∉ L ∧ L.sum = xhs:[a, b] ∈ Sh₂:(¬0 = a ∧ ¬0 = b) ∧ a + b = xc:ℕd:ℕht:[c, d] ∈ Shst:a = c → ¬b = dthis:¬[a, b] = [c, d] → [a, b].Disjoint [c, d] ∧ ∀ L ∈ S, L.length = 2 ∧ ¬L = [] ∧ 0 ∉ L ∧ L.sum = x := hS₂ [a, b] hs [c, d] ht⊢ 4 ≤ x
      simp_all +decide [ List.Disjoint ]right.cons.cons.nil.cons.cons.nilx:ℕS:Finset (List ℕ)a:ℕb:ℕhS₁:2 ≤ S.cardc:ℕd:ℕhS₂:∀ L ∈ S, ∀ M ∈ S, ¬L = M → ∀ ⦃a : ℕ⦄, a ∈ L → a ∉ Mhs:[a, b] ∈ Sh₂:(¬0 = a ∧ ¬0 = b) ∧ a + b = xht:[c, d] ∈ Sthis:((¬a = c ∧ ¬a = d) ∧ ¬b = c ∧ ¬b = d) ∧ ∀ L ∈ S, L.length = 2 ∧ ¬L = [] ∧ 0 ∉ L ∧ L.sum = x⊢ 4 ≤ x
      have := this.2 _ hsright.cons.cons.nil.cons.cons.nilx:ℕS:Finset (List ℕ)a:ℕb:ℕhS₁:2 ≤ S.cardc:ℕd:ℕhS₂:∀ L ∈ S, ∀ M ∈ S, ¬L = M → ∀ ⦃a : ℕ⦄, a ∈ L → a ∉ Mhs:[a, b] ∈ Sh₂:(¬0 = a ∧ ¬0 = b) ∧ a + b = xht:[c, d] ∈ Sthis✝:((¬a = c ∧ ¬a = d) ∧ ¬b = c ∧ ¬b = d) ∧ ∀ L ∈ S, L.length = 2 ∧ ¬L = [] ∧ 0 ∉ L ∧ L.sum = xthis:[a, b].length = 2 ∧ ¬[a, b] = [] ∧ 0 ∉ [a, b] ∧ [a, b].sum = x := this✝.right [a, b] hs⊢ 4 ≤ x
      have := this.2.2right.cons.cons.nil.cons.cons.nilx:ℕS:Finset (List ℕ)a:ℕb:ℕhS₁:2 ≤ S.cardc:ℕd:ℕhS₂:∀ L ∈ S, ∀ M ∈ S, ¬L = M → ∀ ⦃a : ℕ⦄, a ∈ L → a ∉ Mhs:[a, b] ∈ Sh₂:(¬0 = a ∧ ¬0 = b) ∧ a + b = xht:[c, d] ∈ Sthis✝¹:((¬a = c ∧ ¬a = d) ∧ ¬b = c ∧ ¬b = d) ∧ ∀ L ∈ S, L.length = 2 ∧ ¬L = [] ∧ 0 ∉ L ∧ L.sum = xthis✝:[a, b].length = 2 ∧ ¬[a, b] = [] ∧ 0 ∉ [a, b] ∧ [a, b].sum = x := this✝.right [a, b] hsthis:0 ∉ [a, b] ∧ [a, b].sum = x := this✝.right.right⊢ 4 ≤ x
      simp_all +arith +decideright.cons.cons.nil.cons.cons.nilx:ℕS:Finset (List ℕ)a:ℕb:ℕhS₁:2 ≤ S.cardc:ℕd:ℕhS₂:∀ L ∈ S, ∀ M ∈ S, ¬L = M → ∀ ⦃a : ℕ⦄, a ∈ L → a ∉ Mhs:[a, b] ∈ Sh₂:(¬0 = a ∧ ¬0 = b) ∧ a + b = xht:[c, d] ∈ Sthis:((¬a = c ∧ ¬a = d) ∧ ¬b = c ∧ ¬b = d) ∧ ∀ L ∈ S, L.length = 2 ∧ ¬L = [] ∧ 0 ∉ L ∧ L.sum = x⊢ 4 ≤ x
      grindAll goals completed! 🐙
    ·right.cons.cons.nil.cons.cons.consx:ℕS:Finset (List ℕ)a:ℕb:ℕhS₁:2 ≤ S.cardhS₂:∀ L ∈ S, ∀ M ∈ S, ¬L = M → L.Disjoint M ∧ ∀ L ∈ S, L.length = 2 ∧ ¬L = [] ∧ 0 ∉ L ∧ L.sum = xhs:[a, b] ∈ Sh₂:(¬0 = a ∧ ¬0 = b) ∧ a + b = xc:ℕd:ℕt:ℕtail✝:List ℕht:c :: d :: t :: tail✝ ∈ S⊢ 4 ≤ x have := hS₂ _ hs _ htright.cons.cons.nil.cons.cons.consx:ℕS:Finset (List ℕ)a:ℕb:ℕhS₁:2 ≤ S.cardhS₂:∀ L ∈ S, ∀ M ∈ S, ¬L = M → L.Disjoint M ∧ ∀ L ∈ S, L.length = 2 ∧ ¬L = [] ∧ 0 ∉ L ∧ L.sum = xhs:[a, b] ∈ Sh₂:(¬0 = a ∧ ¬0 = b) ∧ a + b = xc:ℕd:ℕt:ℕtail✝:List ℕht:c :: d :: t :: tail✝ ∈ Sthis:¬[a, b] = c :: d :: t :: tail✝ →
  [a, b].Disjoint (c :: d :: t :: tail✝) ∧ ∀ L ∈ S, L.length = 2 ∧ ¬L = [] ∧ 0 ∉ L ∧ L.sum = x := 
  hS₂ [a, b] hs (c :: d :: t :: tail✝) ht⊢ 4 ≤ x
      simp_all +decide [ List.Disjoint ]right.cons.cons.nil.cons.cons.consx:ℕS:Finset (List ℕ)a:ℕb:ℕhS₁:2 ≤ S.cardc:ℕd:ℕt:ℕtail✝:List ℕhS₂:∀ L ∈ S, ∀ M ∈ S, ¬L = M → ∀ ⦃a : ℕ⦄, a ∈ L → a ∉ Mhs:[a, b] ∈ Sh₂:(¬0 = a ∧ ¬0 = b) ∧ a + b = xht:c :: d :: t :: tail✝ ∈ Sthis:((¬a = c ∧ ¬a = d ∧ ¬a = t ∧ a ∉ tail✝) ∧ ¬b = c ∧ ¬b = d ∧ ¬b = t ∧ b ∉ tail✝) ∧
  ∀ L ∈ S, L.length = 2 ∧ ¬L = [] ∧ 0 ∉ L ∧ L.sum = x⊢ 4 ≤ x
      grind +ringAll goals completed! 🐙

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 declaration uses 'sorry'taxicab_for_5_2_2 : answer(sorry) ↔ ∃ x : ℕ, IsTaxicabFor 5 2 2 x := by⊢ True ↔ ∃ x, IsTaxicabFor 5 2 2 x
  sorryAll goals completed! 🐙

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 declaration uses 'sorry'taxicab_for_5_2_n : answer(sorry) ↔ ∃ n : ℕ, n ≥ 2 ∧ (∃ x : ℕ, IsTaxicabFor 5 2 n x)
  := by⊢ True ↔ ∃ n ≥ 2, ∃ x, IsTaxicabFor 5 2 n x sorryAll goals completed! 🐙


end Taxicab