FormalConjectures.OEIS.«159829»

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import FormalConjecturesUtil

Smallest $m$ such that $n^3 + m^3 + 1$ is prime

$a(n)$ is the smallest natural number $m \ge 1$ such that $n^3 + m^3 + 1$ is prime.

References:

A159829

namespace OeisA159829open Classical in

$a(n)$ is the smallest natural number $m \ge 1$ such that $n^3 + m^3 + 1$ is prime, or none if no such $m$ exists.

noncomputable def a (n : ℕ) : Option ℕ :=
  if ∃ m : ℕ, 1 ≤ m ∧ (n ^ 3 + m ^ 3 + 1).Prime then
    some (sInf {m : ℕ | 1 ≤ m ∧ (n ^ 3 + m ^ 3 + 1).Prime})
  else
    none

Value of the sequence a at 1.


@[category test, AMS 11]
theorem a_1 : a 1 = some 1 := by⊢ a 1 = some 1
  have h : IsLeast {m : ℕ | 1 ≤ m ∧ (1 ^ 3 + m ^ 3 + 1).Prime} 1 :=
    ⟨⟨le_rfl, by⊢ Nat.Prime (1 ^ 3 + 1 ^ 3 + 1)h:IsLeast {m | 1 ≤ m ∧ Nat.Prime (1 ^ 3 + m ^ 3 + 1)} 1⊢ a 1 = some 1 decideAll goals completed! 🐙h:IsLeast {m | 1 ≤ m ∧ Nat.Prime (1 ^ 3 + m ^ 3 + 1)} 1⊢ a 1 = some 1⟩, fun m hm => hm.1⟩h:IsLeast {m | 1 ≤ m ∧ Nat.Prime (1 ^ 3 + m ^ 3 + 1)} 1⊢ a 1 = some 1
  have h_ex : ∃ m : ℕ, 1 ≤ m ∧ (1 ^ 3 + m ^ 3 + 1).Prime := ⟨1, h.1⟩h:IsLeast {m | 1 ≤ m ∧ Nat.Prime (1 ^ 3 + m ^ 3 + 1)} 1h_ex:∃ m, 1 ≤ m ∧ Nat.Prime (1 ^ 3 + m ^ 3 + 1)⊢ a 1 = some 1
  rw [a,h:IsLeast {m | 1 ≤ m ∧ Nat.Prime (1 ^ 3 + m ^ 3 + 1)} 1h_ex:∃ m, 1 ≤ m ∧ Nat.Prime (1 ^ 3 + m ^ 3 + 1)⊢ (if ∃ m, 1 ≤ m ∧ Nat.Prime (1 ^ 3 + m ^ 3 + 1) then some (sInf {m | 1 ≤ m ∧ Nat.Prime (1 ^ 3 + m ^ 3 + 1)}) else none) =
  some 1All goals completed! 🐙 if_pos h_ex,h:IsLeast {m | 1 ≤ m ∧ Nat.Prime (1 ^ 3 + m ^ 3 + 1)} 1h_ex:∃ m, 1 ≤ m ∧ Nat.Prime (1 ^ 3 + m ^ 3 + 1)⊢ some (sInf {m | 1 ≤ m ∧ Nat.Prime (1 ^ 3 + m ^ 3 + 1)}) = some 1All goals completed! 🐙 h.csInf_eqh:IsLeast {m | 1 ≤ m ∧ Nat.Prime (1 ^ 3 + m ^ 3 + 1)} 1h_ex:∃ m, 1 ≤ m ∧ Nat.Prime (1 ^ 3 + m ^ 3 + 1)⊢ some 1 = some 1All goals completed! 🐙]All goals completed! 🐙

Value of the sequence a at 2.


@[category test, AMS 11]
theorem a_2 : a 2 = some 2 := by⊢ a 2 = some 2
  have h : IsLeast {m : ℕ | 1 ≤ m ∧ (2 ^ 3 + m ^ 3 + 1).Prime} 2 :=
    ⟨⟨by⊢ 1 ≤ 2h:IsLeast {m | 1 ≤ m ∧ Nat.Prime (2 ^ 3 + m ^ 3 + 1)} 2⊢ a 2 = some 2 decideAll goals completed! 🐙h:IsLeast {m | 1 ≤ m ∧ Nat.Prime (2 ^ 3 + m ^ 3 + 1)} 2⊢ a 2 = some 2, by⊢ Nat.Prime (2 ^ 3 + 2 ^ 3 + 1)h:IsLeast {m | 1 ≤ m ∧ Nat.Prime (2 ^ 3 + m ^ 3 + 1)} 2⊢ a 2 = some 2 decideAll goals completed! 🐙h:IsLeast {m | 1 ≤ m ∧ Nat.Prime (2 ^ 3 + m ^ 3 + 1)} 2⊢ a 2 = some 2⟩, fun m hm => bym:ℕhm:m ∈ {m | 1 ≤ m ∧ Nat.Prime (2 ^ 3 + m ^ 3 + 1)}⊢ 2 ≤ mh:IsLeast {m | 1 ≤ m ∧ Nat.Prime (2 ^ 3 + m ^ 3 + 1)} 2⊢ a 2 = some 2
      by_contra hcm:ℕhm:m ∈ {m | 1 ≤ m ∧ Nat.Prime (2 ^ 3 + m ^ 3 + 1)}hc:¬2 ≤ m⊢ Falseh:IsLeast {m | 1 ≤ m ∧ Nat.Prime (2 ^ 3 + m ^ 3 + 1)} 2⊢ a 2 = some 2
      have hm1 : 1 ≤ m := hm.1m:ℕhm:m ∈ {m | 1 ≤ m ∧ Nat.Prime (2 ^ 3 + m ^ 3 + 1)}hc:¬2 ≤ mhm1:1 ≤ m⊢ Falseh:IsLeast {m | 1 ≤ m ∧ Nat.Prime (2 ^ 3 + m ^ 3 + 1)} 2⊢ a 2 = some 2
      have hm2 : m < 2 := not_le.mp hcm:ℕhm:m ∈ {m | 1 ≤ m ∧ Nat.Prime (2 ^ 3 + m ^ 3 + 1)}hc:¬2 ≤ mhm1:1 ≤ mhm2:m < 2⊢ Falseh:IsLeast {m | 1 ≤ m ∧ Nat.Prime (2 ^ 3 + m ^ 3 + 1)} 2⊢ a 2 = some 2
      obtain ⟨_, hprime⟩ := hmm:ℕhc:¬2 ≤ mhm1:1 ≤ mhm2:m < 2left✝:1 ≤ mhprime:Nat.Prime (2 ^ 3 + m ^ 3 + 1)⊢ Falseh:IsLeast {m | 1 ≤ m ∧ Nat.Prime (2 ^ 3 + m ^ 3 + 1)} 2⊢ a 2 = some 2
      interval_cases m«1»m:ℕhc:¬2 ≤ 1hm1:1 ≤ 1hm2:1 < 2left✝:1 ≤ 1hprime:Nat.Prime (2 ^ 3 + 1 ^ 3 + 1)⊢ Falseh:IsLeast {m | 1 ≤ m ∧ Nat.Prime (2 ^ 3 + m ^ 3 + 1)} 2⊢ a 2 = some 2
      revert hprime«1»m:ℕhc:¬2 ≤ 1hm1:1 ≤ 1hm2:1 < 2left✝:1 ≤ 1⊢ Nat.Prime (2 ^ 3 + 1 ^ 3 + 1) → Falseh:IsLeast {m | 1 ≤ m ∧ Nat.Prime (2 ^ 3 + m ^ 3 + 1)} 2⊢ a 2 = some 2
      decideAll goals completed! 🐙h:IsLeast {m | 1 ≤ m ∧ Nat.Prime (2 ^ 3 + m ^ 3 + 1)} 2⊢ a 2 = some 2⟩h:IsLeast {m | 1 ≤ m ∧ Nat.Prime (2 ^ 3 + m ^ 3 + 1)} 2⊢ a 2 = some 2
  have h_ex : ∃ m : ℕ, 1 ≤ m ∧ (2 ^ 3 + m ^ 3 + 1).Prime := ⟨2, h.1⟩h:IsLeast {m | 1 ≤ m ∧ Nat.Prime (2 ^ 3 + m ^ 3 + 1)} 2h_ex:∃ m, 1 ≤ m ∧ Nat.Prime (2 ^ 3 + m ^ 3 + 1)⊢ a 2 = some 2
  rw [a,h:IsLeast {m | 1 ≤ m ∧ Nat.Prime (2 ^ 3 + m ^ 3 + 1)} 2h_ex:∃ m, 1 ≤ m ∧ Nat.Prime (2 ^ 3 + m ^ 3 + 1)⊢ (if ∃ m, 1 ≤ m ∧ Nat.Prime (2 ^ 3 + m ^ 3 + 1) then some (sInf {m | 1 ≤ m ∧ Nat.Prime (2 ^ 3 + m ^ 3 + 1)}) else none) =
  some 2All goals completed! 🐙 if_pos h_ex,h:IsLeast {m | 1 ≤ m ∧ Nat.Prime (2 ^ 3 + m ^ 3 + 1)} 2h_ex:∃ m, 1 ≤ m ∧ Nat.Prime (2 ^ 3 + m ^ 3 + 1)⊢ some (sInf {m | 1 ≤ m ∧ Nat.Prime (2 ^ 3 + m ^ 3 + 1)}) = some 2All goals completed! 🐙 h.csInf_eqh:IsLeast {m | 1 ≤ m ∧ Nat.Prime (2 ^ 3 + m ^ 3 + 1)} 2h_ex:∃ m, 1 ≤ m ∧ Nat.Prime (2 ^ 3 + m ^ 3 + 1)⊢ some 2 = some 2All goals completed! 🐙]All goals completed! 🐙

Value of the sequence a at 3.


@[category test, AMS 11]
theorem a_3 : a 3 = some 1 := by⊢ a 3 = some 1
  have h : IsLeast {m : ℕ | 1 ≤ m ∧ (3 ^ 3 + m ^ 3 + 1).Prime} 1 :=
    ⟨⟨le_rfl, by⊢ Nat.Prime (3 ^ 3 + 1 ^ 3 + 1)h:IsLeast {m | 1 ≤ m ∧ Nat.Prime (3 ^ 3 + m ^ 3 + 1)} 1⊢ a 3 = some 1 decideAll goals completed! 🐙h:IsLeast {m | 1 ≤ m ∧ Nat.Prime (3 ^ 3 + m ^ 3 + 1)} 1⊢ a 3 = some 1⟩, fun m hm => hm.1⟩h:IsLeast {m | 1 ≤ m ∧ Nat.Prime (3 ^ 3 + m ^ 3 + 1)} 1⊢ a 3 = some 1
  have h_ex : ∃ m : ℕ, 1 ≤ m ∧ (3 ^ 3 + m ^ 3 + 1).Prime := ⟨1, h.1⟩h:IsLeast {m | 1 ≤ m ∧ Nat.Prime (3 ^ 3 + m ^ 3 + 1)} 1h_ex:∃ m, 1 ≤ m ∧ Nat.Prime (3 ^ 3 + m ^ 3 + 1)⊢ a 3 = some 1
  rw [a,h:IsLeast {m | 1 ≤ m ∧ Nat.Prime (3 ^ 3 + m ^ 3 + 1)} 1h_ex:∃ m, 1 ≤ m ∧ Nat.Prime (3 ^ 3 + m ^ 3 + 1)⊢ (if ∃ m, 1 ≤ m ∧ Nat.Prime (3 ^ 3 + m ^ 3 + 1) then some (sInf {m | 1 ≤ m ∧ Nat.Prime (3 ^ 3 + m ^ 3 + 1)}) else none) =
  some 1All goals completed! 🐙 if_pos h_ex,h:IsLeast {m | 1 ≤ m ∧ Nat.Prime (3 ^ 3 + m ^ 3 + 1)} 1h_ex:∃ m, 1 ≤ m ∧ Nat.Prime (3 ^ 3 + m ^ 3 + 1)⊢ some (sInf {m | 1 ≤ m ∧ Nat.Prime (3 ^ 3 + m ^ 3 + 1)}) = some 1All goals completed! 🐙 h.csInf_eqh:IsLeast {m | 1 ≤ m ∧ Nat.Prime (3 ^ 3 + m ^ 3 + 1)} 1h_ex:∃ m, 1 ≤ m ∧ Nat.Prime (3 ^ 3 + m ^ 3 + 1)⊢ some 1 = some 1All goals completed! 🐙]All goals completed! 🐙

Value of the sequence a at 4.


@[category test, AMS 11]
theorem a_4 : a 4 = some 2 := by⊢ a 4 = some 2
  have h : IsLeast {m : ℕ | 1 ≤ m ∧ (4 ^ 3 + m ^ 3 + 1).Prime} 2 :=
    ⟨⟨by⊢ 1 ≤ 2h:IsLeast {m | 1 ≤ m ∧ Nat.Prime (4 ^ 3 + m ^ 3 + 1)} 2⊢ a 4 = some 2 decideAll goals completed! 🐙h:IsLeast {m | 1 ≤ m ∧ Nat.Prime (4 ^ 3 + m ^ 3 + 1)} 2⊢ a 4 = some 2, by⊢ Nat.Prime (4 ^ 3 + 2 ^ 3 + 1)h:IsLeast {m | 1 ≤ m ∧ Nat.Prime (4 ^ 3 + m ^ 3 + 1)} 2⊢ a 4 = some 2 decideAll goals completed! 🐙h:IsLeast {m | 1 ≤ m ∧ Nat.Prime (4 ^ 3 + m ^ 3 + 1)} 2⊢ a 4 = some 2⟩, fun m hm => bym:ℕhm:m ∈ {m | 1 ≤ m ∧ Nat.Prime (4 ^ 3 + m ^ 3 + 1)}⊢ 2 ≤ mh:IsLeast {m | 1 ≤ m ∧ Nat.Prime (4 ^ 3 + m ^ 3 + 1)} 2⊢ a 4 = some 2
      by_contra hcm:ℕhm:m ∈ {m | 1 ≤ m ∧ Nat.Prime (4 ^ 3 + m ^ 3 + 1)}hc:¬2 ≤ m⊢ Falseh:IsLeast {m | 1 ≤ m ∧ Nat.Prime (4 ^ 3 + m ^ 3 + 1)} 2⊢ a 4 = some 2
      have hm1 : 1 ≤ m := hm.1m:ℕhm:m ∈ {m | 1 ≤ m ∧ Nat.Prime (4 ^ 3 + m ^ 3 + 1)}hc:¬2 ≤ mhm1:1 ≤ m⊢ Falseh:IsLeast {m | 1 ≤ m ∧ Nat.Prime (4 ^ 3 + m ^ 3 + 1)} 2⊢ a 4 = some 2
      have hm2 : m < 2 := not_le.mp hcm:ℕhm:m ∈ {m | 1 ≤ m ∧ Nat.Prime (4 ^ 3 + m ^ 3 + 1)}hc:¬2 ≤ mhm1:1 ≤ mhm2:m < 2⊢ Falseh:IsLeast {m | 1 ≤ m ∧ Nat.Prime (4 ^ 3 + m ^ 3 + 1)} 2⊢ a 4 = some 2
      obtain ⟨_, hprime⟩ := hmm:ℕhc:¬2 ≤ mhm1:1 ≤ mhm2:m < 2left✝:1 ≤ mhprime:Nat.Prime (4 ^ 3 + m ^ 3 + 1)⊢ Falseh:IsLeast {m | 1 ≤ m ∧ Nat.Prime (4 ^ 3 + m ^ 3 + 1)} 2⊢ a 4 = some 2
      interval_cases m«1»m:ℕhc:¬2 ≤ 1hm1:1 ≤ 1hm2:1 < 2left✝:1 ≤ 1hprime:Nat.Prime (4 ^ 3 + 1 ^ 3 + 1)⊢ Falseh:IsLeast {m | 1 ≤ m ∧ Nat.Prime (4 ^ 3 + m ^ 3 + 1)} 2⊢ a 4 = some 2
      revert hprime«1»m:ℕhc:¬2 ≤ 1hm1:1 ≤ 1hm2:1 < 2left✝:1 ≤ 1⊢ Nat.Prime (4 ^ 3 + 1 ^ 3 + 1) → Falseh:IsLeast {m | 1 ≤ m ∧ Nat.Prime (4 ^ 3 + m ^ 3 + 1)} 2⊢ a 4 = some 2
      decideAll goals completed! 🐙h:IsLeast {m | 1 ≤ m ∧ Nat.Prime (4 ^ 3 + m ^ 3 + 1)} 2⊢ a 4 = some 2⟩h:IsLeast {m | 1 ≤ m ∧ Nat.Prime (4 ^ 3 + m ^ 3 + 1)} 2⊢ a 4 = some 2
  have h_ex : ∃ m : ℕ, 1 ≤ m ∧ (4 ^ 3 + m ^ 3 + 1).Prime := ⟨2, h.1⟩h:IsLeast {m | 1 ≤ m ∧ Nat.Prime (4 ^ 3 + m ^ 3 + 1)} 2h_ex:∃ m, 1 ≤ m ∧ Nat.Prime (4 ^ 3 + m ^ 3 + 1)⊢ a 4 = some 2
  rw [a,h:IsLeast {m | 1 ≤ m ∧ Nat.Prime (4 ^ 3 + m ^ 3 + 1)} 2h_ex:∃ m, 1 ≤ m ∧ Nat.Prime (4 ^ 3 + m ^ 3 + 1)⊢ (if ∃ m, 1 ≤ m ∧ Nat.Prime (4 ^ 3 + m ^ 3 + 1) then some (sInf {m | 1 ≤ m ∧ Nat.Prime (4 ^ 3 + m ^ 3 + 1)}) else none) =
  some 2All goals completed! 🐙 if_pos h_ex,h:IsLeast {m | 1 ≤ m ∧ Nat.Prime (4 ^ 3 + m ^ 3 + 1)} 2h_ex:∃ m, 1 ≤ m ∧ Nat.Prime (4 ^ 3 + m ^ 3 + 1)⊢ some (sInf {m | 1 ≤ m ∧ Nat.Prime (4 ^ 3 + m ^ 3 + 1)}) = some 2All goals completed! 🐙 h.csInf_eqh:IsLeast {m | 1 ≤ m ∧ Nat.Prime (4 ^ 3 + m ^ 3 + 1)} 2h_ex:∃ m, 1 ≤ m ∧ Nat.Prime (4 ^ 3 + m ^ 3 + 1)⊢ some 2 = some 2All goals completed! 🐙]All goals completed! 🐙

Conjecture 1: For any $k \ge 3$, there are infinitely many primes of the form $n^k + m^k$ for $n, m \ge 1$.

Ulrich Krug, 2009

Answer: No.

Kenta Kitamura, 2026

@[category research solved, AMS 11,
  formal_proof using lean4 at
    "https://github.com/KitaKen1/oeis-a159829-conjecture1-counterexample/blob/6632e626baa7f28ad14045aa7408a84178ec128d/lean/A159829Conjecture1FC.lean#L52-L61"]
theorem conjecture1 :
    answer(False) ↔ ∀ (k : ℕ), 3 ≤ k →
      Set.Infinite {p : ℕ | ∃ n m : ℕ, 1 ≤ n ∧ 1 ≤ m ∧ p.Prime ∧
        p = n ^ k + m ^ k} := by⊢ False ↔ ∀ (k : ℕ), 3 ≤ k → {p | ∃ n m, 1 ≤ n ∧ 1 ≤ m ∧ Nat.Prime p ∧ p = n ^ k + m ^ k}.Infinite
  sorryAll goals completed! 🐙

Conjecture 2: For any $k \ge 3$, there are infinitely many primes of the form $n^k + m^k + 1$ for $n, m \ge 1$.

Ulrich Krug, 2009

@[category research open, AMS 11]
theorem conjecture2 (k : ℕ) (hk : 3 ≤ k) :
    Set.Infinite {p : ℕ | ∃ n m : ℕ, 1 ≤ n ∧ 1 ≤ m ∧ p.Prime ∧ p = n ^ k + m ^ k + 1} := byk:ℕhk:3 ≤ k⊢ {p | ∃ n m, 1 ≤ n ∧ 1 ≤ m ∧ Nat.Prime p ∧ p = n ^ k + m ^ k + 1}.Infinite
  sorryAll goals completed! 🐙

end OeisA159829