1. FormalConjectures
  2. OEIS
  3. «232174»
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Representations with prime conditions

Any integer $n > 1$ can be written as $x + y$ with $x, y > 0$ such that both $x + ny$ and $x^2 + ny^2$ are prime.

Zhi-Wei Sun has offered a $200 prize for the first proof.

References:

    A232174

    Z.-W. Sun, "Conjectures on representations involving primes," in: M. Nathanson (ed.), Combinatorial and Additive Number Theory II: CANT, Springer Proc. in Math. & Stat., Vol. 220, Springer, 2017, pp. 279-310. https://arxiv.org/abs/1211.1588

    D.A. Cox, "Primes of the Form x² + ny²," John Wiley & Sons, 1989.

namespace OeisA232174

The predicate that n can be written as $x + y$ with $x, y > 0$ such that both $x + ny$ and $x^2 + ny^2$ are prime.

def A (n : ℕ) : Prop := ∃ x y : ℕ, 0 < x ∧ 0 < y ∧ n = x + y ∧ (x + n * y).Prime ∧ (x ^ 2 + n * y ^ 2).Prime @[category test, AMS 11] theorem a_2 : A 2 := ⟨1, 1, ⊢ 0 < 1 All goals completed! 🐙, ⊢ 0 < 1 All goals completed! 🐙, ⊢ 2 = 1 + 1 All goals completed! 🐙, ⊢ Nat.Prime (1 + 2 * 1) All goals completed! 🐙, ⊢ Nat.Prime (1 ^ 2 + 2 * 1 ^ 2) All goals completed! 🐙⟩ @[category test, AMS 11] theorem a_3 : A 3 := ⟨2, 1, ⊢ 0 < 2 All goals completed! 🐙, ⊢ 0 < 1 All goals completed! 🐙, ⊢ 3 = 2 + 1 All goals completed! 🐙, ⊢ Nat.Prime (2 + 3 * 1) All goals completed! 🐙, ⊢ Nat.Prime (2 ^ 2 + 3 * 1 ^ 2) All goals completed! 🐙⟩ @[category test, AMS 11] theorem a_4 : A 4 := ⟨1, 3, ⊢ 0 < 1 All goals completed! 🐙, ⊢ 0 < 3 All goals completed! 🐙, ⊢ 4 = 1 + 3 All goals completed! 🐙, ⊢ Nat.Prime (1 + 4 * 3) All goals completed! 🐙, ⊢ Nat.Prime (1 ^ 2 + 4 * 3 ^ 2) All goals completed! 🐙⟩ @[category test, AMS 11] theorem a_5 : A 5 := ⟨3, 2, ⊢ 0 < 3 All goals completed! 🐙, ⊢ 0 < 2 All goals completed! 🐙, ⊢ 5 = 3 + 2 All goals completed! 🐙, ⊢ Nat.Prime (3 + 5 * 2) All goals completed! 🐙, ⊢ Nat.Prime (3 ^ 2 + 5 * 2 ^ 2) All goals completed! 🐙⟩ @[category test, AMS 11] theorem a_6 : A 6 := ⟨5, 1, ⊢ 0 < 5 All goals completed! 🐙, ⊢ 0 < 1 All goals completed! 🐙, ⊢ 6 = 5 + 1 All goals completed! 🐙, ⊢ Nat.Prime (5 + 6 * 1) All goals completed! 🐙, ⊢ Nat.Prime (5 ^ 2 + 6 * 1 ^ 2) All goals completed! 🐙⟩ @[category test, AMS 11] theorem a_8 : A 8 := ⟨5, 3, ⊢ 0 < 5 All goals completed! 🐙, ⊢ 0 < 3 All goals completed! 🐙, ⊢ 8 = 5 + 3 All goals completed! 🐙, ⊢ Nat.Prime (5 + 8 * 3) All goals completed! 🐙, ⊢ Nat.Prime (5 ^ 2 + 8 * 3 ^ 2) All goals completed! 🐙⟩

Zhi-Wei Sun's Conjecture (A232174): Any integer $n > 1$ can be written as $x + y$ with $x, y > 0$ such that both $x + ny$ and $x^2 + ny^2$ are prime.

@[category research open, AMS 11] theorem declaration uses 'sorry'conjecture (n : ℕ) (hn : 1 < n) : A n := n:ℕhn:1 < n⊢ A n All goals completed! 🐙 end OeisA232174