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:

• OEIS 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.

def OeisA232174.HasPrimeRepresentation (n : ℕ) : Prop

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.

Equations

• OeisA232174.HasPrimeRepresentation n = ∃ (x : ℕ) (y : ℕ), 0 < x ∧ 0 < y ∧ n = x + y ∧ Nat.Prime (x + n * y) ∧ Nat.Prime (x ^ 2 + n * y ^ 2)

theorem OeisA232174.hasPrimeRepresentation_2 : HasPrimeRepresentation 2

theorem OeisA232174.hasPrimeRepresentation_5 : HasPrimeRepresentation 5

theorem OeisA232174.hasPrimeRepresentation_8 : HasPrimeRepresentation 8

theorem OeisA232174.conjecture (n : ℕ) (hn : 1 < n) : HasPrimeRepresentation n

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.