Sum of two numbers with prime conditions

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import FormalConjecturesUtil

/-!
# Sum of two numbers with prime conditions

Number of ways to write $n = x+y$, for $x,y > 0$ such that $2^x + y$ is prime.

Zhi-Wei Sun has offered a \$1000 prize for the first proof.

*References:*
- [A231201](https://oeis.org/A231201)
- Zhi-Wei Sun, "Table of n, a(n) for n = 1..10000",
  "Write n = k + m with 2^k + m prime", a message to Number Theory List, Nov. 16, 2013,
  "On a^n+ bn modulo m", arXiv:1312.1166 [math.NT], 2013-2014,
  "Problems on combinatorial properties of primes", arXiv:1402.6641 [math.NT], 2014-2015.
-/

namespace OeisA231201

/-- The predicate that `n` can be written as $x + y$ with $x,y >0$ such that
$2^x + y$ is prime -/
def A (n : ℕ) : Prop :=
  ∃ x y : ℕ, 0 < x ∧ 0 < y ∧ n = x + y ∧ (2^x + y).Prime

@[category test, AMS 11]
theorem a_2 : A 2 :=
  ⟨1, 1, by norm_num, by norm_num, by norm_num, by norm_num⟩

@[category test, AMS 11]
theorem a_3 : A 3 :=
  ⟨2, 1, by norm_num, by norm_num, by norm_num, by norm_num⟩

@[category test, AMS 11]
theorem a_4 : A 4 :=
  ⟨1, 3, by norm_num, by norm_num, by norm_num, by norm_num⟩

@[category test, AMS 11]
theorem a_5 : A 5 :=
  ⟨2, 3, by norm_num, by norm_num, by norm_num, by norm_num⟩

@[category test, AMS 11]
theorem a_8 : A 8 :=
  ⟨3, 5, by norm_num, by norm_num, by norm_num, by norm_num⟩

@[category test, AMS 11]
theorem a_53 : A 53 :=
  ⟨20, 33, by norm_num, by norm_num, by norm_num, by norm_num⟩

/-- The conjecture for sequence A231201: for any $n > 1$, there exist $x, y > 0$ such that $n = x + y$ and $2^x + y$ is prime. -/
@[category research open, AMS 11]
theorem conjecture (n : ℕ) (hn : 1 < n) : A n := by
  sorry

end OeisA231201

• Formalization source ↗

On-site verification needs an executable Lean target. None is attached yet.

A source attachment or link is not a verified proof. The statement’s correspondence with the original question needs human review.