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
• 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.

def OeisA231201.PrimeCondition (n : ℕ) : Prop

The predicate that n can be written as $x + y$ with $x,y >0$ such that $2^x + y$ is prime

Equations

• OeisA231201.PrimeCondition n = ∃ (x : ℕ) (y : ℕ), 0 < x ∧ 0 < y ∧ n = x + y ∧ Nat.Prime (2 ^ x + y)

theorem OeisA231201.primeCondition_8 : PrimeCondition 8

theorem OeisA231201.primeCondition_53 : PrimeCondition 53

theorem OeisA231201.conjecture (n : ℕ) (hn : 1 < n) : PrimeCondition n

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.