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import FormalConjecturesUtilSum 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.
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, ⊢ 0 < 1 All goals completed! 🐙, ⊢ 0 < 1 All goals completed! 🐙, ⊢ 2 = 1 + 1 All goals completed! 🐙, ⊢ Nat.Prime (2 ^ 1 + 1) 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 ^ 2 + 1) 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 (2 ^ 1 + 3) All goals completed! 🐙⟩
@[category test, AMS 11]
theorem a_5 : A 5 :=
⟨2, 3, ⊢ 0 < 2 All goals completed! 🐙, ⊢ 0 < 3 All goals completed! 🐙, ⊢ 5 = 2 + 3 All goals completed! 🐙, ⊢ Nat.Prime (2 ^ 2 + 3) All goals completed! 🐙⟩
@[category test, AMS 11]
theorem a_8 : A 8 :=
⟨3, 5, ⊢ 0 < 3 All goals completed! 🐙, ⊢ 0 < 5 All goals completed! 🐙, ⊢ 8 = 3 + 5 All goals completed! 🐙, ⊢ Nat.Prime (2 ^ 3 + 5) All goals completed! 🐙⟩
@[category test, AMS 11]
theorem a_53 : A 53 :=
⟨20, 33, ⊢ 0 < 20 All goals completed! 🐙, ⊢ 0 < 33 All goals completed! 🐙, ⊢ 53 = 20 + 33 All goals completed! 🐙, ⊢ Nat.Prime (2 ^ 20 + 33) All goals completed! 🐙⟩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 := n:ℕhn:1 < n⊢ A n
All goals completed! 🐙
end OeisA231201