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import FormalConjecturesUtilLemoine's conjectures
[Ki85] Kiltinen, J. and Young P. (1985). Goldbach, Lemoine, and a Know/Don't Know Problem.
namespace Lemoine
def OddPrime (n : ℕ) : Prop :=
n ≠ 2 ∧ n.Prime
For all odd integers $n ≥ 7$ there are prime numbers $p,q$ such that $n = p+2q$.
@[category research open, AMS 11]
theorem lemoine_conjecture (n : ℕ) (hn : 6 < n) (odd : Odd n) :
∃ (p q : ℕ), p.Prime ∧ q.Prime ∧ p + 2 * q = n := n:ℕhn:6 < nodd:Odd n⊢ ∃ p q, Nat.Prime p ∧ Nat.Prime q ∧ p + 2 * q = n
All goals completed! 🐙
For all odd integers $n ≥ 9$ there are odd prime numbers $p,q,r,s$ and natural numbers $a,b$ such that $p+2q = n$, $2+pq = 2^a+r$, $2p+q = 2^b+s$
@[category research open, AMS 11]
theorem lemoine_conjecture_extension (n : ℕ) (hn : 8 < n) (odd : Odd n) :
∃ (p q r s a b : ℕ), OddPrime p ∧ OddPrime q ∧ OddPrime r ∧ OddPrime s ∧
p + 2 * q = n ∧ 2 + p * q = 2 ^ a + r ∧ 2 * p + q = 2 ^ b + s := n:ℕhn:8 < nodd:Odd n⊢ ∃ p q r s a b,
OddPrime p ∧ OddPrime q ∧ OddPrime r ∧ OddPrime s ∧ p + 2 * q = n ∧ 2 + p * q = 2 ^ a + r ∧ 2 * p + q = 2 ^ b + s
All goals completed! 🐙
end Lemoine