FormalConjectures.Paper.Dubner

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import FormalConjecturesUtil

Dubner's conjecture

Reference: Every even number greater than 4208 is the sum of two t-primes by Harvey Dubner.


namespace DubnerConjecture

A twin prime is a prime number that has a prime gap of 2, meaning either p - 2 or p + 2 is also prime.

def IsTwinPrime (p : ℕ) : Prop :=
  p.Prime ∧ ((p - 2).Prime ∨ (p + 2).Prime)


@[category test, AMS 11]
theorem t1 : ¬IsTwinPrime 2 := by⊢ ¬IsTwinPrime 2
  norm_num [IsTwinPrime]All goals completed! 🐙


@[category test, AMS 11]
theorem t2 : IsTwinPrime 3 := by⊢ IsTwinPrime 3
  norm_num [IsTwinPrime]All goals completed! 🐙


@[category test, AMS 11]
theorem t3 : IsTwinPrime 5 := by⊢ IsTwinPrime 5
  norm_num [IsTwinPrime]All goals completed! 🐙


@[category test, AMS 11]
theorem t4 : IsTwinPrime 101 := by⊢ IsTwinPrime 101
  norm_num [IsTwinPrime]All goals completed! 🐙


@[category test, AMS 11]
theorem t5 : ¬IsTwinPrime 100 := by⊢ ¬IsTwinPrime 100
  norm_num [IsTwinPrime]All goals completed! 🐙

Every even number greater than 4208 is the sum of two twin primes.

@[category research open, AMS 11]
theorem declaration uses 'sorry'dubner_conjecture (n : ℕ) (hn : 4208 < n) (h : Even n) :
    ∃ p q : ℕ,
      IsTwinPrime p ∧
      IsTwinPrime q ∧
      p + q = n := byn:ℕhn:4208 < nh:Even n⊢ ∃ p q, IsTwinPrime p ∧ IsTwinPrime q ∧ p + q = n
  sorryAll goals completed! 🐙


end DubnerConjecture