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import FormalConjecturesUtilTug of war score between prime gap increases and decreases
Score at stage $n$ in "tug of war" between prime gap increases vs. prime gap decreases: start with score $= 0$ at $n = 1$ and at stage $k > 1$, increase (resp. decrease) the score by $1$ if the $k$-th prime gap is greater (resp. less) than the previous prime gap.
References:
namespace OeisA92243The $n$-th prime gap $g(n) = p_n - p_{n-1}$ for $n \ge 1$, where $p_i$ is the $i$-th prime (0-indexed).
noncomputable def primeGap (n : ℕ) : ℕ :=
Nat.nth Nat.Prime n - Nat.nth Nat.Prime (n - 1)Score at stage $n$ in "tug of war" between prime gap increases vs. prime gap decreases.
noncomputable def a (n : ℕ) : ℤ :=
if n ≤ 1 then 0
else
∑ k ∈ Finset.Icc 2 n,
((primeGap k : ℤ) - (primeGap (k - 1) : ℤ)).sign
Value of the sequence a at 0.
@[category test, AMS 11]
theorem a_0 : a 0 = 0 := ⊢ a 0 = 0 All goals completed! 🐙
Value of the sequence a at 1.
@[category test, AMS 11]
theorem a_1 : a 1 = 0 := ⊢ a 1 = 0 All goals completed! 🐙
Value of the sequence a at 2.
h2:¬2 ≤ 1h_icc:Finset.Icc 2 2 = {2}h_zero:Nat.nth Nat.Prime 0 = 2h_one:Nat.nth Nat.Prime 1 = 3h_two:Nat.nth Nat.Prime 2 = 5⊢ (↑(5 - 3) - ↑(3 - 2)).sign = 1
All goals completed! 🐙
Value of the sequence a at 3.
h3:¬3 ≤ 1h_icc:Finset.Icc 2 3 = {2, 3}h_zero:Nat.nth Nat.Prime 0 = 2h_one:Nat.nth Nat.Prime 1 = 3h_two:Nat.nth Nat.Prime 2 = 5h_three:Nat.nth Nat.Prime 3 = 7⊢ (↑(5 - 3) - ↑(3 - 2)).sign + (↑(7 - 5) - ↑(5 - 3)).sign = 1
All goals completed! 🐙Is the score $a(n) > 0$ for some $n > 250000$?
@[category research open, AMS 11]
theorem conjecture1 : answer(sorry) ↔ ∃ n > 250000, a n > 0 := ⊢ True ↔ ∃ n > 250000, a n > 0
All goals completed! 🐙Is the score $a(n)$ bounded from below?
@[category research open, AMS 11]
theorem conjecture2 : answer(sorry) ↔ ∃ B : ℤ, ∀ n : ℕ, B ≤ a n := ⊢ True ↔ ∃ B, ∀ (n : ℕ), B ≤ a n
All goals completed! 🐙Is the score $a(n)$ bounded from above?
@[category research open, AMS 11]
theorem conjecture3 : answer(sorry) ↔ ∃ B : ℤ, ∀ n : ℕ, a n ≤ B := ⊢ True ↔ ∃ B, ∀ (n : ℕ), a n ≤ B
All goals completed! 🐙Is the score $a(n) > 0$ for infinitely many values of $n$?
@[category research open, AMS 11]
theorem conjecture4 : answer(sorry) ↔ Set.Infinite {n : ℕ | a n > 0} := ⊢ True ↔ {n | a n > 0}.Infinite
All goals completed! 🐙Is the score $a(n) < 0$ for infinitely many values of $n$?
@[category research open, AMS 11]
theorem conjecture5 : answer(sorry) ↔ Set.Infinite {n : ℕ | a n < 0} := ⊢ True ↔ {n | a n < 0}.Infinite
All goals completed! 🐙end OeisA92243