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import FormalConjecturesUtil
namespace OeisA38098Number of primes strictly less than $n^3$.
def a (n : ℕ) : ℕ := (Nat.primesBelow (n ^ 3)).card
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.
@[category test, AMS 11]
theorem a_2 : a 2 = 4 := ⊢ a 2 = 4
All goals completed! 🐙
Value of the sequence a at 3.
@[category test, AMS 11]
theorem a_3 : a 3 = 9 := ⊢ a 3 = 9
All goals completed! 🐙
Value of the sequence a at 4.
@[category test, AMS 11]
theorem a_4 : a 4 = 18 := ⊢ a 4 = 18
All goals completed! 🐙
Value of the sequence a at 5.
@[category test, AMS 11]
theorem a_5 : a 5 = 30 := ⊢ a 5 = 30
All goals completed! 🐙Conjecture (i): for any integer $k > 2$, the sequence $\pi(n^k)/n^k$ ($n = 2, 3, \ldots$) is strictly decreasing, where $\pi(x)$ denotes the number of primes not exceeding $x$.
Zhi-Wei Sun, Oct 17 2015
@[category research open, AMS 11]
theorem conjecture1 (k : ℕ) (hk : 2 < k) (n : ℕ) (hn : 2 ≤ n) :
(Nat.primeCounting ((n + 1) ^ k) : ℚ) / ((n + 1) ^ k : ℚ) <
(Nat.primeCounting (n ^ k) : ℚ) / (n ^ k : ℚ) := k:ℕhk:2 < kn:ℕhn:2 ≤ n⊢ ↑((n + 1) ^ k).primeCounting / (↑n + 1) ^ k < ↑(n ^ k).primeCounting / ↑n ^ k
All goals completed! 🐙Conjecture (ii): all the numbers $\pi(n^2)/n^2$ ($n = 1, 2, 3, \ldots$) are pairwise distinct. Moreover, we have $\pi(n^2)/n^2 > \pi((n+1)^2)/(n+1)^2$ for all $n > 15646$.
Zhi-Wei Sun, Oct 17 2015
@[category research open, AMS 11]
theorem conjecture2 :
(∀ m n : ℕ, 1 ≤ m → 1 ≤ n →
(Nat.primeCounting (m ^ 2) : ℚ) / (m ^ 2 : ℚ) =
(Nat.primeCounting (n ^ 2) : ℚ) / (n ^ 2 : ℚ) → m = n) ∧
(∀ n : ℕ, 15646 < n →
(Nat.primeCounting ((n + 1) ^ 2) : ℚ) / ((n + 1) ^ 2 : ℚ) <
(Nat.primeCounting (n ^ 2) : ℚ) / (n ^ 2 : ℚ)) := ⊢ (∀ (m n : ℕ), 1 ≤ m → 1 ≤ n → ↑(m ^ 2).primeCounting / ↑m ^ 2 = ↑(n ^ 2).primeCounting / ↑n ^ 2 → m = n) ∧
∀ (n : ℕ), 15646 < n → ↑((n + 1) ^ 2).primeCounting / (↑n + 1) ^ 2 < ↑(n ^ 2).primeCounting / ↑n ^ 2
All goals completed! 🐙end OeisA38098