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import FormalConjecturesUtilNumber of pairs of twin primes between $n^2$ and $(n+1)^2$
$a(n)$ is the number of pairs of twin primes between $n^2$ and $(n+1)^2$. This counts the number of primes $p$ such that $p$ and $p+2$ are both prime, and the entire twin prime pair $(p, p+2)$ lies strictly between $n^2$ and $(n+1)^2$. That is, $n^2 < p$ and $p + 2 < (n+1)^2$.
References:
namespace OeisA91591Number of pairs of twin primes $(p, p+2)$ strictly between $n^2$ and $(n+1)^2$.
def a (n : ℕ) : ℕ :=
(((Finset.Ioo (n ^ 2) ((n + 1) ^ 2)).filter
fun p => p.Prime ∧ (p + 2).Prime ∧ p + 2 < (n + 1) ^ 2)).card
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.
@[category test, AMS 11]
theorem a_2 : a 2 = 1 := ⊢ a 2 = 1 All goals completed! 🐙
Value of the sequence a at 3.
@[category test, AMS 11]
theorem a_3 : a 3 = 1 := ⊢ a 3 = 1 All goals completed! 🐙
Value of the sequence a at 4.
@[category test, AMS 11]
theorem a_4 : a 4 = 1 := ⊢ a 4 = 1 All goals completed! 🐙It is conjectured that $a(n)>0$ for all $n>122$. Proving this would also prove Legendre's conjecture that there is a prime between $n^2$ and $(n+1)^2$. - T. D. Noe, Feb 28 2007
@[category research open, AMS 11]
theorem conjecture :
∀ n : ℕ, n > 122 → a n > 0 := ⊢ ∀ n > 122, OeisA91591.a n > 0
All goals completed! 🐙end OeisA91591