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import FormalConjecturesUtilCatalan-Mersenne numbers
Catalan-Mersenne numbers: $a(0) = 2$; for $n \ge 0$, $a(n+1) = 2^{a(n)} - 1$.
References:
namespace OeisA7013Catalan-Mersenne numbers: $a(0) = 2$, $a(n+1) = 2^{a(n)} - 1$.
def a (n : ℕ) : ℕ :=
match n with
| 0 => 2
| n + 1 => 2 ^ a n - 1@[category test, AMS 11]
theorem a_0 : a 0 = 2 := ⊢ a 0 = 2 All goals completed! 🐙@[category test, AMS 11]
theorem a_1 : a 1 = 3 := ⊢ a 1 = 3 All goals completed! 🐙@[category test, AMS 11]
theorem a_2 : a 2 = 7 := ⊢ a 2 = 7 All goals completed! 🐙@[category test, AMS 11]
theorem a_3 : a 3 = 127 := ⊢ a 3 = 127 All goals completed! 🐙@[category test, AMS 11]
theorem a_4 : a 4 = 170141183460469231731687303715884105727 := ⊢ a 4 = 170141183460469231731687303715884105727 All goals completed! 🐙Catalan-Mersenne conjecture: All terms of the Catalan-Mersenne sequence are prime.
@[category research open, AMS 11]
theorem conjecture (n : ℕ) : (a n).Prime := n:ℕ⊢ Nat.Prime (a n)
All goals completed! 🐙end OeisA7013