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import FormalConjecturesUtilDigital sum of the Fermat number $2^{2^n} + 1$
a n is the digital sum of the Fermat number $2^{2^n} + 1$.
The conjecture asks if there are any prime numbers in this sequence beyond $n=11$.
References:
namespace OeisA108301
The primary defining sequence a.
a n is the digital sum of the Fermat number $2^{2^n} + 1$.
def a (n : ℕ) : ℕ :=
(Nat.digits 10 (2 ^ 2 ^ n + 1)).sumTerm theorems verifying the first few values of the sequence against the official OEIS b-file
@[category test, AMS 11]
theorem a_0 : a 0 = 3 := ⊢ a 0 = 3 All goals completed! 🐙@[category test, AMS 11]
theorem a_1 : a 1 = 5 := ⊢ a 1 = 5 All goals completed! 🐙@[category test, AMS 11]
theorem a_2 : a 2 = 8 := ⊢ a 2 = 8 All goals completed! 🐙@[category test, AMS 11]
theorem a_3 : a 3 = 14 := ⊢ a 3 = 14 All goals completed! 🐙@[category test, AMS 11]
theorem a_4 : a 4 = 26 := ⊢ a 4 = 26 All goals completed! 🐙$a(0)$, $a(1)$, $a(5)$, $a(6)$, $a(7)$ and $a(11)$ are primes.
@[category textbook, AMS 11]
theorem primes_in_a :
(a 0).Prime ∧ (a 1).Prime ∧ (a 5).Prime ∧
(a 6).Prime ∧ (a 7).Prime ∧ (a 11).Prime := ⊢ Nat.Prime (a 0) ∧ Nat.Prime (a 1) ∧ Nat.Prime (a 5) ∧ Nat.Prime (a 6) ∧ Nat.Prime (a 7) ∧ Nat.Prime (a 11)
All goals completed! 🐙$a(0)$, $a(1)$, $a(5)$, $a(6)$, $a(7)$ and $a(11)$ are primes. Are there any more?
@[category research open, AMS 11]
theorem conjecture : answer(sorry) ↔ ∃ n > 11, (a n).Prime := ⊢ True ↔ ∃ n > 11, Nat.Prime (a n)
All goals completed! 🐙end OeisA108301