Digital sum of the Fermat number
Digital sum of the Fermat number
Inferred from the community field; finer classification has not been reviewed.
Problem statement
conjecture
, , , , and are primes. Are there any more?
Lean 4
Formal statement
Lean 4/-
Copyright 2026 The Formal Conjectures Authors.
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you may not use this file except in compliance with the License.
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-/
import FormalConjecturesUtil
/-!
# Digital 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:*
- [A108301](https://oeis.org/A108301)
-/
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)).sum
/-- Term 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 := by native_decide
@[category test, AMS 11]
theorem a_1 : a 1 = 5 := by native_decide
@[category test, AMS 11]
theorem a_2 : a 2 = 8 := by native_decide
@[category test, AMS 11]
theorem a_3 : a 3 = 14 := by native_decide
@[category test, AMS 11]
theorem a_4 : a 4 = 26 := by native_decide
/-- $a(0)$, $a(1)$, $a(5)$, $a(6)$, $a(7)$ 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 := by
sorry
/-- $a(0)$, $a(1)$, $a(5)$, $a(6)$, $a(7)$ and $a(11)$ are primes. Are there any more? -/
@[category research open, AMS 11]
theorem conjecture : answer(sorry) ↔ ∃ n > 11, (a n).Prime := by
sorry
end OeisA108301
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Background
Source snapshot inspected on 10 September 2026. Subsequent research status and the accuracy of source formalizations have not been independently certified.
Fixed commit b82b08faa9006484021c12005ab41287fb2ffb69. This file contains 1 source-labeled research-open declarations. Other declarations in the attached file can be definitions, solved results or tests. The file is preserved as source material; it has not been compiled or proved by this import.
Copyright The Formal Conjectures Authors. Code: Apache-2.0; other original material: CC BY 4.0. Third-party sources retain their own licenses. Original source files and notices preserved.
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