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Copyright 2026 The Formal Conjectures Authors.
Licensed under the Apache License, Version 2.0 (the "License");
you may not use this file except in compliance with the License.
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-/
import FormalConjecturesUtil
namespace AlmostPerfectNumbers
open Nat
open scoped ArithmeticFunction.sigma
A number is almost perfect if the sum of its divisors is equal to $2n - 1$.
def AlmostPerfect (n : ℕ) : Prop :=
1 + σ 1 n = 2 * n
Non-Power-of-2 Almost Perfect Numbers Conjecture. Does there exist an almost perfect number that is not a power of 2?
@[category research open, AMS 11]
theorem exists_almost_perfect_not_power_of_two :
answer(sorry) ↔ ∃ n : ℕ, AlmostPerfect n ∧ ¬ ∃ k : ℕ, n = 2^k := ⊢ True ↔ ∃ n, AlmostPerfect n ∧ ¬∃ k, n = 2 ^ k
All goals completed! 🐙
end AlmostPerfectNumbers