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import FormalConjecturesUtil
namespace BetrothedNumbers
open scoped ArithmeticFunction.sigma
Two natural numbers $m$ and $n$ are betrothed (or quasi-amicable) if $\sigma(m) = \sigma(n) = m + n + 1$, where $\sigma$ is the sum-of-divisors function. Equivalently, the sum of the proper divisors of $m$ equals $n + 1$, and the sum of the proper divisors of $n$ equals $m + 1$.
@[mk_iff]
structure IsBetrothed (m n : ℕ) : Prop where
left : σ 1 m = m + n + 1
right : σ 1 n = m + n + 1The smallest known betrothed pair $(48, 75)$.
@[category test, AMS 11]
theorem betrothed_48_75 : IsBetrothed 48 75 := ⊢ IsBetrothed 48 75
⊢ (σ 1) 48 = 48 + 75 + 1⊢ (σ 1) 75 = 48 + 75 + 1 ⊢ (σ 1) 48 = 48 + 75 + 1⊢ (σ 1) 75 = 48 + 75 + 1 All goals completed! 🐙
IsBetrothed is symmetric.
@[category test, AMS 11]
theorem IsBetrothed.symm {m n : ℕ} (h : IsBetrothed m n) : IsBetrothed n m := m:ℕn:ℕh:IsBetrothed m n⊢ IsBetrothed n m
m:ℕn:ℕh:(σ 1) m = m + n + 1 ∧ (σ 1) n = m + n + 1⊢ (σ 1) n = n + m + 1 ∧ (σ 1) m = n + m + 1
All goals completed! 🐙
Same parity betrothed numbers conjecture. Do there exist betrothed numbers $(m, n)$ where both have the same parity (both even or both odd)?
All known betrothed pairs consist of one even and one odd number.
@[category research open, AMS 11]
theorem same_parity_betrothed :
answer(sorry) ↔ ∃ m n : ℕ, IsBetrothed m n ∧ (Even m ↔ Even n) := ⊢ True ↔ ∃ m n, IsBetrothed m n ∧ (Even m ↔ Even n)
All goals completed! 🐙
Infinitude of betrothed numbers conjecture. Are there infinitely many betrothed number pairs?
@[category research open, AMS 11]
theorem infinitely_many_betrothed :
answer(sorry) ↔ {p : ℕ × ℕ | p.1 < p.2 ∧ IsBetrothed p.1 p.2}.Infinite := ⊢ True ↔ {p | p.1 < p.2 ∧ IsBetrothed p.1 p.2}.Infinite
All goals completed! 🐙
end BetrothedNumbers