BetrothedNumbers.lean`

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5 declarations, 2 contain sorry, 0 declared as axioms

Betrothed numbers

References:

Imports (1)

FormalConjecturesUtil

structure BetrothedNumbers.IsBetrothed 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 line 40 source
Source documentation
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 $m + 1$.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
```
structure IsBetrothed (m n : ℕ) : Prop where
…
```
theorem BetrothedNumbers.betrothed_48_75 The smallest known betrothed pair $(48, 75)$ line 46 source
Source documentation
The smallest known betrothed pair $(48, 75)$.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: BetrothedNumbers.IsBetrothed, OfNat.ofNat, Nat
Proof uses: BetrothedNumbers.IsBetrothed.mk, of_decide_eq_true, Eq, ArithmeticFunction, ArithmeticFunction.sigma, HAdd.hAdd, id, Bool, Decidable.decide, Bool.true, Eq.refl
9 instances, coercions and auxiliary constants left out.

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.
```
theorem betrothed_48_75 : IsBetrothed 48 75 := by
…
```
Mentions: BetrothedNumbers.IsBetrothed
theorem BetrothedNumbers.IsBetrothed.symm IsBetrothed is symmetric line 51 source
Source documentation
IsBetrothed is symmetric.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, BetrothedNumbers.IsBetrothed
Proof uses: Eq.mpr, And, Eq, ArithmeticFunction, ArithmeticFunction.sigma, OfNat.ofNat, HAdd.hAdd, id, congrArg, propext, BetrothedNumbers.isBetrothed_iff, Decidable.byContradiction, Not, Eq.mp
10 instances, coercions and auxiliary constants left out.

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.
```
theorem IsBetrothed.symm {m n : ℕ} (h : IsBetrothed m n) : IsBetrothed n m := by
…
```
Mentions: BetrothedNumbers.IsBetrothed
theorem BetrothedNumbers.same_parity_betrothed Same parity betrothed numbers conjecture sorry line 67 source
Source documentation
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.
The requirement $m \neq n$ is part of the question: $\mathrm{IsBetrothed}\ n\ n$ says $\sigma(n) = 2n + 1$, i.e. that $n$ is quasiperfect, which is the separate open problem QuasiperfectNumbers.exists_quasiperfect.

depends on sorry what this means
Statement uses: Iff, True, Exists, Nat, And, Ne, BetrothedNumbers.IsBetrothed, Even
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat
3 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem same_parity_betrothed :
    answer(sorry) ↔ ∃ m n : ℕ, m ≠ n ∧ IsBetrothed m n ∧ (Even m ↔ Even n) := by
…
```
Mentions: BetrothedNumbers.IsBetrothed
theorem BetrothedNumbers.infinitely_many_betrothed Infinitude of betrothed numbers conjecture sorry line 77 source
Source documentation
Infinitude of betrothed numbers conjecture. Are there infinitely many betrothed number pairs?

depends on sorry what this means
Statement uses: Iff, True, Set.Infinite, Prod, Nat, Set.ofPred, And, LT.lt, Prod.fst, Prod.snd, BetrothedNumbers.IsBetrothed
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat
3 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem infinitely_many_betrothed :
    answer(sorry) ↔ {p : ℕ × ℕ | p.1 < p.2 ∧ IsBetrothed p.1 p.2}.Infinite := by
…
```
Mentions: BetrothedNumbers.IsBetrothed