1061.lean`

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Project sources: Pinned project documentation, Formal Conjectures paper, arXiv v1

2 declarations, 1 contain sorry, 0 declared as axioms

Erdős Problem 1061

References:

erdosproblems.com/1061
[Gu04] Guy, Richard K., Unsolved problems in number theory. (2004), Problem B15.

Imports (1)

FormalConjecturesUtil

abbrev Erdos1061.S Let S x count the number of ordered pairs of positive integers (a, b) with a + b ≤ x such that σ(a) + σ(b) = σ(a + b), where σ is the sum of divisors function line 40 source
Source documentation
Let S x count the number of ordered pairs of positive integers (a, b) with a + b ≤ x such that σ(a) + σ(b) = σ(a + b), where σ is the sum of divisors function.
In particular, (a, b) and (b, a) are counted separately; an unordered variant could be obtained by additionally requiring a ≤ b.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Real
Proof uses: Finset.card, Prod, Nat, Finset.filter, Erdos1061.S.match_1, And, LE.le, HAdd.hAdd, Eq, ArithmeticFunction, ArithmeticFunction.sigma, OfNat.ofNat, Prod.fst, Prod.snd, SProd.sprod, Finset, Finset.Icc, Nat.floor
23 instances, coercions and auxiliary constants left out.
```
noncomputable abbrev S (x : ℝ) : ℝ :=
…
```
theorem Erdos1061.erdos_1061 How many (ordered) solutions are there to σ(a) + σ(b) = σ(a + b) with a + b ≤ x? sorry line 49 source
Source documentation
How many (ordered) solutions are there to σ(a) + σ(b) = σ(a + b) with a + b ≤ x? Is it true that this number is asymptotic to c * x for some constant c > 0?

depends on sorry what this means
Statement uses: Iff, True, Exists, Real, And, LT.lt, OfNat.ofNat, Asymptotics.IsEquivalent, Filter.atTop, Erdos1061.S, HMul.hMul
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, Nat
23 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_1061 : answer(sorry) ↔ ∃ c : ℝ, 0 < c ∧ S ~[atTop] (fun x : ℝ ↦ c * x) := by
…
```
Mentions: c, Erdos1061.S