49.lean`

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

6 declarations, 4 contain sorry, 0 declared as axioms

Erdős Problem 49

References:

erdosproblems.com/49
[Er95] Erdős, Paul, Some of my favourite problems in number theory, combinatorics, and geometry. Resenhas (1995), 165-186.
[Er95c] Erdős, Paul, Some problems in number theory. Octogon Math. Mag. (1995), 3-5.
[Ta24d] Tao, Terence, Monotone nondecreasing sequences of the Euler totient function. Matematica (2024), 793-820.

Imports (1)

FormalConjecturesUtil

def Erdos49.f The largest size of a set $A\subseteq\{1,\ldots,N\}$ on which Euler's totient function is strictly increasing line 40 source
Source documentation
The largest size of a set $A\subseteq\{1,\ldots,N\}$ on which Euler's totient function is strictly increasing.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: SupSet.sSup, Set.ofPred, Exists, Finset, And, LE.le, Finset.Icc, OfNat.ofNat, StrictMonoOn, Nat.totient, Eq, Finset.card
9 instances, coercions and auxiliary constants left out.
```
noncomputable def f (N : ℕ) : ℕ :=
…
```
def Erdos49.g The largest size of a set $A\subseteq\{1,\ldots,N\}$ on which Euler's totient function is non-decreasing line 46 source
Source documentation
The largest size of a set $A\subseteq\{1,\ldots,N\}$ on which Euler's totient function is non-decreasing.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: SupSet.sSup, Set.ofPred, Exists, Finset, And, LE.le, Finset.Icc, OfNat.ofNat, MonotoneOn, Nat.totient, Eq, Finset.card
9 instances, coercions and auxiliary constants left out.
```
noncomputable def g (N : ℕ) : ℕ :=
…
```
theorem Erdos49.erdos_49 sorry line 67 source
depends on sorry what this means
Statement uses: Iff, True, Real, LT.lt, OfNat.ofNat, Filter.Eventually, Nat, LE.le, Erdos49.f, HMul.hMul, HAdd.hAdd, Nat.primeCounting, Filter.atTop
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
27 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem Erdos49.erdos_49.variants.little_o sorry line 74 source
depends on sorry what this means
Statement uses: Asymptotics.IsLittleO, Nat, Real, Filter.atTop, Erdos49.f
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat
9 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem Erdos49.erdos_49.variants.tao sorry line 85 source
depends on sorry what this means
Statement uses: Exists, Real, And, LE.le, OfNat.ofNat, Nat, Erdos49.g, HMul.hMul, HAdd.hAdd, HDiv.hDiv, HPow.hPow, Real.log, Nat.primeCounting
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
40 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem Erdos49.erdos_49.variants.primes_maximal Are the primes the largest possible example, i.e. is $f(N) = \pi(N)$ for all $N \geq 2$? sorry line 92 source
Source documentation
Are the primes the largest possible example, i.e. is $f(N) = \pi(N)$ for all $N \geq 2$?

depends on sorry what this means
Statement uses: Iff, True, Nat, LE.le, OfNat.ofNat, Eq, Erdos49.f, Nat.primeCounting
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
4 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_49.variants.primes_maximal : answer(sorry) ↔
    ∀ N : ℕ, 2 ≤ N → f N = Nat.primeCounting N := by
…
```
Mentions: Erdos49.f