105565.lean`

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

11 declarations, 1 contain sorry, 0 declared as axioms

Indicator sequence for 5 Fibonacci numbers with n digits

$a(n) = 1$ if exactly 5 Fibonacci numbers exist with exactly $n$ digits, otherwise $0$. For the partial sums $S(n) = \sum_{k=1}^n a(k)$, it is conjectured that $\beta-2 < S(n)-\alpha n < \beta-1$, where $\alpha = \log(10)/\log(\phi) - 4$ and $\beta = \log(5)/(2\log(\phi)) - 1$.

References:

A105565

Imports (1)

FormalConjecturesUtil

def OeisA105565.a The primary defining sequence a line 44 source
Source documentation
The primary defining sequence a. $a(n) = 1$ if exactly 5 Fibonacci numbers exist with exactly $n$ digits, otherwise $0$. That is, $a(n) = 1$ if the number of indices $k \in \mathbb{N}$ such that $10^{n-1} \le \mathrm{fib}(k) < 10^n$ is 5.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: ite, GT.gt, OfNat.ofNat, HPow.hPow, HSub.hSub, HAdd.hAdd, HMul.hMul, Finset.card, Finset.filter, And, LE.le, Nat.fib, LT.lt, Finset.range, Eq
17 instances, coercions and auxiliary constants left out.
```
def a (n : ℕ) : ℕ :=
…
```
theorem OeisA105565.a_1 Term theorems verifying the first few values of the sequence against the official OEIS b-file line 63 source
Source documentation
Term theorems verifying the first few values of the sequence against the official OEIS b-file

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA105565.a, OfNat.ofNat
Proof uses: of_decide_eq_true, id, Bool, Decidable.decide, Bool.true, Eq.refl
3 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem a_1 : a 1 = 0 := by decide
…
```
Mentions: OeisA105565.a
theorem OeisA105565.a_2 line 66 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA105565.a, OfNat.ofNat
Proof uses: of_decide_eq_true, id, Bool, Decidable.decide, Bool.true, Eq.refl
3 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem OeisA105565.a_3 line 69 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA105565.a, OfNat.ofNat
Proof uses: of_decide_eq_true, id, Bool, Decidable.decide, Bool.true, Eq.refl
3 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem OeisA105565.a_4 line 72 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA105565.a, OfNat.ofNat
Proof uses: of_decide_eq_true, id, Bool, Decidable.decide, Bool.true, Eq.refl
3 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem OeisA105565.a_5 line 75 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA105565.a, OfNat.ofNat
Proof uses: of_decide_eq_true, id, Bool, Decidable.decide, Bool.true, Eq.refl
3 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
def OeisA105565.phi The golden ratio $\phi = (1 + \sqrt{5})/2$ line 78 source
Source documentation
The golden ratio $\phi = (1 + \sqrt{5})/2$.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Real
Proof uses: Real.goldenRatio
```
noncomputable def phi : Real := goldenRatio
```
def OeisA105565.alphaConst The constant $\alpha = \log(10)/\log(\phi) - 4$ line 81 source
Source documentation
The constant $\alpha = \log(10)/\log(\phi) - 4$.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Real
Proof uses: HSub.hSub, HDiv.hDiv, Real.log, OfNat.ofNat, OeisA105565.phi
9 instances, coercions and auxiliary constants left out.
```
noncomputable def alphaConst : Real := Real.log 10 / Real.log phi - 4
```
Mentions: OeisA105565.phi
def OeisA105565.betaConst The constant $\beta = \log(5)/(2\log(\phi)) - 1$ line 84 source
Source documentation
The constant $\beta = \log(5)/(2\log(\phi)) - 1$.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Real
Proof uses: HSub.hSub, HDiv.hDiv, Real.log, OfNat.ofNat, HMul.hMul, OeisA105565.phi
13 instances, coercions and auxiliary constants left out.
```
noncomputable def betaConst : Real := Real.log 5 / (2 * Real.log phi) - 1
```
Mentions: OeisA105565.phi
def OeisA105565.s The partial sum $S(n) = \sum_{k=1}^n a(k)$ line 87 source
Source documentation
The partial sum $S(n) = \sum_{k=1}^n a(k)$.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Real
Proof uses: Finset.sum, Finset.Icc, OfNat.ofNat, OeisA105565.a
6 instances, coercions and auxiliary constants left out.
```
noncomputable def s (n : ℕ) : Real :=
…
```
theorem OeisA105565.conjecture sorry line 100 source
depends on sorry what this means
Statement uses: Nat, LE.le, OfNat.ofNat, And, LT.lt, Real, HSub.hSub, OeisA105565.betaConst, OeisA105565.s, HMul.hMul, OeisA105565.alphaConst
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
28 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.