11545.lean`

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L4DEL: Lean4 Domain Explanation Language

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Credit: The upstream authors and contributors retain credit for the project and mathematics. lean4-explainer only renders their pinned source and adds the separately disclosed explanation layer.

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

7 declarations, 2 contain sorry, 0 declared as axioms

$a(n)$ is the integer whose decimal digits are the first $n+1$ decimal digits of $\pi$

References:

A011545

Imports (1)

FormalConjecturesUtil

def OeisA11545.a a n is the integer whose decimal digits are the first $n+1$ decimal digits of $\pi$ line 34 source
Source documentation
a n is the integer whose decimal digits are the first $n+1$ decimal digits of $\pi$.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: Int.toNat, Int.floor, Real, HMul.hMul, Real.pi, HPow.hPow, OfNat.ofNat
14 instances, coercions and auxiliary constants left out.
```
noncomputable def a (n : ℕ) : ℕ :=
…
```
theorem OeisA11545.a_0 line 38 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA11545.a, OfNat.ofNat
Proof uses: id, of_eq_true, Int.toNat, Int.floor, Real, HMul.hMul, Real.pi, HPow.hPow, Eq.trans, True, congrFun', congrArg, Int, Int.toNat.match_1, Mathlib.Meta.NormNum.IsNat.to_eq, Mathlib.Meta.NormNum.isNat_pow, Eq.refl, Mathlib.Meta.NormNum.isNat_ofNat, Mathlib.Meta.NormNum.instAtLeastTwo, Mathlib.Meta.NormNum.isNat_natCast and 18 more
47 instances, coercions and auxiliary constants left out.

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.
theorem OeisA11545.a_1 line 44 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA11545.a, OfNat.ofNat
Proof uses: Eq.mpr, Int.toNat, Int.floor, Real, HMul.hMul, Real.pi, id, congrFun', congrArg, Int, HPow.hPow, Eq.trans, Nat.cast_one, pow_one, congr_arg, Iff.mpr, And, LE.le, LT.lt, HAdd.hAdd and 97 more
99 instances, coercions and auxiliary constants left out.

Tactic trace: 5 steps
Open to load the checked before-and-after goal states.
theorem OeisA11545.a_2 line 51 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA11545.a, OfNat.ofNat
Proof uses: id, Eq.trans, Int.toNat, Int.floor, Real, HMul.hMul, Real.pi, HPow.hPow, Int, congr_arg, Iff.mpr, And, LE.le, LT.lt, HAdd.hAdd, Int.floor_eq_iff, And.intro, Mathlib.Tactic.LinearCombination.le_of_lt, OfScientific.ofScientific, Bool.true and 101 more
99 instances, coercions and auxiliary constants left out.

Tactic trace: 6 steps
Open to load the checked before-and-after goal states.
theorem OeisA11545.a_3 line 58 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA11545.a, OfNat.ofNat
Proof uses: Eq.symm, Eq.mpr, Int.toNat, Int.floor, Real, HMul.hMul, Real.pi, Mathlib.Meta.NormNum.instAtLeastTwo, id, congrArg, Int, HPow.hPow, Mathlib.Meta.NormNum.IsNat.to_eq, Mathlib.Meta.NormNum.isNat_pow, Eq.refl, Mathlib.Meta.NormNum.isNat_ofNat, Mathlib.Meta.NormNum.isNat_natCast, Mathlib.Meta.NormNum.IsNatPowT.run, Mathlib.Meta.NormNum.IsNatPowT.bit1, congr_arg and 95 more
99 instances, coercions and auxiliary constants left out.

Tactic trace: 5 steps
Open to load the checked before-and-after goal states.
theorem OeisA11545.conjecture1 Wolfgang Haken (1977) conjectured that no term of this sequence is a perfect square, and estimated the probability that this conjecture is false to be smaller than $10^-9$ sorry line 70 source
Source documentation
Wolfgang Haken (1977) conjectured that no term of this sequence is a perfect square, and estimated the probability that this conjecture is false to be smaller than $10^-9$.

depends on sorry what this means
Statement uses: Nat, Not, IsSquare, OeisA11545.a
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 conjecture1 : ∀ n, ¬ IsSquare (a n) := by
…
```
Mentions: OeisA11545.a
theorem OeisA11545.conjecture2 Number of collisions occurring in a system consisting of an infinitely massive, rigid wall at the origin, a ball with mass m stationary at position sorry line 85 source
Source documentation
Number of collisions occurring in a system consisting of an infinitely massive, rigid wall at the origin, a ball with mass m stationary at position $x_1 > 0$, and a ball with mass $(10^2n)m$ at position $x_2 > x_1$ and rolling toward the origin, assuming perfectly elastic collisions and no friction.
Strictly speaking, this property, which is equivalent to the statement that the interval $(m\pi, \pi/\textrm{arctan}(1/m))$ does not contain an integer for all $m = 10^n$, is not known to be true for sure. In other words, we do not know for certain that A332045 does not contain a power of $10$.

depends on sorry what this means
Statement uses: Nat, Not, Exists, Int, And, LT.lt, Real, HMul.hMul, Real.pi, HPow.hPow, OfNat.ofNat, HDiv.hDiv, Real.arctan
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
42 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem conjecture2 :
    ∀ n : ℕ, ¬ ∃ (k : ℤ),
      (Real.pi * (10 : ℝ) ^ n.cast < k.cast) ∧
      (k.cast < Real.pi / Real.arctan (1 / (10 : ℝ) ^ n.cast)) := by
…
```