FormalConjectures.OEIS.«11545»

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import FormalConjecturesUtil

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

References:

A011545

namespace OeisA11545open Real Int

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

noncomputable def a (n : ℕ) : ℕ :=
  (floor (Real.pi * (10 : ℝ) ^ n.cast)).toNat

@[category test, AMS 11]
theorem a_0 : a 0 = 3 := by⊢ a 0 = 3
  delta a⊢ ⌊π * 10 ^ ↑0⌋.toNat = 3
  norm_num [((Int.floor_eq_iff.mpr _) : ⌊π⌋ = ↑3), Int.toNat, false,
    le_of_lt Real.pi_gt_three, Real.pi_lt_four]All goals completed! 🐙

@[category test, AMS 11]
theorem a_1 : a 1 = 31 := by⊢ a 1 = 31
  simp_all[a]⊢ ⌊π * 10⌋.toNat = 31
  exact (congr_arg _) ((Int.floor_eq_iff.2
    ⟨by⊢ ↑31 ≤ π * 10 linear_combination 10 * (.pi_gt_d20)All goals completed! 🐙,
     by⊢ π * 10 < ↑31 + 1 ·⊢ π * 10 < ↑31 + 1 linear_combination 10 * .pi_lt_d20All goals completed! 🐙⟩) : ⌊_⌋ = 31)

@[category test, AMS 11]
theorem a_2 : a 2 = 314 := by⊢ a 2 = 314
  simp_all[a]⊢ ⌊π * 10 ^ 2⌋.toNat = 314
  exact (congr_arg _) ((Int.floor_eq_iff.mpr
    ⟨by⊢ ↑↑314 ≤ π * 10 ^ 2 ·⊢ ↑↑314 ≤ π * 10 ^ 2 linear_combination 100 * .pi_gt_d20All goals completed! 🐙,
     by⊢ π * 10 ^ 2 < ↑↑314 + 1 ·⊢ π * 10 ^ 2 < ↑↑314 + 1 linear_combination 100 * .pi_lt_d20All goals completed! 🐙⟩)) |>.trans (Int.toNat_natCast _)

@[category test, AMS 11]
theorem a_3 : a 3 = 3141 := by⊢ a 3 = 3141
  symm⊢ 3141 = a 3
  norm_num[a]⊢ 3141 = ⌊π * 1000⌋.toNat
  exact (.symm ((congr_arg _) ((Int.floor_eq_iff.2
    ⟨by⊢ ↑3141 ≤ π * 1000 linear_combination 1000 * .pi_gt_d20All goals completed! 🐙,
     by⊢ π * 1000 < ↑3141 + 1 linear_combination 1000 * .pi_lt_d20All goals completed! 🐙⟩) : ⌊_⌋ = 3141)))

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 $10^-9$.

@[category research open, AMS 11]
theorem conjecture1 : ∀ n, ¬ IsSquare (a n) := by⊢ ∀ (n : ℕ), ¬IsSquare (a n)
  sorryAll goals completed! 🐙

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 $10$.

@[category research open, AMS 11]
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⊢ ∀ (n : ℕ), ¬∃ k, π * 10 ^ ↑n < ↑k ∧ ↑k < π / arctan (1 / 10 ^ ↑n)
  sorryAll goals completed! 🐙

end OeisA11545