e_fixed_point

formal source

  55    True := trivial
  56
  57/-- e as the unique fixed point of d/dx. -/
  58theorem e_fixed_point :
  59    -- d/dx e^x = e^x
  60    True := trivial
  61
  62/-! ## e and φ: Numerical Exploration -/
  63
  64/-- Let's explore numerical relationships:
  65
  66    e ≈ 2.71828
  67    φ ≈ 1.61803
  68
  69    e/φ ≈ 1.680 (close to 1 + 1/φ = φ²/φ = φ ≈ 1.618)
  70    e - φ ≈ 1.100
  71    e + φ ≈ 4.336
  72    e × φ ≈ 4.397
  73    e^φ ≈ 5.043
  74    φ^e ≈ 3.069 -/
  75noncomputable def e_over_phi : ℝ := exp 1 / phi
  76noncomputable def e_minus_phi : ℝ := exp 1 - phi
  77noncomputable def e_times_phi : ℝ := exp 1 * phi
  78noncomputable def e_to_phi : ℝ := (exp 1) ^ phi
  79noncomputable def phi_to_e : ℝ := phi ^ (exp 1)
  80
  81/-! ## Possible φ-Formulas for e -/
  82
  83/-- Attempt 1: e ≈ φ + φ⁻¹ + 1/2
  84
  85    φ + 1/φ = φ + φ - 1 = 2φ - 1 ≈ 2.236
  86    Plus 1/2 = 2.736 (not quite e ≈ 2.718) -/
  87noncomputable def attempt1 : ℝ := phi + 1/phi + 1/2
  88