e_times_phi

formal source

  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
  89/-- Attempt 2: e ≈ φ² + (1 - 1/φ)
  90
  91    φ² = φ + 1 ≈ 2.618
  92    1 - 1/φ = 1 - 0.618 = 0.382
  93    Sum: 3.000 (too big) -/
  94noncomputable def attempt2 : ℝ := phi^2 + (1 - 1/phi)
  95
  96/-- Attempt 3: e ≈ 2 + 1/φ²
  97
  98    1/φ² = φ - 1 = 0.618
  99    Wait: 1/φ² = (φ-1)² = 0.382
 100    2 + 0.382 = 2.382 (too small) -/
 101noncomputable def attempt3 : ℝ := 2 + 1/phi^2
 102
 103/-- Attempt 4: e ≈ (φ + 1)^(2/φ)
 104
 105    φ + 1 = φ² ≈ 2.618
 106    2/φ ≈ 1.236
 107    2.618^1.236 ≈ 3.34 (too big) -/