I am delving into Python development, particularly with Flask, to build backend applications. This program serves as a refresher on Python's core foundations, emphasizing terminal-controlled outputs.

This program calculates Fibonacci numbers up to 22 and Lucas numbers up to 20. It employs the Sieve of Eratosthenes to find prime numbers from 2 to the array limit, including all perfect numbers within that range. Additionally, it establishes a relationship to estimate the golden ratio. If identical values are set for both Fibonacci and Lucas calculations, it computes the golden ratio to the power of n, compares it to a combination of the Lucas and Fibonacci theorems, and provides a mathematical limit proof as values approach infinity.

• Fibonacci_n = Fibonacci_{n - 1} + Fibonacci_{n - 2}
• Lucas_n = Lucas_{n - 1} + Lucas_{n - 2}
• Golden Ratio (φ) = 1+√52
  
  
• F_n ⁄F_{n - 1} ≅ φ
• L_n ⁄L_{n - 1} ≅ φ
• φⁿ ≅ (L_n+F_n * √5)/2
• L_2n = 5(F_n)² + 2(-1)^{n - 1}
• lim(n→∞) L_n ⁄F_n = √5