The golden ratio as a number base
6 hours ago
- #Fibonacci
- #Golden Ratio
- #Number Theory
- Fibonacci numbers are a sequence where each number is the sum of the two preceding ones (1, 1, 2, 3, 5, 8, etc.).
- Every positive integer can be uniquely represented as a sum of distinct, non-consecutive Fibonacci numbers (Zeckendorf’s Theorem).
- The golden ratio (φ ≈ 1.618) is closely related to Fibonacci numbers, with Fₙ = (φⁿ – (-φ)⁻ⁿ)/√5.
- George Bergman proved that every positive real number can be written as a sum of distinct powers of φ, with uniqueness conditions.
- Jeffrey Shallit and Ingrid Vukusic studied φ-representations of integers, focusing on φ-anti-palindromic numbers.
- Clark Kimberling conjectured that φ-anti-palindromic numbers are the only ones where doubling exponents in φ-representation yields another integer.
- Lucas numbers (2, 1, 3, 4, 7, etc.) also relate to the golden ratio, with ratios converging to φ.
- Shallit and Vukusic used Walnut to connect φ-representations with Lucas numbers, proving properties about exponents.