👉 Lucas mathematics, named after mathematician John Lucas, is a family of number systems and computational methods that extend beyond the traditional base-10 (decimal) system. It is based on the binary numeral system but introduces unique properties and operations that make it particularly useful in computer science and theoretical mathematics. Lucas numbers, a sequence defined by the recurrence relation \( L_n = 4L_{n-1} + L_{n-2} \) with initial conditions \( L_0 = 1 \) and \( L_1 = 3 \), form the foundation of this system. These numbers exhibit interesting properties like being relatively prime to powers of 2 and having a rich structure in modular arithmetic. Lucas mathematics also includes operations such as Lucas addition, which behaves similarly to binary addition but with a twist that preserves certain number-theoretic properties, making it valuable in cryptography and algorithmic design. The system's elegance lies in its ability to bridge number theory, combinatorics, and computer science, offering insights and tools that are both theoretically profound and practically applicable.
