The relationship between $a$ and Fibonacci numbers is defined as follows: if $a$ divides $F_n$ for every $d \geq 1$, then $a$ is said to be a divisor of Fibonacci numbers.
Sure, for example, if $a=3$, then $3$ is a divisor of Fibonacci numbers because $3$ divides $F_n$ for every $n \geq 1$. This is because the Fibonacci sequence is defined recursively as $F_n = F_{n-1} + F_{n-2}$, so each term is the sum of the two previous terms, making it divisible by $3$.
The statement $a$ divides Fibonacci $F_n$ for every $d \geq 1$ means that the divisibility holds for all terms of the Fibonacci sequence, starting from the first term ($n=1$) and continuing indefinitely. This is important because it shows that the divisibility holds for all possible values of $n$, not just a specific subset.
No, if $a$ is a divisor of Fibonacci numbers, then it must divide every term in the sequence. This is because the divisibility is defined for all $d \geq 1$, so if $a$ does not divide a certain term, it will not hold for all values of $d$.
This property is related to the fact that the Fibonacci sequence exhibits a specific pattern based on the previous terms, making it predictable and allowing for the divisibility by $a$. This can also be seen in the closed form expression for the Fibonacci sequence, $F_n = \frac{\varphi^n - \psi^n}{\sqrt{5}}$, where $\varphi$ and $\psi$ are the roots of the characteristic equation $x^2-x-1=0$, and any divisor of $F_n$ must also divide the numerator.