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# Fibonacci Numbers

1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, ... They show up in nature, they show up in math, and they've got some beautiful properties.

# Fibonacci Numbers: Level 4 Challenges

Find the least positive integer $$d$$ for which there exists an infinite arithmetic progression satisfying the following properties:

1. Each term of the progression is a positive integer.
2. The common difference of the progression is $$d$$.
3. No term of the progression appears in the Fibonacci sequence.

Details and assumptions

The Fibonacci sequence is defined by $$F_1 = 1, F_2 = 1$$ and $$F_{n+2} = F_{n+1} + F_{n}$$ for $$n \geq 1$$.

The arithmetic progression has infinitely many terms.

How many values of $$k$$ are there such that $$F_k$$ is of the form $$2^n$$?

Note that both $$k$$ and $$n$$ are non-negative integers.

Also $$F_0=0,F_1=1$$ and $$F_m=F_{m-1}+F_{m-2}$$

How many Fibonacci numbers $$F_n$$ divide $$F_{100}$$, where $$n$$ is a positive integer greater than 1?

Find the sum of all primes $$p$$, such that $$p$$ divides $$u_p$$, where $$u_p$$ is the $$p$$-th Fibonacci number.

Details and assumptions

The Fibonacci sequence is defined by $$F_1 = 1, F_2 = 1$$ and $$F_{n+2} = F_{n+1} + F_{n}$$ for $$n \geq 1$$.

I have a Monster, in which when I enter two positive integers $$(a,b)$$, where $$b>a$$, it gives out two positive integers, $$(b,a+b)$$.

I then enter these two numbers again into the Monster, and get two more numbers.

I do this process continuously $$10$$ times, and then add my final two numbers, to get $$3935$$.

What is the product of the two numbers I entered at first?

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