Discrete Mathematics
# Fibonacci Numbers

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

- Each term of the progression is a positive integer.
- The common difference of the progression is \(d\).
- 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 \(F_m=F_{m-1}+F_{m-2}\)

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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