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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.

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.

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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 \(F_m=F_{m-1}+F_{m-2}\)

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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 \( n \geq 1 \).

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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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