Discrete Mathematics

Fibonacci Numbers

Fibonacci Numbers: Level 4 Challenges

         

Find the least positive integer dd 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 dd.
  3. No term of the progression appears in the Fibonacci sequence.

Details and assumptions

The Fibonacci sequence is defined by F1=1,F2=1F_1 = 1, F_2 = 1 and Fn+2=Fn+1+Fn F_{n+2} = F_{n+1} + F_{n} for n1 n \geq 1 .

The arithmetic progression has infinitely many terms.

How many values of kk are there such that FkF_k is of the form 2n2^n?

Note that both kk and nn are non-negative integers.

Also F0=0,F1=1F_0=0,F_1=1 and Fm=Fm1+Fm2F_m=F_{m-1}+F_{m-2}

How many Fibonacci numbers FnF_n divide F100F_{100} , where nn is a positive integer greater than 1?

Find the sum of all primes pp, such that pp divides upu_p, where upu_p is the pp-th Fibonacci number.

Details and assumptions

The Fibonacci sequence is defined by F1=1,F2=1F_1 = 1, F_2 = 1 and Fn+2=Fn+1+Fn F_{n+2} = F_{n+1} + F_{n} for n1 n \geq 1 .

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

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

I do this process continuously 1010 times, and then add my final two numbers, to get 39353935.

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

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