Fibonacci and 7

Number Theory Level pending

Let $a_n$ be the $n$ . item of the Fibonacci sequence. How many solutions are there for the next equation, where $k$ is a positive integer?

$7^k=a_m$

What is the Fibonacci sequence? If you don't know then read this wiki: The Fibonacci sequence

5 2 4 0 1 More than 5, but not infinite 3 Infinite

1 solution

Áron Bán-Szabó
Jun 7, 2017

It is not a real solution, but with it we get the answer:

We can notice that if $7|a_m$ , then $8|m$ and if $3|a_m$ , then $4|m$ . So the sequence's every item which is divisible by $7$ is also divisible by $3$ , so $a_m$ never will be a power of $7$ .

Fibonacci and 7

1 solution

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