How to take the interval where the nth Fibonacci number f(n) is the lower limit of the number of

0. References

[1]https://youtu.be/YNmRhCNvBbg Lujandre Conjecture "College Mathematics and Physics" studied in a prep school groove.

1. review of the Lujandre conjecture

It is explained in detail in reference [1]. Please take a look at it. The conjecture is as follows.

"For any natural number n, if we take the interval [n^2,(n+1)^2], then the interval will contain at least one prime number."

This conjecture is an unsolved problem. I have considered it before, but that is all I know about it. For now. 2.

2. about the Fibonacci numbers

Fibonacci numbers, by definition, are

f(n+2)=f(n+1)+f(n)

which can be expressed as

Here, the interval [f(n+1)^2, (f(n+1)+1)^2] contains at least one prime number if the Lujandre conjecture is correct.

Similarly, the intervals [(f(n+1)+1)^2, (f(n+1)+2)^2], [(f(n+1)+2)^2, (f(n+1)+3)^2], [(f(n+1)+3)^2, (f(n+1)+4)^2]... each contain at least one prime.

Continuing the above, we eventually get the following interval.

[(f(n+1)+f(n)-1)^2, (f(n+1)+f(n))^2].

Putting these intervals together, we get the following interval.

[f(n+1)^2, f(n+2)^2].

So far, we have obtained at least f(n) prime numbers.