Fibonacci sequence primes - math

Question

I wasnt able to find any topic on this despite multiple searches. Apologies if this was already covered here.
Can anyone point me in the right direction for further research on the below topic:
I have played recently with the Fibonacci sequence and prime numbers. I have noticed that, at least for certain initial items in that sequence, if you mark all of the primes in the sequence (I did it for numbers: 2(odd prime),3,5,13,89,233,1597,28657) and check their place in the sequence, it also turns out to be a prime. I'm asssuming that 0 is the 0th item. To give some examples: for 2(odd prime) - which is the 3rd number of the sequence - 3 is also a prime, for 13 being 7th number of the sequence - 7 is also prime, for 233 - being 17th number of the sequence - 17 is also a prime. This seems to be true for all primes up to 17 item in the sequence, it then starts to diverge as 19th (19 being a prime) number of the sequence is 4181 which is not a prime.
To give a visual example:
item number Fib number
0 0
1 1
2 1
prime 3 2 prime !
not prm 4 3 prime
prime 5 5 prime !
6 8
prime 7 13 prime !
8 21
9 34
10 55
prime 11 89 prime !
12 144
prime 13 233 prime !
14 377
15 610
16 987
prime 17 1597 prime !
18 2584
prime 19 4181 not prime
20 6765
21 10946
22 17711
prime 23 28657 prime !
24 46368
25 75025
26 121393
27 196418
28 317811
prime 29 514229 prime !
30 832040
prime 31 1346269 not prime
32 2178309
33 3524578
34 5702887
35 9227465
36 14930352
prime 37 24157817 not prime
38 39088169
39 63245986
40 102334155
41 165580141
Despite there being certain numbers in the sequence which are primes but their sequence order number not being a prime and vice versa, it's still quite interesting to know why is there such a pattern and if it's true for majority of Fibonacci sequence numbers.
Again, apologies if this is something obvious.
TIA for any clarification on this!

Answer 1

You might want to look at A001605 and follow the links from there. The Online Encyclopedia of Integer Sequences is a fantastic resource for things like this.
I also discuss fibonacci primes at my blog.

Answer 2

You seem to be mixing two different thoughts here:
If I index the Fibonacci series with a prime, will I find a prime
there?
If I find a prime in the Fibonacci series, will its index be a
prime?
According to the Fibonacci Prime article in Wikipedia:
Except for the case n = 4, all Fibonacci primes have a prime index,
..., but not every prime is the index of a Fibonacci prime.
Which means 1 above doesn't hold, as you demonstrated with prime index 19, but 2 above always holds except for Fibonacci prime 3 whose Fibonacci index is 4.
To bring this back to programming, my Python code to reproduce your table (sorta):
def is_prime(n):
if n > 1 and n % 2 != 0 or n == 2:
for i in range(3, int(n ** 0.5) + 1, 2):
if n % i == 0:
break
else:
return n
return '*' * len(str(n))
print('n', 'f', sep='\t')
f, p, n = 0, 1, 0
while True:
print(is_prime(n), is_prime(f), sep='\t')
f, p, n = f + p, f, n + 1
OUTPUT
n f
* *
* *
2 *
3 2
* 3
5 5
* *
7 13
* **
* **
** **
11 89
** ***
13 233
** ***
** ***
** ***
17 1597
** ****
19 ****
** ****
** *****
** *****
23 28657
** *****
** *****
** ******
** ******
** ******
29 514229
** ******
31 *******
** *******
** *******
** *******
** *******
** ********
37 ********
** ********
** ********
** *********
41 *********
** *********
43 433494437
** *********
** **********
** **********
47 2971215073
** **********
** **********
** ***********
** ***********
** ***********
53 ***********
** ***********
** ************
** ************
** ************
** ************
59 ************
** *************
61 *************
** *************
** *************
** **************
** **************
** **************
67 **************
** **************
** ***************
** ***************
71 ***************
** ***************
73 ***************
** ****************
** ****************
** ****************
** ****************
** ****************
79 *****************
** *****************
** *****************
** *****************
83 99194853094755497
** ******************
** ******************
** ******************
** ******************
** *******************
89 *******************
** *******************
** *******************
** *******************
** ********************