A topic that I found particularly fascinating in my research is undoubtedly related to the figure of Leonardo Fibonacci, a mathematician from Pisa, who lived between the twelve and thirteenth centuries and who was considered one of the greatest mathematicians of all time.
Fibonacci is best known for the sequence of numbers he identified and known, precisely, as
"Fibonacci sequence" 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89 ... where each term is the sum of the
two preceding it. A peculiarity of the Fibonacci sequence or succession is that the relationship
between the pairs of successive terms tends very quickly to the number 1.61803 ..., known by the name of Golden Ratio or Golden Section. Without going into the innumerable properties of the sequence I simply investigated the relationship of this sequence and others derived from it with the probable prime numbers (highlighted through the application of Fermat's Small Theorem) for this purpose I developed programs with the Python programming language compiled to be executable both under Windows and Mac OS.
Below I will illustrate results obtained through the use of these programs subsequently processed through the Excel spreadsheet.
Let's start with the real Fibonacci sequence:
The program generates Fibonacci type series of n numbers with its probable primes starting from a variable number with a variable rate
For initial number 0 and initial rate 1 we have the Fibonacci series
0 Published by Gasbion 04/2020
1 How many numbers of the Fibonacci type series?: 30
2 Starting number?: 0
3 Initial rate?: 1
4 3 prime
5 5 prime
6 8
7 13 prime
8 21
9 34
10 55
11 89 prime
12 144
13 233 prime
14 377
15 610
16 987
17 1597 prime
18 2584
19 4181
20 6765
21 10946
22 17711
23 28657 prime
24 46368
25 75025
26 121393
27 196418
28 317811
29 514229 prime
30 832040
Above I reported the sequence for the first 30 numbers. It is a property of this sequence that each number prime of it has also the order number prime! (with the only exception of the number 3 that has position number equal to 4, first column equal to order number).The opposite does not apply:
That each prime order number corresponds to a sequence number that is prime. We do not
know for sure if the prime numbers belonging to the sequence are infinite or not.
The idea was that to verify the relationship of other similar successions by construct it equal to
that of the Fibonacci’s one but starting from different numbers and with an initial rate different
from 1:
The program generates Fibonacci type series of n numbers with its probable primes starting from a variable number with a variable rate. For initial number 0 and initial rate 1 we have the Fibonacci series.
How many numbers of the Fibonacci type series?: 30
Starting number?: 1
1 Initial rate?: 3
2 4
3 7 prime
4 11 prime
5 18
6 29 prime
7 47 prime
8 76
9 123
10 199 prime
11 322
12 521 prime
13 843
14 1364
15 2207 prime
16 3571 prime
17 5778
18 9349 prime
19 15127
20 24476
21 39603
22 64079
23 103682
24 167761
25 271443
26 439204
27 710647
28 1149851
29 1860498
30 3010349 prime
This sequence has several prime numbers in but does not retain the fact that the prime numbers in it correspond to order positions also prime. I found other sequences with a number of primes similar for example for initial number 1 and rate 4, initial number 1 and rate 20, but also 11 and 3, 11 and 4, 11 and 8, 2 and 1, 7 and 11 etc. But none that maintains the admirable relationship between prime numbers and order position of the original Fibonacci sequence.
I recommend not to test your computers with excessively large numbers, you just need to actually see the trend perhaps for the first thousand numbers.