The both number sets can be shown on Fibonacci spiral.

If you observe a spiral and calculate a little, You may find few interesting behavior.

The sequence of the Fibonacci numbers is the main set in this solution.

Let's treat two Fibonacci numbers as a border for subset of primes. Then we will see some results that easy to find.

Trivial as some probably would like to say.

Each next subset of primes include more numbers except first 3 numbers (2,3,5).

Each prime number has unique place in this 2D space on spiral, and this confirm randomness.

Non trivial for me is that:

Each prime numbers subset sum divided by the quantity of primes in this subset and divided by Fibonacci number give us in result approximation of half of golden number Phi. This is checked on my own application written in C language, and perhaps it's true for all primes greater than 5.

Hope this few images may help to find something more. If someone would like to play with these numbers on spiral I can share the app.

Small description what you see on screen.

Primes are in white color.

QTY - quantity of prime numbers between consecutive numbers of the Fibonacci sequence, i.e. in the given square.

SUM - the sum of prime numbers between consecutive Fibonacci numbers.

AVG - average calculated by dividing the sum by quantity, i.e. SUM / QTY.

SUM / F - the sum of prime numbers divided by the Fibonacci number.

AVG / F - shows the average of the prime numbers in a given square field, divided by the Fibonacci number. Note that. It’s look like it is always an approximation equal to half the Golden number PHI / 2 ≈ 0.80 ... except for the first three numbers (2,3,5), which are equal to Fibonacci numbers, and a good approximation increases with higher values.

Hope this be useful. I never seen before drawn Fibonacci spiral with indicated Prime numbers, so decided to share my own. Sorry for low quality of pics, but this is result of screenshot taken from running app, and spiral was scaled in HD resolution to fit the screen. This is only tested on approx. 20 000 natural numbers. So… Now If I can I would like to ask for help to confirm or deny this my nontrivial theorem. I need also some help to find formula to mathematically describe this finding.