Eyebrow-raising pattern of number of primes between terms of the Fibonacci number sequence?

So, 1,1,2,3,5,8,13,21… Any connection to primes?…it appears not. However, in between the Fibonacci numbers are how much primes? Let’s see:

  • 1 and 1: 0
  • 1 and 2: 0
  • 2 and 3: 0
  • 2 and 3: 0
  • 5 and 8: 1
  • 8 and 13: 1
  • 13 and 21: 2
  • 21 and 34: 3
  • 34 and 55: 5
  • 55 and 89: 8
  • 89 and 144: 13

Huh. What could this imply? Let me just close with the same annoying (but wonderful) pattern. 1,2,3,5,8,13,21…

Answer

Eyebrow raising indeed, though the pattern does not continue as you suggest. I get
0,1,1,2,3,5,7,10,16,23,37,55,84,125,198

Remember that the the number of primes has a well known growth rate (https://en.wikipedia.org/wiki/Prime_number_theorem). Since the Fibonacci numbers are relatively spread out, using n/logn to approximate the number of primes less than n will cause the number of primes between them to behave like the growth rate of the primes.

Attribution
Source : Link , Question Author : HyperLuminal , Answer Author : Martin Sleziak

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