51
$\begingroup$

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 ZERO
  • 1 and 2 NADA
  • 2 and 3 ZILCH
  • 3 and 5 ZIP
  • 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 pattern. $$1,2,3,5,8,13,21...$$

$\endgroup$
  • 11
    $\begingroup$ @HyperLuminal as the rest of people said, it seems to be the "strong law of small numbers", but I like very much when people tries to think different and find new patterns, and I enjoyed this one very much. $\endgroup$ – iadvd May 29 '15 at 0:22
  • $\begingroup$ love this question/thread. if you count the integers that are parts of a pythagorean triple between the fibonaccis, it appears there might be a "fibbonaci-esque" sequence, but frequently off by one. [3 to 5: 4, count = 1] [5 to 8; 7, count = 1] [8 to 13; 9 11 12, count = 3] [13 to 21; 15 16 17 19 20, count = 5] subsequent counts= 9, 15, 25, 40, 66, 107, 174, 282, 457 $\endgroup$ – don bright May 30 '15 at 23:26
  • $\begingroup$ From an old number theory book: the Fibonacci numbers, divisible by a given prime, are evenly spread in the sequence, for example every third Fibonacci number is even: 2,8,34,144... $\endgroup$ – DVD Jun 4 '15 at 5:21
56
$\begingroup$

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 $n/\log n$ 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.

$\endgroup$
  • 14
    $\begingroup$ This has been considered before. Searching with your data in OEIS gives A052011. $\endgroup$ – Jeppe Stig Nielsen May 29 '15 at 10:19
28
$\begingroup$

This is a fun observation... but I think you have a mistake. I wrote a quick python script to generate the fibonacci numbers and primes and make the counts and this is what I get:

Between 5 and 8 there is 1 prime: 7

Between 8 and 13 there is 1 prime: 11

Between 13 and 21 there are 2 primes: 17, 19

Between 21 and 34 there are 3 primes: 23, 29, 31

Between 34 and 55 there are 5 primes: 37, 41, 43, 47, 53

Between 55 and 89 there are 7 primes: 59, 61, 67, 71, 73, 79, 83

Between 89 and 144 there are 10 primes: 97, 101, 103, 107, 109, 113, 127, 131, 137, 139

Between 144 and 233 there are 16 primes: 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229

Between 233 and 377 there are 23 primes: 239, 241, 251, 257, 263, 269, 271, 277, 281, 283, 293, 307, 311, 313, 317, 331, 337, 347, 349, 353, 359, 367, 373

Between 377 and 610 there are 37 primes: 379, 383, 389, 397, 401, 409, 419, 421, 431, 433, 439, 443, 449, 457, 461, 463, 467, 479, 487, 491, 499, 503, 509, 521, 523, 541, 547, 557, 563, 569, 571, 577, 587, 593, 599, 601, 607

$\endgroup$
18
$\begingroup$

You can make sense of this pattern (which as yoann points out does not go on very far) as saying that the density of primes is a constant $\phi^{-3}\approx0.236$, since the gap length is $F_{n+1}-F_n=F_{n-1}$ and the conjectured prime count in this gap is $F_{n-4}\approx F_{n-1}\phi^{-3}$.

Unfortunately, this goes against the prime number theorem or various weak versions of it that say that the density of primes goes to $0$ for large $n$, so even without working it out it is clear that the pattern must be short-lived.

(Note: Because the Fibonacci numbers diverge too quickly, the conjectured pattern is not itself sufficient to show that the density of primes is $\phi^{-3}$. But it is enough to prove that the upper density is at least this, which is enough to violate the density zero result. And this is easy to show - considering numbers not divisible by $2,3,5,7$ already gives density $8/35<\phi^{-3}$.)

$\endgroup$
11
$\begingroup$

The pattern does not seem to go on: there are only 17 primes between 144 and 233.

More generally, using the Prime number theorem $\pi(x) \sim_{x \to \infty} \frac x {\ln x}$ (where $\pi(x)$ is and the number of primes lower or equal to $x$), and the formula for the n-th Fibonacci number $F_n \sim_{n \to \infty} \frac{\phi^n}{\sqrt 5}$, we can show that: $$\pi(F_n) \sim_{n \to \infty} \frac{\phi^n}{n \sqrt 5 \ln \phi}$$ The number of primes between two consecutive Fibonacci numbers is therefore: $$\pi(F_{n+1}) - \pi(F_n) \sim_{n \to \infty} \frac{\phi^n (\phi - 1)}{n \sqrt 5 \ln \phi} \sim_{n \to \infty} \frac{\phi - 1}{n\ln \phi} F_n$$.

Which contradicts what you would like to have by a factor of $\frac{\phi - 1}{n\ln \phi}$.

$\endgroup$
  • $\begingroup$ Why does your figure of 16 primes in [144,233] disagree with @TravisJ’s result of 17? $\endgroup$ – PJTraill May 29 '15 at 9:39
  • 1
    $\begingroup$ @PJTraill Because 233 is prime. He considered (144,233) while I considered [144,233]. $\endgroup$ – yoann May 29 '15 at 9:50

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.