5

I recently found out that Mathematica can plot polar coordinates, and I really liked polar plotting. Today, I found something surprising. The $n$ th prime gap is defined as $$g(n)=p_{n+1}-p_n$$ I made a polar plot of $(g(n),n)$ and this is the result:

enter image description here

This surprised me, because prime gaps tend to be in some diagonals more than others, and there are some areas left untouched. Here is a plot with from $-10000$ to $10000$:

enter image description here

And this pattern continues to hold. My question is:

Why do prime gaps tend to be in some "lines" in the plot more than others? What is special in them?

Another thing:

Does this tell us anything about prime gaps?

russian bot
  • 421
  • 2
  • 11
  • At least the angle is integer (in degree or in radian measure?). – Yuval Dec 22 '20 at 04:27
  • 2
    @Yuval Presumably the angle is in radians, which is why there are $22$ spokes (since $\pi \approx 22/7$). – Théophile Dec 22 '20 at 04:30
  • Why is the sky blue? I've often noted the similarities between number theory and the study of the primes, on the one hand, and stargazing or the study of heavenly bodies, on the other. These are ultimately religious questions, or some such. –  Dec 22 '20 at 05:45
  • 5
    @ChrisCuster I don't see why they are "religious". See the answer below. – russian bot Dec 22 '20 at 05:49

1 Answers1

4

First, note that there are $22$ spokes since $\pi \approx 22/7$. If you were to zoom out (much) farther, you'd see another pattern emerge with $355$ spokes.

With that established, now observe that every increase by $2$ in the prime gap progresses $7$ spokes counterclockwise, since $$2 \cdot \frac{180}\pi \approx 2 \cdot \frac{180}{\frac{22}7} = 7 \cdot \frac{360}{22}.$$

If you were to lay out your data in a more standard histogram, you would see that there are many small gaps and fewer larger gaps. See this question, for example: Prime gaps distribution.

Théophile
  • 24,627
  • Ah, okay. What about the second question I asked? – russian bot Dec 22 '20 at 04:48
  • 1
    It probably doesn't tell us much except that it's fun to make pictures. :) You might be interested in this video, though: https://youtu.be/EK32jo7i5LQ (related to this question: https://math.stackexchange.com/questions/885879/meaning-of-rays-in-polar-plot-of-prime-numbers) – Théophile Dec 22 '20 at 04:49