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In this experiment, I have checked how many times different gapped primes occur out of the first 10000, 100000, 1000000 first primes.

Please view the following as ($X$:$Y$) where $X$ represents the gap and $Y$ represents how many times it occurs.

Out of first 1000000 primes:

2: 40405 4: 40233 6: 68311 8: 28746 10: 36766 12: 44113 14: 23569 16: 16658 18: 29065 20: 14409 22: 12833 24: 17680 26: 7979 28: 8493 30: 13773 32: 4048 34: 4263 36: 6344 38: 2749 40: 3313 42: 4424 44: 1717 46: 1387 48: 2235

Out of first 100000 primes:

2: 10251 4: 10213 6: 15989 8: 7067 10: 8873 12: 10158 14: 5353 16: 3661 18: 6304 20: 3043 22: 2826 24: 3538 26: 1543 28: 1631 30: 2114 32: 742 34: 756 36: 1032 38: 455 40: 563 42: 661 44: 250 46: 219 48: 290

Out of first 10000 primes:

2: 1271 4: 1263 6: 2012 8: 801 10: 953 12: 1008 14: 512 16: 353 18: 537 20: 249 22: 235 24: ///22/// 26: 91 28: 102 30: 154 32: 35 34: 36 36: 55 38: 20 40: 28 42: ///20/// 44: 5 46: 6 48: ///3///

You will notice the following:

  • A peak occurs in almost all $X \mod 6 =0$ compared to the prior 2. The only exceptions are in the first first 10000 primes and only in $X=24$, $X=42$ and $X=48$

  • There are always more shorter gaps than longer gaps

My question is broken into 3 parts:

A) Is it fair to assume that probably there are more shorter gaps than longer gaps?

B) Is it fair to assume that probably there are more $X \mod 6=0$ gaps than any other gaps?

C) If B) is assumed to be fair, is it fair to assume that SEXY primes (primes with a gap of 6) are the most popular gap between primes?

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    $\begingroup$ The Hardy–Littlewood prime tuples conjecture (see, for example, this post on Tao's blog for the statement: terrytao.wordpress.com/2013/06/03/…) gives conjectural asymptotic densities for all prime tuples, and fixed gaps between primes are a special case. Does this address some of your questions? $\endgroup$
    – Daniel Hast
    Commented Feb 2, 2021 at 7:59
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    $\begingroup$ It's generally believed that the most common gap between consecutive primes increases without bound the farther out you go – it's just that you have to go out pretty far (much farther than a mere million primes) to see it happen. $\endgroup$
    – Gerry Myerson
    Commented Feb 2, 2021 at 8:55
  • $\begingroup$ @Gerry Myerson The article explains A) and C) but there is not much mentioning in regards to the other $X \ mod 6 = 0$ ‘s overall increase $\endgroup$
    – Isaac Brenig
    Commented Feb 5, 2021 at 20:28
  • $\begingroup$ Asked as a variant on math.stackexchange.com/questions/4014327/… $\endgroup$
    – Isaac Brenig
    Commented Feb 5, 2021 at 22:07

1 Answer 1

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Andrew Odlyzko, Michael Rubinstein, and Marek Wolf, Jumping champions, Experimental Mathematics 8 (1999), 107–118 suggest that somewhere around $x=1.7427\times10^{35}$, the most common gap between consecutive primes less than $x$ switches from $6$ to $30$.

See The Most Common Prime Gaps, posted by John Baez, at https://golem.ph.utexas.edu/category/2016/03/the_most_common_prime_gaps.html

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  • $\begingroup$ The article explains A) and C) but there is not much mentioning in regards to the other $X \mod 6 = 0$ ‘s overall increases (asked in B)) $\endgroup$
    – Isaac Brenig
    Commented Feb 5, 2021 at 20:28
  • $\begingroup$ If $p>3$ is prime and $x\equiv0\bmod6$ then $p+x$ is guaranteed not to be divisible by three which makes it more likely to be prime than $p+y$ for $y$ close to $x$ but not a multiple of six. I'd guess that for large $p$ a gap of six is more common than four or eight, $12$ is more common than $10$ or $14$, $18$ is more common than $16$ or $20$, and so on. There might be an exception for a gap of $x\not\equiv0\bmod6$ if $x$ is divisible by enough small primes, if $\sum_{p\mid x}p^{-1}>1/3$. $\endgroup$
    – Gerry Myerson
    Commented Feb 5, 2021 at 22:34
  • $\begingroup$ Many gratitudes for commenting back. And i just wanted to updated you that I have started a variant question here (while incorporating your answers) : math.stackexchange.com/questions/4014327/… which seems to be more of a suitable forum $\endgroup$
    – Isaac Brenig
    Commented Feb 5, 2021 at 22:48

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