Why is “jump” between two primes (almost) always prime or 1 up to 1000

number theoryprime numbers

While looking at prime numbers between 1 to 1000 I noticed that the number of non-primes between prime numbers are almost always also prime or 1. In other words, if we take the prime gap,
$$
g_n=p_{n+1}-p_n,
$$
for prime numbers larger than 2, and subtract 1 from it, we almost always get a prime number or 1. In the cases (between 1 to 1000) where $g_n-1$ is not equal to a prime number or 1 it is instead equal to 9. Below is a figure I made to better explain:

enter image description here

What is the reason that the "jump length" (i.e. numbers in red or blue above) between primes are either 1, 9 or also prime when looking at 1 to 1000?

Best Answer

This phenomenon was studied by Odlyzko, Rubinstein and Wolf (with whom I have written a paper on primes).

Definition: An integer $d$ is called a jumping champion for a given $x$ if d is the most common gap between consecutive primes up to $x$.

They postulated that most common gap between primes or the jumping champions are the product of prime numbers i.e. the first jumping champion is $2$. The after some time, $2*3 = 6$ takes over as the next jumping champion. At around $1.74*10^{35}$, the number $2*3*5 = 30$ takes over as the next jumping champion. At around $10^{425}$, the number $2*3*5*7 = 210$ takes over as the next jumping champion and so on.

This means that per your definitions, if we subtract $1$ from the gap between primes then the most common results would be $1$ initially. This would be overtaken by $5$ which would in turn be overtaken by $29$ and this would be overtaken by $209$ as you go higher up the number line.

So it is a co-incidence that $5, 29$ are prime but $209 = 11*19$ is not. This means if you looked large enough to reach the range of $209$, you would actually have the opposite conclusion based on experimental evidence that gap between prime minus $1$ is often a non prime.

Equivalent question: With the above arguments, your question is equivalent to

How often is the primorial number $2.3.5.7\cdots p - 1$ a prime?

Reference:

  1. Odlyzko, A.; Rubinstein, M.; and Wolf, M. "Jumping Champions." Experiment. Math. 8, 107-118, 1999.
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