GAPS BETWEEN PRIME PAIRS (Twin Primes)

[mart_r]

Originally Posted by Bobby Jacobs

It is weird that we know all twin prime gaps up to 4426 in the numbers up to 1.05e15. However, we have searched up to 1.84467e18 for regular prime gaps, and we have still not found a gap of 716. That seems strange because twin primes are a lot rarer than prime numbers.

Naturally, twin primes are rarer, so the gaps between them are more numerous.

I was a bit lazy to finish my range, but new results should arrive this week.

Looking at arXiv: 1807.08777 (J. P. Sorenson and J. Webster: Two algorithms to find primes in patterns), (also jzakiya's program in Misc.Math) I should consider writing my own algorithm in C sooner or later, there has to be a faster method than the one I'm currently using.

[mart_r]

Originally Posted by mart_r

Naturally, twin primes are rarer, so the gaps between them are more numerous.

I knew something was odd about this phrasing...

There's a larger variety in the size of the gaps.

Is that somewhat understandable an explanation?

[mart_r]

.

[pinhodecarlos]

Taking 1.15e16 to 1.19e16.

[Bobby Jacobs]

Originally Posted by mart_r

I knew something was odd about this phrasing...

There's a larger variety in the size of the gaps.

Is that somewhat understandable an explanation?

It still seems counterintuitive, but OK, it is a good explanation. It seems like since prime numbers are more common than twin primes, the gaps between primes would fill up first. In the beginning, the first missing gap between primes is greater than the first missing gap between twin primes. However, at 1.05e16, the first missing gap between twin primes is greater than the first missing gap between primes. There must be a point where the first n such that 6n is a missing gap between twin primes passes the first m such that 2m is a missing gap between primes. I wonder when that point is. By the way, I meant 1.05e16 in my earlier post.

[mart_r]

Taking 1.19e16 to 1.2e16.

[mart_r]

Not a single new gap in this range. Oy.

Taking 1.2e16 to 1.22e16.

[mart_r]

Taking 1.22e16 to 1.25e16.