So we would naturally expect there to be about 4.17% X 5424 (approx 255) 2p semi-primes nestled in between those twin primes.

Instead there is EXACTLY one!

And that one 2p semi-prime between two twin primes is the number 4, which is the very first composite number. Above 4–nothing.

It makes you wonder whether #1. I made some horrible mistake in tallying the 2p semi-prime positions or #2. There is some property of the number in between twin primes that somehow prevents it from being a 2p semi-prime.

]]>Thanks for all the comments! I’m working through some associated ideas this week in class, with the goal of having students come up with and explore their own conjectures. What you’ve provided here–analysis, conjecture, and process–is a great resource for us as we explore!

And thanks for alerting me to the potential error in undercounting the prime-semiprime pairs (I swear I thought I addressed the before- and after- possibilities!). I’ll re-run my analysis and update the post accordingly.

Thanks again!

]]>For example, from the above discussion we know there are a number of prime/semi-prime triplets, like 13-14-15 and 2017-2018-2019.

And there is an even longer example contiguous stretch of primes & semi-primes in the very low integers: 2-3-4-5-6-7. So that is a sextuplet. Are there any more sextuplets, or any longer sequences (septuplets, octoplets, etc)? Are there any quadruplets or quintuplets to go along with the (known-to-be numerous) triplets and (one known) sextuplet?

]]>My guess is because neither primes nor 2p semi-primes are randomly and equally distributed across the integers. Rather, both are relatively more common among smaller integers and less common the higher we go.

What that means is that both the set of primes and the set of 2p semi-primes are bunched up closer to the 0 end of the number line. Since they are both bunched up in more or less the same area that makes them more likely to collide with each other than if they were both randomly and equally scattered across the entire number line.

]]>Using the spreadsheet I created (link below) that lists every 1p, 2p, and 3p prime & semi-prime up through 611,953, I thought it would be interesting to see how many 1-2-3 prime/semi-prime triplets exist in that range.

It turns out that 2016-2017-2018 is the TENTH 1-2-3 triplet, if you start counting with 1-2-3 as the first.

There are 381 1-2-3 triplets through 611,953 (representing the first 50,000 primes).

Here are the first 11 1-2-3 triplets in order:

#1. 1-2-3

#2. 13-14-15 (1X13 – 2X7 – 3X5)

#3. 37-38-39

#4. 157-158-159

#5. 541-542-543

#6. 877-878-879

#7. 1201-1202-1203

#8. 1381-1382-1383

#9. 1621-1622-1623

#10. 2017-2018-2019

#11. 2557-2558-2559

So it has been almost 400 years since we experienced our last 1-2-3 triplet set of years, and it will be another 550 years before we have another one.

]]>I think the discrepancy is probably that your software was only counting half the semi-primes adjacent to primes. It was (likely) counting just the ones right before OR just the ones right after a prime, and missing the other possibility.

I count 2450 semi-primes just after a prime, and 2468 just before a prime, through 500,000. So that is in very close agreement with what you found.

It is interesting that the proportion of semi-primes just before & just after primes is very close to 50/50.

And this brings up the question–are the any primes with a 2p semi-prime just before AND just after the prime? I’m guessing yes . . .

]]>I put them into Excel and did a little analysis on them. Here is what I found:

2885 2p semi-primes were one greater than a prime

2897 2p semi-primes were one smaller than a prime

Total 5781 2p semi-primes adjacent to a prime. (There is exactly ONE semi-prime that is both one greater and one smaller than a prime–4 is 2×2 and adjacent to both 3 and 5. So our total is 2885+2897-1=5781.)

With 50000 primes, there are approx. 100,000 possible spots for a semi-prime to possibly inhabit (one before and one after each prime). But there are 5424 sets of twin primes below 611953, so the total number of spots available is 100,000-5424 = 94,576.

So we have 5781 of 94.576 spots, or 6.11% of possible spots, inhabited by a 2p semi-prime.

The first 50,000 primes stretch from 1 through 611953, but 50,000 of those spots are taken by the primes themselves. So that leaves 561953 spots remaining where a 2p semi-prime could be.

I counted 26,488 2p semi-primes below 611953. So 26,488 of 561,953 gives 4.71% of available spots are filled by 2p semi-primes.

So the interesting thing is that 2p semi-primes fill just 4.71% of ALL available spots, but 6.11% of spots adjacent to primes.

So 2p semi-primes are actually ***MORE likely by a pretty subsantial amount (6.11% vs 4.71%) to be located adjacent to a prime than random chance would suggest***.

]]>This sequence is of the form 1p, 2q, 3r where each of p, q, r is prime.

So this brings you a few questions:

* Is this the smallest possible 1-2-3 triplet (aside from 1,2,3 of course!) (We can argue later about whether or not p, q, and r can equal ¹ . . . )

* We know that all 2017-2018 pairs must be of the form 1p,2q or 2p,1q. So must all semi-primes triplets be of the form 1p,2q,3r or 3p,2q,1r? Or are forms such as 1p,2q,5r or 1p,2q,7r possible? I would conjecture that triplets of the form 1p,2q,Xr exist for *every* odd prime X.

* Furthermore, I would conjecture that there is not just one triplet but an infinite number of them for every X.

* And finally, since I seen to be in a conjecturing mood today, I would conjecture that both of the above conjectures are true iff the twin prime conjecture is true. They all have to do with the idea that the prime distribution is smooth in some sense, and if the twin prime connector is true it will show the distribution to be pretty much as smooth as possible given the definition of prime.

]]>