0

I'm learning. Trying to figure out what this is a variation of. It's cool and I want to learn more about it. It's been fun to watch as my app runs into them.

Any way, in a search for twin primes, I noticed that if I

  1. Take a positive number n
    • (that's greater than 1. I do not check the first n it's n ± 1 for primality.)
  2. multiply n by 3
  3. check for primes at ± 1
  4. if we have a twin prime pair
    • take n, double the multiplier (ie 3 becomes 6), multiply it by n.
    • repeat 3 and 4 - break when no twin prime pair is found.
  5. n+1, repeat from 1

Here's a few listed as n, 3n, 6n, 12n, 24n, ... and check n ± 1 for primality after the first n (its not being counted).

An example of my format... I'm trying my best.

n=4, (3n = 12, 6n = 72)

4, (12, 72)
6, (18, 108)
9450, (28350, 170100, 2041200)
11490, (34470, 206820, 2481840)
157570, (472710, 2836260, 34035120, 816842880)
860640, (2581920, 15491520, 185898240, 4461557760)

Thanks! I hope it's okay I'm only listing the number between the twin primes - it's easier for me to watch and keep track of as stuff is processing.

1 Answers1

0

These are not prime k-tuplets. That term, and the related term prime k-tuples are most commonly used for primes that occur close together. See, for example, this glossary entry at the PrimePages.

What we have is a kind of chain of twin primes. We set some rules to calculate an integer sequence and check if twin primes occur next to those integers. We can do some basic analysis for these particular chains.

Although the OP states the multipliers are $3n, 6n, 12n, 24n,\dots$, this is not the case and the multipliers used are: $3n, 6\cdot 3n=18n, 12\cdot 18n=216n, 24\cdot 216n=5184n, \dots$

$1,3,18,216,5184,\dots$ is OEIS sequence A132727, though the connection is almost certainly coincidental.

With the exception of the twin prime pair $(3,5)$, all twin prime pairs are of the form $(6m-1,6m+1)$ for some positive integer $m$. Since the first check is $3n$ we can note that $n$ must be even.

We can consider just the last decimal digit of the integers we will check, starting from each of $n=0,2,4,6,8$:

0: 0, 0, 0, 0, 0, ...
2: 6
4: 2, 2, 4
6: 8, 8, 6
8: 4

The sequences end when the last decimal digit is either $4$ or $6$. When we reach those values one of the $\pm 1$ integers that we check will end in $5$ so it will not be prime and the chain will end. So to find a longer chain $n$ must be a multiple of $10$.

We can use this to speed up the search a little to find the first sequences of $5$ and $6$ terms, which are:

2009759780 [6029279340, 36175676040, 434108112480, 10418594699520, 500092545576960]
3962369240 [11887107720, 71322646320, 855871755840, 20540922140160, 985964262727680, 94652569221857280]
nickgard
  • 4,116