As I think much about primes and usually that ends up with just a formulation of some new/old conjectures I came to the following idea.

First, an example. Take a prime $1277$. Now increment every digit of that prime by $2$ to obtain $3499$, which is also a prime. We can do this even if some digit is $8$ or $9$, for example a number $4899$ would be by this procedure of incrementation of digits by $2$ be incremented to $6101111$.

If we have some prime $p$ that after this procedure of incrementation of digits by $2$ gives another prime $q$ then we could call $p$ a 2-incrementable prime.

It is evident that this can be further generalized to arrive at the concept of 2m-incrementable primes for every natural number $2m$.

But, are there an infinite or finite number of 2-incrementable primes?

Edit 1: Trivial note: This procedure can be specialized to some number of digits or to only one digit (another generaliization). If we do the incremention of only the last digit then we are in the domain of twin primes, that is, twin primes are 2-incrementable primes with respect to only the last digit.


closed as off-topic by Andy Putman, Charles, Steven Landsburg, David Handelman, Stefan Kohl Dec 30 '17 at 9:47

This question appears to be off-topic. The users who voted to close gave this specific reason:

  • "This question does not appear to be about research level mathematics within the scope defined in the help center." – Andy Putman, Charles, Steven Landsburg
If this question can be reworded to fit the rules in the help center, please edit the question.

  • $\begingroup$ Very nice, I decide to share an idea with you and you just downvote it. $\endgroup$ – user114642 Dec 29 '17 at 22:32
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    $\begingroup$ I have not downvoted, not have I voted to close, but let me repeat a comment I left on your meta.MO answer/post: "Sometimes users vote to close a question as "not research level" whereas what is really meant, I think, is something like the following: in mathematics, especially in number theory, it is very easy to come up with a myriad of questions whose statements are elementary, whose truth or falsehood seems very hard to determine, BUT whose solutions do not seem to offer us any insight into the wider contexts/structures that research mathematicians are really interested in" $\endgroup$ – Yemon Choi Dec 29 '17 at 22:46
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    $\begingroup$ @AntoinePalAdeen To find unexpected connections you need to come with some mathematical background, results, ideas, problems, which is not what you have here. Did you hear about the random model for the primes ? $\endgroup$ – reuns Dec 29 '17 at 23:10
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    $\begingroup$ It's at oeis.org/A068990 (with a contribution from @Robert from a few months back). $\endgroup$ – Gerry Myerson Dec 30 '17 at 3:08
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    $\begingroup$ Since this question does not seem to fall into the "useful research-level question" but is a well-written good question about math, how about moving this question to Math.SE instead of starting a downvote/upvote fight? $\endgroup$ – Pedro A Dec 30 '17 at 3:38

Here is the graph of the number of 2-incrementable primes among first $n$ primes, as a function of $n$.

enter image description here

And here, on request by OP, number of 2-incrementable primes up to $n$ divided by the total number of primes up to $n$, as a function of $n$.

enter image description here

Disclaimer: I did not downvote it, and I do not think this is a meaningful question.

  • $\begingroup$ I did upvote it. Work must be rewarded. Can you add to the answer what fraction of primes is 2-incrementable. From the graph it looks like 1/7, on the average. $\endgroup$ – user114642 Dec 29 '17 at 22:36
  • $\begingroup$ Here you are. Looks like going to zero $\endgroup$ – მამუკა ჯიბლაძე Dec 29 '17 at 22:42
  • $\begingroup$ Thank you. It looks like it has "jumps" quite often..If only I could zoom your graph. $\endgroup$ – user114642 Dec 29 '17 at 22:48
  • $\begingroup$ It is easy to prove that the ratio of these primes to the total number of primes goes to 0. $\endgroup$ – Charles Dec 29 '17 at 22:51
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    $\begingroup$ @AntoinePalAdeen Drmota, Mauduit, and Rivat have methods that seem relevant. The problem is hard. $\endgroup$ – Charles Dec 29 '17 at 23:09