I just thought of an interesting problem when discussing prime numbers with a friend.

Some numbers are prime, but even fewer numbers preserve their primality when we reverse their digits.

So for example:

$13$ is prime and $31$ is also prime.

I'll call these 'reversible primes' informally.

I was wondering, is there a way of determining the $n^{th}$ 'reversible prime' or an efficient sieve for them?

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    $\begingroup$ The concept you're looking for is called an Emirp: en.wikipedia.org/wiki/Emirp $\endgroup$ – Jasper Apr 29 '16 at 15:25
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    $\begingroup$ Looking up the references from mathworld.wolfram.com/Emirp.html and of the wikipedia link, together with oeis.org/A006567 - I think there is no known algorithm better than brute-force. $\endgroup$ – Jasper Apr 29 '16 at 15:30
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    $\begingroup$ Study binary emirps (3, 5, 7 are trivially reversible, 11, 13, 23, 29 are a little more interesting). Either you will get an insight or grow bored of the topic. $\endgroup$ – Robert Soupe Apr 29 '16 at 16:44
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    $\begingroup$ If your notion of reversible primes includes the palindomes $11, 101, 131,\ldots$, the appropriate OEIS entry is oeis.org/A007500 $\endgroup$ – Barry Cipra Apr 29 '16 at 17:30
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    $\begingroup$ @fleablood You're right, they're not emirps in base 10. $\endgroup$ – Robert Soupe Apr 30 '16 at 2:40

what i have observed from

The concept you're looking for is called an Emirp: en.wikipedia.org/wiki/Emirp – Jasper 4 mins ago

comment is as follows:

NOTE: this is purely on the basis of the observation of the number given in the given list

let there be such a number which is 'reversible prime'. Let $j$ be the sum of all the digits in that number

from observation $j$ is divisible by 2 or 3 or 5 or is itself a prime. well that could be a test but this could not be the only test. let it be called the test A. actually $j$ was coming again and again 4,5,8,10,13,14,16,17,19 and let it be called list A and other interesting thing to be noted is that there is not any single number in list A which is divisible by 2 and 3 simultaneously i.e. it is not divisible by 6

now from observation again we can see that these numbers are ended and started with 1,3,7,9 and one thing can also be noted that any of the number at the first place is not repeated at the last hence many possibility has been removed(this observation can also be understood as we are checking the 'reversible primes'). let is be called 'B'

now for numbers that are coming in middle of the first number and the last number, they are 0,1,3,4,5,6,7,8,9 but not 2.

If we use Test A and 'B' then we can by trial and error method guess the weather the number $n$ is a 'reversible prime' or not by remembering that the sum of the digit of that number should be one of the number given in the list A and that number should contain 1 or 3 or 7 or 9 at the first place but at the last place it, same digit(which is selected for the first place) should not be repeated.

For example you select a number(which should be a prime), suppose, 149 then the sum of the digit is 14 which is in list A and is divisible by 2 but not by 2 and 3 simultaneously and the number between the first digit and last digit is not 2 and the first digit and last digit is from 1, 3, 7, 9 and are not equal hence there are many-many chances that this number is 'reversible prime'


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