Yesterday, I was playing around with prime numbers just for amusement and thus became aware that sometimes 'attaching' two prime numbers will result in another prime number. Let me explain my idea by an example. The numbers $3$ and $7$ are prime numbers, and when we write them in succesive order, we obtain the prime number $37$. Similarly, $7$ and $19$ are prime numbers, as well as $719$. There can be found many more such examples.
I am wondering if there is some kind of pattern or rule, by which one can see whether attaching two given prime numbers will result in a prime number or not? Does anyone know something in this direction?
Best wishes
To reiterate what was already said in the comments: what you're calling "attaching" is usually called "concatenation".
The results of a concatenation depend on the base being used. I suspect that concatenation of primes in binary comes closest to having a satisfying "general rule." In any other base, what you get from concatenating primes is likely to feel like an accident of the base.
In decimal, for example, consider the prime quadruplet 11, 13, 17, 19. Of course 1 was never a prime number, though we didn't realize that until very recently. And 9 is obviously not prime. So there you have four primes in a row that are not the concatenation of smaller primes.
And then we get 23. Nice, huh? The next one is 37, which of course you already found. The next one after that is 53... do you care if the concatenated primes are in ascending order? If you require that, then you're not going to like 73 or 113.
I'm regarding 113 as the concatenation of 11 and 3, but it's also possible to regard it as the concatenation of 11 and 13. I disagree with that. My point is that a lot of situations arise in which subtle differences of opinion lead to different interpretations of concatenations.
Or maybe not so much in this particular instance? I see 137 as the concatenation of 13 and 7, but you might see it as 13 and 37.
This might be enough for a decent OEIS search. I get six results, the first of which is http://oeis.org/A105184 But neither in that result nor the other five results do I find what I would call a "general rule". Do you?