can a Car Registration Number, a combination of prime, be prime?

This is true for the prime numbers 3, 7, 109 and 673 that if you concatenate any two of these numbers in any order , the resulting number will be a prime ; as in this case Concatenating 7 at the end of 673 resulting in 6737 which is a prime . Concatenating 7 in the front of 673, which gives 7673 , is also a prime . So is 1093, 1097, 3109, 7109.

Like pointed out in the comments by @naveen dankal, primes and patterns are not "best friends"*. Or at least we do not know of many patterns concerning primes. In general, finding out if a given number is prime or not can be a very difficult question to answer. If a given number is the concatenation of other primes or not, is just as irrelevant - as far as I know.

Nonetheless, I asked Mathematica to find some pairs of primes that, when concatenated, yield another prime number.

I started with just the prime numbers with 2 digits. There are 20 of them:

$$11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, \ 79, 83, 89, 97$$

And there are 73 concatenations that are prime, which is $73/400 = 18.25\%$ of the concatenations. The ones found are listed here:

$$[1117, 11, 17], [1123, 11, 23], [1129, 11, 29], [1153, 11, 53], [1171, 11, 71], [1319, 13, 19], [1361, 13, 61], [1367, 13, 67], [1373, 13, 73], [1723, 17, 23], [1741, 17, 41], [1747, 17, 47], [1753, 17, 53], [1759, 17, 59], [1783, 17, 83], [1789, 17, 89], [1913, 19, 13], [1931, 19, 31], [1973, 19, 73], [1979, 19, 79], [2311, 23, 11], [2341, 23, 41], [2347, 23, 47], [2371, 23, 71], [2383, 23, 83], [2389, 23, 89], [2917, 29, 17], [2953, 29, 53], [2971, 29, 71], [3119, 31, 19], [3137, 31, 37], [3167, 31, 67], [3719, 37, 19], [3761, 37, 61], [3767, 37, 67], [3779, 37, 79], [4111, 41, 11], [4129, 41, 29], [4153, 41, 53], [4159, 41, 59], [4337, 43, 37], [4373, 43, 73], [4723, 47, 23], [4729, 47, 29], [4759, 47, 59], [4783, 47, 83], [4789, 47, 89], [5323, 53, 23], [5347, 53, 47], [5923, 59, 23], [5953, 59, 53], [6113, 61, 13], [6131, 61, 31], [6143, 61, 43], [6173, 61, 73], [6719, 67, 19], [6737, 67, 37], [6761, 67, 61], [6779, 67, 79], [7129, 71, 29], [7159, 71, 59], [7331, 73, 31], [7919, 79, 19], [7937, 79, 37], [8311, 83, 11], [8317, 83, 17], [8329, 83, 29], [8353, 83, 53], [8389, 83, 89], [8923, 89, 23], [8929, 89, 29], [8941, 89, 41], [8971, 89, 71]$$

On the other hand, there are $1061$ primes with 4 digits, so the percentage of 4-digit prime numbers which are a concatenation of two 2-digit primes is $73/1061$ which is less than $7\%$. Funnily enough, there are $237$ unique 4-digit primes that are the concatenation of a 1-digit prime i.e. $2,3,5,7$ and a 3-digit prime. This makes up for a total of more than $22\%$ of 4-digit prime numbers being a concatenation of one prime digit with a 3-digit prime. In total, we conclude that there are $310$ concatenations that yield 4-digit prime numbers, for a total of around $29.2\%$. This means that one out of every 3 cars with prime numbers in the license plate has a number made of concatenating 2 primes! (if the license plate is like the one in the picture)

After that, I concatenated, in order, the first $1000$ prime numbers with themselves. The way I did it was simple: let us say $h_1$ and $h_2$ are the two prime numbers that are going to be concatenated. We will want to check if $h_1h_2$ is prime. I fixed $h_1$ and made $h_2$ iterate over the $1000$ said prime numbers, to check if $h_1h_2$ was prime. Then changed $h_1$ to the next prime and repeated. And so on and so forth until I checked all $1000^2$ concatenations of the first $1000$ prime numbers. (By the way, the $1000^{th}$ prime number is $104729$)

Along the way I found $107850$ concatenations that turned out to also be prime numbers, which is roughly $10.8\%$ of all concatenations tested. To me this turns out to be a seemingly high percentage! I sure was not expecting it. (Please notice this does not mean I found exactly $107850$ unique primes. I did not test for repetitions nor removed them, so the actual prime count should be a little lower than that).

(If requested, I can extend the bounds of the test to more than $1000$ prime numbers)

Some of the concatenations I found are listed here (I use $a : b$ to mean "$a$ concatenated with $b$")

$2 : 3 = 23$ (obvious one, I guess)

$2 : 29 = 229$

...

$7919 : 7109 = 79197109$

$7919 : 7907 = 79197907$

Funnily enough, I found no concatenation that started with a prime number after $7919$ (up until my bound which was the prime $104729$). Maybe this means that there are less concatenated primes that the result of concatenating two big prime numbers? Who knows...