We can call a prime intertwined if it consists of two smaller primes having their digits alternating in it, for example:

$$1433 = 13 \oplus 43\\ 1873 = 17 \oplus 83$$

If we let $\oplus$ mean this alternating operation.

Is there likely to exist a largest intertwined prime?

**Own work**

I have considered the prime distribution from the prime number theorem

$$\pi(N) = \frac{N}{\log(N)}$$

Maybe it would be possible to consider probabilities two prime numbers of $k$ digits each forming a prime number of $2k$ digits taking into account approximately how many prime numbers there are on each interval $[10^k,10^{k+1}]$.