Extending primes forever I found a big number 357686312646216567629137 with the characteristic that if we cut it from the left, we always get a prime number, meaning that 7 is prime, 37 is prime, 137 is prime and so on..
Thinking about this i got up with the question, if it is possible to continue this and if it possible to find always a bigger number with this characteristic.
I found this question making it reasonable that it is kind of unlikely that we can continue with this process forever, but that means not that it is not possible. Obviously we could brute force all 9 possibilities for the next number on the far left, but at numbers of this size I personally have no tool to check if the resulting number is prime or not.
So I would appreciate if anyone have any knowledge on this. I also have no idea what I should google to find any results on this question.. Maybe this problem have a name or something
 A: Here is a summary of the above comments. The numbers you are describing are called left-truncatable primes, and it is easy to see that the set of such primes is finite - since it has finitely many endpoints, see the video, or this reference:
Proof for finite number of truncatable primes
There is an OEIS page for it here. The final list has $4260$ primes and starts with
$$
2, 3, 5, 7, 13, 17, 23, 37, 43, 47, 53, 67, 73, 83, 97, 113, 137, 167,\cdots , 357686312646216567629137.
$$
A: The probability of hitting a prime number reduces to approximately ln N when N is large. If we already have a prime of m digits then adding a (base 10) digit to the left between 1 and 9 will achieve a multiple between 2 and 91 of your m digit number. Now using ‘the law of averages’ there will be (ln 91 - ln 2)= approximately average of 3.8177 prime numbers in that whole zone of possible adding an extra digit.
The probability, however that simply adding one digit (1-9) to the left will hit one of those prime numbers in amongst the more than 10m numbers in that zone will diminish and the probability of hitting a prime number every time will diminish to zero. If one allowed the possibility of adding more than one digit to the left, then the sequence will progress longer, possibly indefinitely. However if the number of added digits was limited to say 2, then, again, I expect the sequence to terminate.
