To solve this problem :

1)

Download PARI/GP

2)

Calculate e as accurate as you want, even 1 million digits do not take very long. You just have to establish enough memory with allocatemem()

3)
Convert each block of length 21 that does not begin with a 0

with the command isprime(,2) which is a rigorous primality test. The best way is probably to create an array with the digits-command.

4)

The first such block is the answer.

I did not post the full code, but I mentioned the necessary commands. I want to leave a little work to the OP.
source: https://math.stackexchange.com/questions/3513323/what-is-the-first-21-digit-prime-in-consecutive-digits-of-e



Have fun and please consider PM'ing me for the tip payment.