Hi!

Today I was surfing Wikipedia and came across this article — Palindromic prime

This article says that the largest known palindromic prime is $$$10^{474500}$$$ + $$$999 * 10^{237249} + 1$$$.

Well it is easy to see that this number is palindrome, but... why is it prime?

I don't find any proof, and I am really curios — how to proof that this number is prime, when number is quite big?

In general, it's quite hard. There is a result from 2002 for which later authors got the Godel prize which proves that primality testing can in fact be done in polynomial time, check this wikipedia link: AKS primality test. There is a lot of advanced research, especially for the uses in cryptography, and the answer to your question is likely not simple (the prime you mention is huge). I am curious if someone has a more optimistic answer...

Primality Tests for Large Primes : https://primes.utm.edu/prove/.