How on Earth does ^.?$|^(..+?)\1+$ produce primes?

This strange string is like a super-smart detective who finds prime numbers, those special numbers that only have two friends: 1 and themselves.

Imagine you're playing with blocks. A prime number is like a block that can't be broken into smaller blocks unless you use just 1 or itself. For example, 5 is a prime because you can’t split it evenly into smaller whole-number blocks (except for 1 and 5).

Now think of the string ^.(.+?)\1+$ as a clever detective who checks if a number has only two friends. It starts with ^ (like saying "beginning of the story") and ends with $ (like "end of the story"). The .` is like looking at one block, and (.+?) is like trying to find a way to split that block into smaller ones, but it’s lazy and picks the smallest possible split first. Then \1+ means checking if that same split can be used again and again.

If the detective finds that you can split the number evenly more than once using the same small piece, then it's not a prime. But if no such split works, then that number is prime, like 7 or 11!

It’s like having a special block-checking rule that only lets primes pass through, and this detective rule does it all with just letters and symbols!