My approach to the first 100 Euler eventually involved a bitmap of possible primes (only odd numbers, there's only one even prime) which was initially seeded by a 'magic number' set representing the lowest primes and would grow up to a target number of cached small / really fast primes.

Then the algorithms I understand less kick in. Most of those involve some form of modo math to wrap the number space based on one or more origami like folds and use well known and test algorithms which were previously exhaustively proven to cover the entire 32 bit number-space when utilized in a composite fashion.

I've always wondered if Stirling(n) can be used to arrive quickly in the vicinity of n!, and then use a search of some kind to get to the exact target.

That one is an approximation rather than returning all millions of exact big integer digits though (the approximation is more useful for real life statistics etc..., but doesn't look like what this article is targeting)