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There are different answers depending on the end goal.
Mere survival: Isolated human populations have been bottlenecked to as few as a few hundred individuals and survived, IIRC.
A quick search says biologists like to see 25+ breeding pairs to maintain an animal species (if I'm reading that correctly). So 50-100 seems like pretty close to the minimum.
Long-term colony building with full genetic diversity needs a lot more: At least one estimate is as high as 40,000 people. The high number is for Earth-like diversity in the population, and with no need for any overarching breeding program, so it's really kind of an outlier scenario. That 40k figure can be pared down significantly if you have strict protocols, or accept some loss of diversity.
So anywhere from 50 people to 40,000 people, but the end result will look wildly different at the extremes.
I like your answer because I feel the only real answer you can have is “more is better, less is worse”
How many should you take for good diversity? All 8 billion, more if you can find them.
It seems very plausible that if you can make a generation ship, you could make something like CRISPR to artificially increase genetic diversity/eliminate potential birth defects. Or perhaps just store genetic material for more people than the actual colonists on the ship. You'd probably want to hedge your bets in case there's a low survival rate. There's a lot that could go wrong. Ecosystem failure, negative effects of radiation, or just good old fashioned murder.
just bringing a ton of frozen sperm and eggs is honestly the real answer i feel, why worry about bringing a bunch of living people who need resources and stuff, when you can just bring a quarter of earth's diversity with you in the freezers?
The two downsides to this is that people can't really have kids with each other, and if the freezers malfunction you're a bit in the shit.
On a scale from Hapsburg to Earth, how do you want your new colony to go?
We can have a little bit of hapsburg, as a treat.
I've seen the jaw, I don't think you can have a little hapsburg.
Thanks, ChatGPT.
LOL. I mainly got the idea of using genetic modification to increase diversity of a small population from the book Seveneves.