Does 'some' necessarily imply 'not all'?

This is bordering on logic rather than language, but the answer is definitely no: Some is "an indeterminate amount", which means it can be all. If I say I have some red M&Ms in my bag, it could be that all of them are red.

But then, depending on inflection, as @codelegant pointed out, I could be using emphasis on some to indicate that not all are red...like if you asked for some red ones and I said I had some, which turned out to be more than one but less than all.

Yes. If I offered you some peanut M & M's, I would feel that you misunderstood me if you took them all. :)

For straight English prose, yes. It implies more than one, but not all.

However, for a discrete math homework or test question, I think it would be synonymous with "one or more". It could be just one, or it could be all of them. So if you are asking this question to try to get a couple of points back from your math instructor, sorry. :-)

Some suggests (but does not require) that there are counterexamples. The reason is that if you say "Some X's are Y's" instead of just saying "X's are Y's", you presumably had cause to use the extra word. The natural reason for you to do so is if saying "X's are Y's" was actually not true--some are not Y's, but some are. So "Some X's are Y's" is then true.

In conventional discourse, if you want to emphasize that you are using "some" because you are not sure (or do not wish to check) that the statement applies to all, then you can use a phrase like "at least some X's are Y's". This longer form no longer carries the implication that some X's are not Y's. (In logic and mathematics, the short form carries no implication; allowing oneself to not check every case is very useful in math and logic. Then again, the phrasing then is usually "there exists an X that is a Y".)