Gee, 'fake' seems so harsh....well, is it time to bite the bullet, guys?
It090, if you haven't done this already, can you take a pair of scissors (or anything medium sharp and sturdy, but manageable for your grip to control) and gently scratch a pearl (near the clasp!) at the hole where it's knotted? See if the pearl layer peels off.
Remember, a sturdy but not too sharp edge. You don't want to accidentally slice the knot, and then need to repair the necklace so it's wearable... Or your fingers.
This information about genuine vs artificial pearls is coming to you in a piecemeal fashion. If you google (generic term, you could use another search engine) this topic, you'll find the info in a more organized fashion. Sorry I can't remember more specific directions - Senior Moment.
Pearls are somewhat contradictory; under a magnifying glass they look very smooth, yet rubbed against the teeth or rubbed against each other they feel gritty. That's why I suggest the google.
My son went with his Sr German HS class on a tour of Germany, Austria, the old Czekoslovakia and brought home a set of artificial (not knowing it) pearls for me. He bought them in Prague I think, but they had the eagle on the clasp-tag. The stories behind family pearls can make them lovely, sentimental treasures.
Will you let us know what comes of the scratch test?