18,446,744,073,709,551,616 is 2^64. I assume that this means the planet generation algorithm is based on a random seed that is a 64-bit number (e.g. the long type in many programming language). I don't know how (or even if; how would anyone check?) they guarantee that all possible inputs are used and that none are repeated.
EDIT: While it's still true that this number is 2^64 (and it still appears in some statements in the lore, so could be said to be the number of planets that exist in-universe), there are not this many planets in the game. Since the Atlas Rises update, there are 256 (257? It's not entirely clear) galaxies, and each planet has a 12-digit hex coordinate that identifies it within the galaxy. This coordinate is divided into parts:
- 3 digits ea. X and Z coordinates of region (a region is a cube-shaped volume of space)
- 2 digits Y coordinate of region
- 3 digits system within region (known regions have 533 to 553 systems, so not all values are used)
- 1 digit planet within system (0 is not used, largest known system is 6 planets and 2 moons)
So, putting that all together, there are about 600 trillion systems. If eight bodies per system is typical (in practice this is a high estimate), that's about 5 quadrillion, still far from the 18 quintillion being discussed here.
In practice it doesn't matter, because if a new system were discovered every second, it would take 74,000 years to reach all of the systems in one galaxy.
The Atlas Stone are used during the Atlas Path of the game.
Atlas Stone (spoilers, proceed with caution)
After getting 10 Atlas stones and going to the next Atlas Interface you can make a new star, this will take all of your Atlas stones and say that you have created a new star. Current importance of this matter is unknown and it is unknown about what the effects of these actions are.
Additional Information
Whatever you do, DO NOT SELL. A buddy of mine made that mistake early on in the game, and sold two Atlas stones. When he made it to the final Atlas location, he was unable to proceed due to only have 8/10. Afterwards he was not able to locate any more Atlas locations to gather more stones. This may be a bug in the game, but no more information at this time
Best Answer
These are trade routes. They connect the local space station with on-planet Trading Posts, giving you a reliable means of finding them, so you can sell your cargo (or buy commodities) without having to return all the way to the space station or hope to find one on-planet by chance.