Gems are a secondary resource generated as you play and cashed in when abdicating. Once cashed-in, these gems increase your production and Faction coin find chance. At each abdication the gems are carried over and added to the previous ones, making your production stronger and letting you progress further.
Gems are also the resource necessary to reincarnate. Each reincarnation will cost a defined minimum and increasing amount of gems. In this process, all your gems will get reset to zero in exchange for powerful reincarnation perks.
The first gem is obtained at 1 T (1E12) coins, then 2 T (2E12) more, then 3 T (3E12) more, etc. The formula for the total number of coins you need to have produced is n*(n+1)*5E11 (500B), where n is the number of gems you want.
|1 T (1E12)||1|
|5.005 Qi (5.005E18)||1 K|
|500 Dc (5E35)||1 T (1E12)|
To double your gems you will need to gain 4x the "Coins Gained (Max)" number visible in the Stats menu.
Gems also increase your Faction Coin find chance. The rate of increase to the Faction Coin find chance steeply decreases the more gems you accrue, with every additional "100*Current Gems" adding only a percent or two extra chance as the gems reaches more than 1 Qi (1E18).
The Gem Power upgrade is rewarded when you get your first gem. This upgrade makes each gem give you a production boost and boosts your faction coin chance. Further upgrades improve this effect. If you don't buy the first Gem Power upgrade your gems won't do anything. The difference it brings can be clearly felt when aiming for the few secret trophies requiring to play without using it.
|Gem Power||1 Coin||1 Gem||Any||Unlocks Gem Power, each of which will Increase your overall production by 2% and slightly improve your chance of obtaining faction coins.|
|Gem Potency||10 Sp
|1 M (1E6) Gems||Any||Gem production bonus is Increased by an additional 1%.|
|Gem Supremacy||10 Dvg
|1 Dd (1E39) Gems||Any||Gem production bonus is Increased by an additional 1%.|
|Gem Sovereignty||10 Qitg
|1 Vg (1E63) Gems||Any||Gem production bonus is Increased by an additional 1%|
The actual gem cap the game supports is 896e148 gems (and 1.797e308 gold coins). Going over this amount will cause the game to bug. The Ascension soft reset, which can be reached at R39 and 1 Spqag (1E144) gems, was implemented to overcome this issue, and shifts the numbers to a lower range. (Source)