My two month old weighed 10 lbs, 10 oz three weeks and two days ago..., now he weighs 13 lbs, 10 oz.…

if his doctor isn't concerned about it, don't worry about it!

We don't have an appointment for another month... Is it necessary to make one early and get him checked out?

I don't see why my son was about the same weight when he was 2 months old

How many inches is he ? Because my daughter is 12 pounds and 23 inches and my doctor said normally babies don't double their weight until six months but if their long it's fine.

He's about 22.8 inches

And you feed them when they are hungry, they show you signs of hunger

Yes