What Size Neutral Wire For 125 Amp Service

The wire size chart below shows allowable ampacities of insulated conductors rated up to.
What size neutral wire for 125 amp service. The run will be about 100 feet and the breaker boxes are 125 amp. I understand i could use a 125 amp breaker but i do not need the additional amperage. I assume i need to run 4 wires 2 hot 1 neutral and 1 ground.
As usual it depends. 200 a services are usually 3 0 copper. I will be installing a 100amp breaker in my main panel.
Distance from the main service panel type of insulation coating the wire and amperage dictate the size of the conductor. I am installing a 125 amp load center in my house to service an addition. Do these sound right i ve looked at some of the online calculators and they all say to run 3 for everything.
I m looking for suggestions on wire size this is what was given to me by a friend of mine. You can use 14 gauge wire for a 15 amp breaker but you must use 12 gauge which is thicker for a 20 amp 10 gauge for a 30 amp and 8 gauge for a 50 amp breaker. Size the electric wire for the circuits according to the size of the circuit breaker.
Other conduits for circuits in the subpanel will be 6 20 amp cbs and two double width 240 volt cbs. It is important to pick the correct size of wire so that the wire doesn t overheat. The insulation type is normally embossed on the insulation along with the conductor size.
Three thhn 3 wires and a 8 egc wire will be run through 1 1 4 inch pvc. If allowed the local co. No neutral current on these two circuits.