Place to start

[Gtown78]

I have a .680 choke for my Remington 870. I'm looking for a good shot size to start patterning. I have read that bigger shot often patterns better in a open choke. I'm not looking for super tight patterns just a good killing pattern.

[Farmboy27]

longbeard 4s or 5s. 

[Philippe]

Yepppp as stated above, the Longbeard 4 and 5's will be just fine for that purpose.