(I know these aren't authentic and I don't claim them to be. This is just for fun.)
Sticks of gum used to come in 17-packs, as seen in the picture. (Why 17? Why not 18?) But recently, a switch has been made to 5x3, 15 packs. (For the same price.) Unfortunately, that makes this Instructable is a little late, but it's still fun. I've included the dimensions of the paper used, so you can make your own, if you don't have the old-style gum packs lying around.
Step 1: Folding your first link
If you don't have gum wrappers like this, the dimensions are 53mm x 58mm.
Tear the paper in half the long way. Fold it in half to find the middle, then unfold it. (Pictures 3 and 4) Fold both sides in the long way, then fold from the middle again. The paper will be 1/4 size. (Pictures 5 and 6)
Fold this in half the short way, then fold the ends in. (Pictures 7 and 8)
Step 2: Linking the links
Basically, the arms from one link will be stuck through the other. Here's a hint: Remember how the link is folded into quarters? One side will have the folded edge, while the other will have the two sides of the fold. When inserting the arms of one through the other, try to do so from the side with the folded edge. It's much easier. (Does that make sense?)
It may help to use the nail file from a pocket knife to pry the arm holes open for easy insertion. Don't use the knife, though!