Prepare the jam jars by washing and sterilizing both the jars and lids. Sterilizing can be done by submerging the jars and lids in a gentle boiling water bath for 5 minutes. (Don't have the water boiling vigorously because it could cause you jars to bang against each other and chip).
To prepare the papaya, cut in half, then scrape out and discard the seeds with a spoon. Remove the pulp from the skin with a spoon and discard the skins. Weigh the fruit and put it in a large deep cooking pot.
Add the sugar to the pot. Wash the 3 limes well, cut in half, and hand juice the limes directly into the pot. Add squeezed limes (rinds) to the pot.
Cook on medium-low heat for about 20 minutes, stirring frequently.
Mash the fruit pulp in the pot until you get the desired texture. Alternately, you can remove the fruit pulp and blend in a blender until smooth (do not put limes in the blender).
Increase the heat, stirring constantly, and bring to a rolling boil (it should continue to boil as you stir). At this point, you should see foam forming at the top. Cook the jam for approximately 5 minutes. If you are using the temperature method to check if your jam has reached the "set" stage, use a candy thermometer; the temperature should reach 220°F (this is the jam setting temp). Cook longer if needed.**See notes for the cold plate method to see if the jam is set (I prefer this method because I find it more accurate).
Once the jam is at its "set" point, turn off the heat and remove the limes.
Ladle finished jam into prepared jars. Wipe the rims of the jar with a clean, damp cloth and screw lids on tightly. Let the jars rest, undisturbed for 12 to 24 hours. After the jars have rested, check that the jars have sealed by pressing down on the center of the lid. If there is no give, the jar is sealed. Store in a cool, dry place for up to one year. Once opened, keep in the refrigerator for up to one month.