What is required of someone who prayed for several months wiping over his socks for longer than the Islamically prescribed period, out of ignorance, knowing that he does not know the number of prayers he performed in this manner?

Wiping over the socks is permissible under certain conditions. The duration for wiping is one day and one night for a resident, and three days and nights for a traveler. This period begins from the first nullification of ablution (hadath) after wearing them. If the wiping period ends while you are still in a state of purity (tahara), your prayers are valid. However, if you nullify your ablution after the period has ended, the prayers performed are invalid and must be re-offered. If you cannot ascertain their number, then pray until you are reasonably certain that you have fulfilled your obligation. Sheikh al-Islam Ibn Taymiyyah, however, holds the view that you do not need to re-offer them.