Is it permissible for a traveler to shorten and combine prayers for three weeks in a country where he intends to reside for tourism?

You should not have shortened the prayer if you intended to stay for more than four days. Shortening the prayer is for the traveler, or for one who intends to stay for less than four days, or for one who does not know when he will depart. If you shortened the prayer based on a fatwa from a trustworthy scholar, then there is no harm upon you and no need for redoing the prayers, because some scholars permit shortening for the traveler as long as he does not intend a permanent stay. However, if you shortened the prayer without following any of the people of knowledge, then it is obligatory to redo those prayers.