Is the foundational principle regarding the carpet in an apartment in Germany that it is pure, or must it be purified before praying on it, considering that its previous occupants were non-Muslim students and there is no visible impurity on it, nor can it be removed?

It is permissible to pray and sit on carpets, because the default state of things is purity. This is the principle of "the original purity of substances," as Shaykh al-Sa'di said in his poetic composition: "And the default for our waters is purity... And for the earth, clothes, and stones." If this default principle contradicts an apparent observation, such as the possibility of impurity in the clothes of disbelievers that touch their private parts, some scholars have elaborated on the matter. They rule that what touches the private parts is deemed impure due to the strong presumption, while what does not touch them remains pure. In summary, if you are certain of impurity, it is impure; if you are certain of purity, it is pure. Doubt is not considered. Since you do not see any trace of impurity on the carpet, there is no harm in praying on it. If you wish to use a special prayer mat, that is good.