Is it permissible to write Qur’anic verses, such as Surat Al-Baqarah, on paper for memorization, and to transfer them from a mobile application, without ablution?

It is permissible to write the Qur'an for someone who is not in a state of purity, as long as they do not touch what is written. The فقهاء (jurists) have explicitly stated this, such as Al-Bahuti Al-Hanbali and Al-Abbadi Al-Hanafi. Some scholars, like the Malikis, have permitted a teacher or a student to touch the Qur'an for one who is in a state of minor impurity (حدث أصغر).