I prayed with Abu Bakr, ‘Umar, and ‘Uthmān and I did not hear any one of them recite "In the name of Allah, the Most Compassionate, the Most Merciful"

Anas ibn Mālik (may Allah be pleased with him) mentioned that in spite of his long companionship with the Prophet (may Allah's peace and blessings be upon him) and staying with him and his Rightly-Guided Successors, he never heard any of them recite "In the name of Allah, the Most Compassionate, the Most Merciful" in prayer, neither at the start, nor at the end. Rather, they used to start the prayer with {[All] praise is [due] to Allah, the Lord of the worlds} [Sūrat al-Fātihah: 1].

The scholars held different opinions on the ruling of reciting the Basmalah (In the name of Allah, the Most Compassionate, the Most Merciful) out loud, and the sound opinion is that the praying person recites the Basmalah secretly before reciting Sūrat al-Fātihah in every Rak‘ah of the prayer, whether it is a prayer with silent recitation or one that is recited out loud.