Prophet Muhammad ﷺ predicted Persians will not only accept Islam but will become great scholars of Islam and worldly sciences.
The Prophet ﷺ sent letter of invitation to Islam for all regional rulers, in particular the Kings of Byzantine and Persian empires, the 2 greatest empires of the time. He invited Khosrow II Pervez  , King of Persia by dispatching a letter. The following are the contents of the letter mentioned in one of his biography called Al Raheeq Al Makhtum (The Sealed Nectar):
“In the Name of Allâh, the Most Beneficent, the Most Merciful.
From Muhammad, the Messenger of Allah to Chosroes, king of Persia. Peace be upon him who follows true guidance, believes in Allah and His Messenger and testifies that there is no god but Allah Alone with no associate, and that Muhammad is His slave and Messenger. I invite you to accept the religion of Allah. I am the Messenger of Allah sent to all people in order that I may infuse fear of Allah in every living person, and that the charge may be proved against those who reject the Truth. Accept Islam as your religion so that you may live in security, otherwise, you will be responsible for all the sins of the Magians.”
The biography further states that “The proud monarch was enraged by the style of the letter as the name of the Prophet (Peace be upon him) had been put above his own name. He tore the letter into shreds and forthwith dictated a command to his viceroy in Yemen to send a couple of troopers to arrest the Prophet and bring him to his presence. The governor, Bazan by name, immediately sent two men to Madinah for the purpose. As soon as the men reached Madinah, the Prophet (Peace be upon him) was informed by a divine revelation that Pervez, the emperor of Persia, had been murdered by his son. The Prophet (Peace be upon him) disclosed to them the news and they were stunned. He added asking them to tell their new monarch that Islam would prevail everywhere and outstrip the sovereignty of Chosroes himself. They hurried back to Bazan and communicated to him what they heard. Meanwhile, Sherweh, the new monarch sent a letter to Bazan confirming the news and bidding him to stop any procedures as regards the Prophet till further notice. Bazan, together with the Persians in Yemen, went into the folds of Islam, and gladly signified his adhesion to the Prophet.” This event of Khosrow Pervez being murdered by his son and overthrowing his father’s rule is also recorded in non-Islamic historical sources. At the time when Prophet ﷺ made this prediction of Islam spreading in the lands of Al Faris (Persia), the Muslims had not even taken control of Arabian Peninsula, their rule was only limited to two cities, Makkah and Madinah. Within the twelve years after Prophet's passing away Persia was conquered by his companions under the rule of his second successor and Caliph Umar ibn Al Khattab (r.a). Hence came true the Prophet's prediction of Muslim conquest of Persia.
Additionally the Prophet ﷺ also foretold an extraordinary prediction about the knowledge of Persians. One of the foremost companions of Prophet Muhammad ﷺ named Salman Al-Farsi (r.a) was a Persian. In the early years of Islam, the Prophet ﷺ had predicted something very amazing looking at Salman and through him about the Persian nation when the following Quranic verse was revealed: “He it is Who sent among the unlettered ones (i.e. Arabs) a Messenger from among themselves, reciting to them His Ayat (Verses), purifying them, and teaching them the Book and the Hikmah (Wisdom). And verily, they had been before in manifest error. And (he is also sent to) others (i.e. non-Arabs) who have not yet joined them. And He is the Almighty, the All-Wise.” Referring to "others who have not yet joined" the companions inquired: “Who are they, O Allah's Messenger?” At that time, the Prophet ﷺ put his hand on Salman Al Farsi, saying, “If Faith were at (the place of) Ath-Thuraiya (pleiades, the highest star), even then (some men or man from these people (i.e. Salman's folk) would attain it.” It is a historical fact that great scientists, Islamic scholars, poets, geographers, mathematicians and saints in the medieval period, majority of them, who made contributions in various fields of science and religion were Persian Muslims. One can just appreciate this fact by clicking here and viewing list of Persian scientists and scholars throughout the medieval times. Islamic philosophers like Al Kindi, Ibn Sina (Avicenna), Al Razi (Rhazes), saints like Bayazid Bistami, Abdul Qadir Gilani, scientist & mathematicians like Al Khwarizmi (Al Gorithm), Al Tusi, Islamic scholars like Al Ghazali, Ibn Taymiyyah, were all Persians who revolutionized various fields of science and religion. It is the impact of their efforts and discoveries in the medieval age that gave birth to the modern scientific and technological era. This amazing prediction of the Prophet ﷺ came true after Persians abandoned Zoroastrianism and accepted Islam.
 Chosroes or Khusrow (also Kisra) was a title for Kings of Persia during Sassanian era.
 Last great king of the Sasanian Empire who ruled from 590 to 628 CE before Muslim conquest of Persia.
 al-Mubarakpuri, Safi-ur-Rahman. 1996. Al-Raheeq Al-Makhtum (The Sealed Nectar). Dar-us-Salam Publications - al-Mubarakpuri 1996
 al-Mubarakpuri 1996
 Al Quran 62:2–3
 Sahih Al Bukhari Vol. 6, Book 60, Hadith 420 – Sunnah.com