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There's two real long-term consequences of an Islamic conquest of Persia:
- Islamic culture. I think that, if they took and held Persia for centuries, the Arab/Islamic conquerors would manage to succeed at establishing Islam as a major if not dominant faith, at least among the elite. However, they could not help but be strongly influenced by Persian culture. As it was OTL, Islamic culture has been VERY strongly influenced by Byzantine culture - much Islamic art was derived from Greek, Roman, and/or or Jewish styles,
- India. Islam largely limited its expansion - at least conquest-wise - to the Levant, parts of Asia Minor, and Africa, with a few forays into Southern Europe, mostly in Iberia and Sicily. In OTL, a small but significant Islamic population exists in the Indian coastal cities due to the prominence of the Indian Ocean trade route, or from travellers heading east through Persia, and from the occasional missionaries. If Persia were conquered by the early muslims, its possible that they would go further and cross the Indus, into northern/western India, making the religious situation further convoluted (as if the Hindus, Buddhists, Jains, and the occasional converts from Christian or Muslim missionaries weren't complicated enough...). This ight also speed the spread of ideas between India and the West - in OTL it took many centuries before Indian numerals were adopted as widespread, first spreading to Persia, then into Asia Minor thanks to the semi-Farsified Kuiunlu Turks, then from there into Europe during the late Renaissance.... an Arabic conquest in India might spread the ideas much sooner...
(OOC: this TL probably uses a different number system than OTL, "Indian" numerals having evolved somewhat via Arabic... the original Indian numerals are, courtesy of Wikipedia:
० १ २ ३ ४ ५ ६ ७ ८ ९).