The most efficient documentary evidence of Shankaracharya’s period is the carefully preserved date-wise list of all the succeeding Shankaracharyas who sat on that religious throne which was established by Adi (the original) Shankaracharya thousands of years ago; and that list goes back up to 477 BC. Adi Shankaracharya lived only 32 years so his birth date is 477 + 32 = 509 BC. He had established four maths. (Math is a religious throne, which is used as a center for propagating dharm, and whoever sits on that throne, holds the title of Shankaracharya.) In his last days, Adi Shankaracharya lived in Kanchi Kamkoti so it is also considered as a math. Dwarika Sharda Math and Kanchi Kamkoti Math, both have the complete date-wise record of all the succeeding Shankaracharyas for the last 2,500 years, but the records of Kanchi Math are more detailed.
Shankaracharya established four peeth (math) and appointed four sanyasi disciples at those peeth as acharyas.
Jyotishpeeth at Badrikashram(also called Jyotirmath)
Totakacharya
Sharda Peeth at Dwarika(also called Dwarika Sharda Math)
Sureshwaracharya
Sringeri Peeth, South India(also called Sringeri Math)
Hastamalakacharya
Govardhan Peeth at Puri(also called Govardhan Math)
Padmpadacharya
Shankaracharya, after establishing the four maths and spreading the greatness of Sanatan Dharm, came back to South India and, for the last four to six years of his life, he lived in Kanchi Kamkoti. Thus, Kanchi Kamkoti is also called the fifth math.
Thus, according to the records of Kanchi Kamkoti Math, Adi Shankaracharya was born on 2593 Kali era and left this earth planet on 2625 Kali era which comes to (3102 - 2593) 509 BC and (3102 - 2625) 477 BC. The same dates are mentioned in the records of Dwarika Sharda Math except that they are written in Yudhishthir era.
