Was Korea founded in 2333 BC?

Koreans often say that the first Korean kingdom was formed in 2333 BC:

The people of Gojoseon or the oldest kingdom of Korea are recorded as Dongi, "eastern bowmen" or "eastern barbarians." They propagated in Manchuria, the eastern littoral of China, areas north of the Yangtze River, and the Korean Peninsula. The eastern bowmen had a myth in which the legendary founder Dangun was born of a father of heavenly descent and a woman from a bear-totem tribe. He is said to have started to rule in 2333 B.C., and his descendants reigned in Gojoseon, the "Land of Morning Calm," for more than a millennium.
Korea.net: History: Gojoseon

You would wonder why such a remote date is in detail. How to calculate the date?

The date is computed based on the description of Dongguk Tonggam (Annals of the Eastern Kingdom) (1485): Dangun took over the throne in the 25th year of Emperor Yao's reign, the year of wuchen (the 5th year of the 60-year cycle Ganzhi). It is to be noted that another history book Samguk Yusa (1284) points to the different date: the 50th year of Emperor Yao's reign, the year of gengyin (the 27th year of Ganzhi). (*) Yao (Tang Yao) is a legendary Chinese sage emperor, whose reign was said to have extended from 2357 to 2256 BC (over 100 years!) So the 25th year of the reign is 2333 BC.

Koreans weigh in with the legend of Dangun to maintain their ethnic identity, but ironically it depends on the Chinese legend.

Korea.net alters the legend, probably to make it plausible. According to Samguk Yusa, Dangun's mother is not a woman from a bear-totem tribe but a bear transformed into a woman called Ungnyeo (bear woman), and Dangun, not his descendants, ruled Gojeoson for 1,500 years.

The legend of Dangun was first recorded in Samguk Yusa (1284). Before that no literature had mentioned it, so it was considered to be built from the 10th century to the 13th century under the oppression of Khitan and Mongol.

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