As for whether Lao Tzu enlightened Shakyamuni, let's first look at it from the perspective of time.

Lao Tzu was born in the Spring and Autumn Period, of course, the Spring and Autumn Period and the Warring States are two different things, some people believe that the dividing line between the Spring and Autumn Period and the Warring States is in 479 BC, this year Confucius died, Zhu Xi once said: Heaven does not give birth to Zhongni, the eternal is like a long night; the heavens take away Zhongni, and the princes are destroyed.
Others believe that the dividing line between the Spring and Autumn Period and the Warring States was in 481 BC; this year coincided with the year of the spring and autumn book.
Lao Tzu was born around 571 BC.
The birth of Shakyamuni is also controversial, southern Buddhism and northern Buddhism have different accounts, but there is no specific documentary record, so we can only calculate according to the historical records, according to the Chinese translation of the "Good Seeing Law Vishnu Sha" out of the Vinaya inferred to be 565 BC, so it seems that the two are indeed people of the same period.
So how did Lao Tzu enlighten Shakyamuni come about?
According to the "Records of History", Lao Tzu went out of the Valley Pass and left the Tao Te Ching, but he disappeared.
However, at that time, the Tao Te Ching was not called the Tao Te Ching, the book was called "Lao Tzu", in the pre-Qin period, the naming method of the book was to mark the author's surname, followed by a sub-character; for example: Mengke's book was called "Mencius", Zhuang Zhou's book was called "Zhuangzi", etc.; As for why Laozi's book was called "Lao Tzu" instead of Plum, it was mainly because Lao Tzu was the first son of the hundreds of sons.
The name of Lao Tzu was changed to The Tao Te Ching in the early years of the Western Han Dynasty, and the famous Lao Tzu scholar Heshang Gong added its name to it. In addition, the current "Tao Te Ching" is not the original of "Lao Tzu", in 1973, our archaeological community excavated two groups of "Lao Tzu" books in the mawangdui Han tomb in Hunan, the first sentence of the book is: Dao Ke Dao, non-Heng Dao; name can be named, non-Heng name; not the Tao Ke Dao we are familiar with, very Dao; name can be named, very famous; and so on, there are many subtle differences.
Lao Tzu enlightens Shakyamuni, History. The Biography of Lao Tzu gives us a dumb riddle, but in the Book of the Later Han Dynasty, we are given the answer, which proposes that Lao Tzu went out of the Hangu Pass to the Western Regions to preach, and on the way to India, he met a young man sitting under a Bodhi tree, and when he saw that this person had a lot of wisdom and roots, he gave a word and nodded, and since then, Shakyamuni has attained enlightenment and become a Buddha! This is also roughly the same as described in the Buddhist scripture The Floating Tu Sutra.
The Floating Tu Sutra records that Shakyamuni left the Kingdom of Lin'er (that is, his birthplace), went through a lot of difficult and incomprehensible experiences, and was finally enlightened by a high person, and the road was complete!
But then again, as the official history book of the Later Han Dynasty, the Southern Dynasty and Laozi were too long apart, and it is unknown whether all of them coincided.
At present, there is still no conclusive evidence that Lao Tzu enlightened Shakyamuni.
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