The manuscript was sold on Tuesday by Christie's in Paris to an anonymous buyer, Xinhua news agency reported.
Christie's said on its website that the 54-page working manuscript, co-written by Einstein and Michele Besso between June 1913 and early 1914, was a crucial stage in the development of the General Theory of Relativity.
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It is one of only two surviving manuscripts documenting the genesis of the theory, and provides a fascinating glimpse into the mind of the scientist, Christie's added.
The auction house had previously estimated the value of the manuscript at between 2.4 million and 3.5 million dollars.
In 2019, an exhibition marking the 140th anniversary of Einstein's birth was held in Shanghai, China. Handwritten pages of the General Theory of Relativity and famous equation E=MC2 (squared) were displayed.
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