Chinese hypothesis

In number theory, the Chinese hypothesis is a disproven conjecture stating that an integer n is prime if and only if it satisfies the condition that 2n−2 is divisible by n. In other words, that integer n is prime if and only if 2^n \equiv 2 \pmod{n}\,. It is true that if n is prime, then 2^n \equiv 2 \pmod{n}\, (this is a special case of Fermat's little theorem). However, the converse (if \,2^n \equiv 2 \pmod{n} then n is prime) is false, and therefore the hypothesis as a whole is false. The smallest counter example is n = 341 = 11×31. Composite numbers n for which 2n−2 is divisible by n are called Poulet numbers. They are a special class of Fermat pseudoprimes.

History

The Chinese hypothesis is commonly attributed to Chinese scholars more than 2500 years ago. However, this oft-quoted attribution is a myth originating with James Jeans (1898), who wrote that "a paper found among those of the late Sir Thomas Wade and dating from the time of Confucius" contained the theorem. This assertion was refuted by Needham, who attributes the misunderstanding to an incorrect translation of a passage in a well-known book The Nine Chapters on the Mathematical Art. Qi (1991) attributed the hypothesis to Chinese mathematician Li Shanlan (1811–1882), communicated the statement to his collaborator in the translation of Western texts, and the collaborator then published it. Li subsequently learned that the statement was wrong, and hence did not publish it himself, but Hua Heng-Fang published the statement as if it were correct in 1882.

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