Chinese Hypothesis

The hypothesis that an integer is prime iff it satisfies the condition that is divisible by . Dickson (2005, p. 91) stated that Leibniz believe to have proved that this congruence implies that is prime. In actuality, this condition is necessary but not sufficient for to be prime since, for example, is divisible by 341, but is composite.

Composite numbers (such as 341) for which is divisible by are called Poulet numbers, and are a special class of Fermat pseudoprimes. The Chinese hypothesis is a special case of Fermat's little theorem.

The "Chinese hypothesis," "Chinese congruence," or "Chinese theorem," as it is sometimes called, is commonly attributed to Chinese scholars more than 2500 years ago. However, this oft-quoted attribution (e.g., Honsberger 1973, p. 3) is a myth originating with Jeans (1897-98), 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 of Mathematical Art (Ribenboim 1996, p. 104). Qi (1991) attributed the hypothesis to Chinese mathematician Li Shan-Lan (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 (Ribenboim 1996, pp. 104-105).

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