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 of 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.

References

• Dickson, L. E. (2005), History of the Theory of Numbers, Vol. 1: Divisibility and Primality, New York: Dover, ISBN 0486442322 .
• Erdos, P. (1949), "On the Converse of Fermat's Theorem", American Mathematical Monthly 56 (9): 623–624, doi:10.2307/2304732 .
• Honsberger, R. (1973), "An Old Chinese Theorem and Pierre de Fermat", Mathematical Gems, I, Washington, DC: Math. Assoc. Amer., pp. 1–9 .
• Jeans, J. H. (1898), "The converse of Fermat's theorem", Messenger of Mathematics 27: 174 .
• Needham, J., ed. (1959), "Ch. 19", Science and Civilisation in China, Vol. 3: Mathematics and the Sciences of the Heavens and the Earth, Cambridge, England: Cambridge University Press .
• Qi, H. (1991), Transmission of Western Mathematics during the Kangxi Kingdom and Its Influence Over Chinese Mathematics, Beijing: Ph.D. thesis .
• Ribenboim, P. (1996), The New Book of Prime Number Records, New York: Springer-Verlag, pp. 103–105, ISBN 0387944575 .
• Shanks, D. (1993), Solved and Unsolved Problems in Number Theory (4th ed.), New York: Chelsea, pp. 19–20, ISBN 0828412979 .
• Yan, L. & Shiran, D. (1987), Chinese Mathematics: A Concise History, Oxford, England: Clarendon Press, ISBN 0198581815 .