Chinese hypothesis

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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.

[edit] 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 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. 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.

Despite the fact that the hypothesis is partially incorrect, it is noteworthy that it may have been known to ancient mathematicians. Some, however, claim that the widely propagated belief that the hypothesis was around so early sprouted from a misunderstanding, and that it was actually developed in 1872.

[edit] References

  • Dickson, L. E. History of the Theory of Numbers, Vol. 1: Divisibility and Primality. New York: Dover, 2005.
  • Erdos, P. "On the Converse of Fermat's Theorem." Amer. Math. Monthly 56, 623-624, 1949.
  • Honsberger, R. "An Old Chinese Theorem and Pierre de Fermat." Ch. 1 in Mathematical Gems I. Washington, DC: Math. Assoc. Amer., pp. 1-9, 1973.
  • Jeans, J. H. Messenger Math. 27, 1897-98.
  • Needham, J. (Ed.). Ch. 19 in Science and Civilisation in China, Vol. 3: Mathematics and the Sciences of the Heavens and the Earth. Cambridge, England: Cambridge University Press, 1959.
  • Qi, H. Transmission of Western Mathematics during the Kangxi Kingdom and Its Influence Over Chinese Mathematics. Ph.D. thesis. Beijing, 1991.
  • Ribenboim, P. The New Book of Prime Number Records. New York: Springer-Verlag, pp. 103-105, 1996.
  • Shanks, D. Solved and Unsolved Problems in Number Theory, 4th ed. New York: Chelsea, pp. 19-20, 1993.
  • Yan, L. and Shiran, D. Chinese Mathematics, A Concise History. Oxford, England: Clarendon Press, 1987.