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Re: Chinese theorem?
Posted:
Jun 25, 1997 12:14 AM


After ManKeung Siu reminded me that he had posted the response to my query, I found his post and attach it below.
Han Sah, sah@math.sunysb.edu
********* From @math.maa.org:mathhistorylistowner@maa.org Thu Mar 28 04:35:08 1996 Organization: Department of Mathematics, HKU To: conway@math.Princeton.EDU XFile: PRIME XFinfo: DOS,"PRIME",,,,WordStar Subject: Chinese theorem? Cc: mathhistorylist@maa.org Priority: normal XMailer: Pegasus Mail v3.22 Sender: mathhistorylistowner@maa.org Comments: send subscribe/unsubscribe messages to majordomo@maa.org
} You are correct in doubting the existence of any Chinese knowledge on "pseudoprime in base 2", i.e. the (false) fact that n is a prime if n divides 2^n2. ( The first counterexample was offered by F. Sarrus in 1819, viz. 341= 11*31 divides 2^3412. ) The myth seems to originate in a paper by J.H.Jeans, in the magazine MESSENGER OF MATHEMATICS (vol.27,1897/8), who wrote that "a paper found among those of the late Sir Thomas Wade and dating from the time of Confucius" contained the theorem that 2^n = n (mod n) holds if and only if n is a prime number. In a footnote of the monumental work of Joseph Needham ( "Science and Civiliza tion in China", Volume 3, Chapter 19 ) the author dispels Jean's assertion, which is due to an erroneous translation of a passage in Chapter One of the very famous ancient Chinese mathematical text JIUZHANG SUANSHU ( "The Nine Chapters of Mathematical Art" ). It is commonly believed that JIUZHANG was compiled some time between 100 BC and AD 100, but the content definitely dated to a few centuries earlier. The passage just mentioned is actually giving a method of computation same as the Euclidean algorithm. That will explain why the Chinese could still achieve a lot even though they did not have the notion of prime numbers and thus never had the notion of the Fundamental Theorem of Arithmetic. This mistake has been perpetuated by several Western schol ars, among them Leonard Eugene Dickson who quoted Jean's remark in his "History of the Theory of Numbers", Volume I (p.91), and also quoted that Leibniz believed to have proved it. In a talk given at University of Western Australia in the summer of 1984 I mentioned in passing this myth. Paul Erdos was in the audience and he introduced me a couple of years later to Paulo Ribenboim who was then working on his book "The Book of Prime Number Re cords". Thus you can find the remark above on p.86 of that book (2nd edition, 1989). After 1989 I happened to find out more about this issue and told Paulo Ribenboim about it. Maybe the addition al information will appear in his 3rd edition of the book to be published in 1995/6. I gleaned the additional information from the PhD thesis of HAN Qi at the Institute for the History of Natural Science, Acadmia Sinica, Beijing (1991). The thesis is on the transmission of Western mathematics in the reign of the Qing Emperor Kangxi ( c. late 17th century to early 18th century ) and its influence on subsequent development of Chinese mathematics. In Section 4 of Chapter 3 HAN mentions the work of LI Shanlan (18111882). LI believed that he obtained a criterion for testing primality of n, viz. n divides 2^n2 or not. He told his friend and collaborator in translating Western texts, the English mis sionary Alexander Wylie, about it in 1869 when they were in Shanghai. Apparently Wylie regarded this as a significant discov ery, but did not understand the mathematics well enough. Soon after Wylie travelled south to Hong Kong and wrote up a note on it, submitting it to the magazine "Notes and Queries on China and Japan". It appeared in the May issue of 1869 with the title "A Chinese Theorem". In subsequent issues some readers discussed the problem and some pointed out the fallacy of the assertion. Appar ently LI Shanlan himself was aware of this, after discussion with his colleague at Tongwenguan ( an institute set up then to trans late and study Western science ), a German named J.von Gumpach. his "criterion" for testing primality. However, he had also told his younger friend and colleague HUA Hengfang (18331902), who put down this "criterion" in his 1882 book on number theory. HUA did not notice the fallacy and ascribed it to LI. I do not know whether the myth originating with Jean has anything to do with the article of Wylie or not. But it seems that the name "Chinese Theorem" did stick with the (wrong) result for a time in the mid 19th century. This is about all I can tell for the moment.
Sincerely, ManKeung SIU Department of Mathematics University of Hong Kong Hong Kong



