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Topic: Chinese theorem?
Replies: 2   Last Post: Jun 25, 1997 12:49 AM

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Chih-Han sah

Posts: 75
Registered: 12/3/04
Re: Chinese theorem?
Posted: Jun 25, 1997 12:14 AM
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After Man-Keung Siu reminded me that he had posted the
response to my query, I found his post and attach it below.

Han Sah,

From Thu Mar 28 04:35:08 1996
Organization: Department of Mathematics, HKU
To: conway@math.Princeton.EDU
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Subject: Chinese theorem?
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} 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^n-2. ( The first counter-example was
offered by F. Sarrus in 1819, viz. 341= 11*31 divides 2^341-2. )
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
(1811-1882). LI believed that he obtained a criterion for testing
primality of n, viz. n divides 2^n-2 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 (1833-1902), 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.

Man-Keung SIU
Department of Mathematics
University of Hong Kong
Hong Kong

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