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In [[number theory]], the '''Chinese hypothesis''' is a disproven [[conjecture]] stating that an [[integer]] ''n'' is [[Prime number|prime]] [[if and only if]] it satisfies the condition that 2<sup>''n''</sup>&minus;2 is divisible by ''n''. In other words, that integer ''n'' is prime if and only if <math>2^n \equiv 2 \pmod{n}\,</math>. It is true that if ''n'' is prime, then <math>2^n \equiv 2 \pmod{n}\,</math> (this is a special case of [[Fermat's little theorem]]). However, the converse (if <math>\,2^n \equiv 2 \pmod{n}</math> 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 number]]s ''n'' for which 2<sup>''n''</sup>&minus;2 is divisible by ''n'' are called [[Poulet number]]s. They are a special class of [[Fermat pseudoprime]]s.

==History==
The Chinese hypothesis is commonly attributed to [[Chinese mathematics|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 on the 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==
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*{{Citation |last=Honsberger |first=R. |chapter=An Old Chinese Theorem and Pierre de Fermat |title=Mathematical Gems |volume=I |location=Washington, DC |publisher=Math. Assoc. Amer. |pages=1–9 |year=1973 |isbn= }}.
*{{Citation |last=Jeans |first=J. H. |title=The converse of Fermat's theorem |journal=Messenger of Mathematics |volume=27 |issue= |pages=174 |year=1898 }}.
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*{{Citation |last=Qi |first=H. |title=Transmission of Western Mathematics during the Kangxi Kingdom and Its Influence Over Chinese Mathematics |publisher=Ph.D. thesis |location=Beijing |year=1991 |isbn= }}.
*{{Citation |last=Ribenboim |first=P. |title=The New Book of Prime Number Records |location=New York |publisher=Springer-Verlag |pages=103–105 |year=1996 |isbn=0-387-94457-5 }}.
*{{Citation |last=Shanks |first=D. |title=Solved and Unsolved Problems in Number Theory |edition=4th |location=New York |publisher=Chelsea |pages=19–20 |year=1993 |isbn=0-8284-1297-9 }}.
*{{Citation |last=Yan |first=L. |lastauthoramp=yes |last2=Shiran |first2=D. |title=Chinese Mathematics: A Concise History |location=Oxford, England |publisher=Clarendon Press |year=1987 |isbn=0-19-858181-5 }}.

[[Category:Pseudoprimes]]
[[Category:Conjectures about prime numbers]]
[[Category:Disproved conjectures]]