In
number theory
Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and integer-valued functions. German mathematician Carl Friedrich Gauss (1777–1855) said, "Mat ...
, the Chinese hypothesis is a disproven
conjecture stating that an
integer
An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the languag ...
''n'' is
prime
A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways ...
if and only if
In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false.
The connective is b ...
it satisfies the condition that
is
divisible
In mathematics, a divisor of an integer n, also called a factor of n, is an integer m that may be multiplied by some integer to produce n. In this case, one also says that n is a multiple of m. An integer n is divisible or evenly divisible by ...
by ''n''—in other words, that an integer ''n'' is prime if and only if
. It is true that if ''n'' is prime, then
(this is a special case of
Fermat's little theorem), however the
converse
Converse may refer to:
Mathematics and logic
* Converse (logic), the result of reversing the two parts of a definite or implicational statement
** Converse implication, the converse of a material implication
** Converse nonimplication, a logical c ...
(if
then ''n'' is prime) is false, and therefore the hypothesis as a whole is false. The smallest
counterexample
A counterexample is any exception to a generalization. In logic a counterexample disproves the generalization, and does so rigorously in the fields of mathematics and philosophy. For example, the fact that "John Smith is not a lazy student" is a ...
is ''n'' = 341 = 11×31.
Composite number
A composite number is a positive integer that can be formed by multiplying two smaller positive integers. Equivalently, it is a positive integer that has at least one divisor other than 1 and itself. Every positive integer is composite, prime, ...
s ''n'' for which
is divisible by ''n'' are called
Poulet numbers. They are a special class of
Fermat pseudoprime
In number theory, the Fermat pseudoprimes make up the most important class of pseudoprimes that come from Fermat's little theorem.
Definition
Fermat's little theorem states that if ''p'' is prime and ''a'' is coprime to ''p'', then ''a'p''− ...
s.
History
Once, and sometimes still, mistakenly thought to be of ancient Chinese origin, the Chinese hypothesis actually originates in the mid-19th century from the work of
Qing dynasty
The Qing dynasty ( ), officially the Great Qing,, was a Manchu-led imperial dynasty of China and the last orthodox dynasty in Chinese history. It emerged from the Later Jin dynasty founded by the Jianzhou Jurchens, a Tungusic-spea ...
mathematician
Li Shanlan
Li Shanlan (李善蘭, courtesy name: Renshu 壬叔, art name: Qiuren 秋紉) (1810 – 1882) was a Chinese mathematician of the Qing Dynasty.
A native of Haining, Zhejiang, he was fascinated by mathematics since childhood, beginning with the '' ...
(1811–1882).
He was later made aware his statement was incorrect and removed it from his subsequent work but it was not enough to prevent the false proposition from appearing elsewhere under his name;
[ a later mistranslation in the 1898 work of Jeans dated the conjecture to Confucian times and gave birth to the ancient origin myth.][ (all of footnote d)]
References
Bibliography
*
*
*
*
*
*
*
*
*
{{Prime number conjectures
Pseudoprimes
Conjectures about prime numbers
Disproved conjectures