Chinese hypothesis

The Chinese hypothesis states that a number $n$ is prime if and only if $2^{n}-2$ is a multiple of $n$ . By Fermat’s little theorem we have that $2^{p}\equiv 2\mod p$ , so that means $n$ does divide $2^{n}-2$ if $n$ is prime. However, if $n$ is composite Fermat’s little theorem does not rule out that $n$ could divide $2^{n}-2$ . The Chinese hypothesis checks out for the small powers of two. The first counterexample is $n=341$ , but since $2^{341}$ has more than a hundred digits, it wasn’t easy to check it back in the 18th century when this test was first proposed. Though back then they attributed it to ancient Chinese mathematicians, hence the name.

Title	Chinese hypothesis
Canonical name	ChineseHypothesis
Date of creation	2013-03-22 18:11:08
Last modified on	2013-03-22 18:11:08
Owner	FourDozens (21006)
Last modified by	FourDozens (21006)
Numerical id	6
Author	FourDozens (21006)
Entry type	Definition
Classification	msc 11A51