Chinese hypothesis

In number theory, the Chinese hypothesis is a disproven conjecture stating that an integer n is prime if and only if it satisfies the condition that 2ⁿ−2 is divisible by n—in other words, that integer n is prime if and only if $2^n \equiv 2 \pmod{n}\,$. It is true that if n is prime, then $2^n \equiv 2 \pmod{n}\,$ (this is a special case of Fermat's little theorem). However, the converse (if $\,2^n \equiv 2 \pmod{n}$ 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 numbers n for which 2ⁿ−2 is divisible by n are called Poulet numbers. They are a special class of Fermat pseudoprimes.

References

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