Bertrand

Bertrand's postulate (also called Bertrand's conjecture, Chebyshev's theorem, Chebyshev's postulate, and Chebyshev's bias) is a statement about prime numbers named after Joseph Bertrand. It asserts that for any integer n > 1, there is always at least one prime number p with n < p < 2n.

History

Joseph Bertrand first conjectured this in 1845, and it was proven by Pafnuty Chebyshev in 1852. Despite the name "postulate", it is not something that is assumed; it is a statement that has been proven to be true.

Proof

The proof by Chebyshev used properties of the factorial function and the binomial coefficients. Later, simpler proofs were found by Sylvester and Erdős. The proof by Erdős is particularly simple, using the probabilistic method.

Generalizations

There have been many generalizations of Bertrand's postulate. For example, it has been shown that for any integer n > 1, there is always at least one prime number p with n < p < n + 315.

See also

• Prime number
• Prime gap
• List of conjectures

References

Bertrand Resources
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