Math CentralQuandaries & Queries


Question from Mike, a student:


I was wondering if there was some kind of guaranteed interval that ensures that if you start at some number n, you will always have passed a prime number p.

I remember having once read something that there is always at least one prime between n and 2n, but I cannot for the life of me remember where I read that, or what that kind of rule would be called so I can look it up online...

thank you for your help,



We have two responses for you

You are right. For any positive integer n, there is always a prime between n and 2n. This is known as Bertrand's Postulate.



It is certainly true, but not trivial to prove, that there is a prime between n and 2n for n > 1 - this statement was first conjectured in 1845 by Joseph Bertrand (1822–1900). Bertrand himself verified his statement for all numbers up to 3 million. His conjecture was proved by Chebyshev (1821–1894) in 1850 and so the postulate is often known as Chebyshev's theorem. There are much stronger results for n large enough but of course, the proofs get more and more difficult.


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