Bertrand's postulate states that for any positive integer , there is a prime between and . Despite its name, it is, in fact, a theorem. A more widely known version states that there is a prime between and .
It is similar to the proof of Chebyshev's estimates in the prime number theorem article but requires a closer look at the binomial coefficient . Assuming that the reader is familiar with that proof, the Bertrand postulate can be proved as follows.
Note that the power with which a prime satisfying appears in the prime factorization of is . Thus,
The first product does not exceed and the second one does not exceed . Thus,
The right hand side is strictly greater than for , so it remains to prove the Bertrand postulate for . In order to do it, it suffices to present a sequence of primes starting with in which each prime does not exceed twice the previous one, and the last prime is above . One such possible sequence is .
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