Difference between revisions of "Bertrand's Postulate"
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Revision as of 14:42, 13 June 2007
Formulation
Bertrand's postulate states that for any positive integer , there is a prime between
and
. Despite its name, it is, in fact, a theorem.
Proof
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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