Bertrand's postulate

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Despite it's name, Bertrand's postulate is actually a theorem rather than a postulate:

Theorem (Bertrand's Postulate): For every integer [math]n \geq 2[/math], there is a prime [math]p[/math] satisfying [math]n \lt p \lt 2n[/math].

The relevance of the postulate for the finding primes problem is that it guarantees the existence of a [math]k[/math]-digit prime for any [math]k[/math]. Brute force search thus yields a [math]k[/math]-digit prime after about [math]O(10^k)[/math] steps; this can be considered the "trivial bound" for the problem.

This result was apparently first proved by Chebyshev. For large [math]n[/math], the claim follows as a consequence of the prime number theorem. We will give an elementary proof due to Erdos.

Our proof of Bertrand's postulate starts with a result independent interest, due to Chebyshev.

Lemma (Chebyshev bound): For integers [math]n \geq 2[/math],

[math] \prod_{p \leq n} p \leq 4^n, [/math]

where the product is over all primes [math]p[/math] less than or equal to [math]n[/math].