Are there any interesting or lesser known proofs related to Bertrand's Postulate - MathOverflow

Are there any interesting or lesser known proofs related to Bertrand's Postulate

2012-08-01T22:54:28Z

There are 3 standard proofs of Bertrand's Postulate:

(1) Chebyshev's original proof

(2) Ramanujan's simplification of Chebyshev's proof

(3) Erdos's proof

I recently learned about the very simple proof that if the Goldbach conjecture is true, then Bertrand's postulate follows (see here).

Does anyone know of any other proofs? There are recent proofs that extend Bertrand's postulate to show that there is always a prime in $2n$/$3n$ and $3n$/$4n$.

I am wondering if there aren't other lesser known proofs that take a different approach to establish the existence of a prime between $n$ and $2n$.

Thanks,

-Larry

Answer by Benjamin Dickman for Are there any interesting or lesser known proofs related to Bertrand's Postulate

2012-08-02T00:35:48Z

Bertrand's Postulate follows as a direct consequence of the following theorem of J.J. Sylvester:

Theorem (Sylvester, 1892): Let $k$ be a positive integer. Then at least one of any $k$ consecutive integers greater than $k$ is divisible by a prime greater than $k$.

(For comparison: Chebyshev's analytic proof dates to 1850; Erdos' elementary proof dates to 1932.)

See Theorem 6 (p. 6) in http://www.math.sc.edu/~filaseta/papers/schurpaper.pdf, from which I quote:

"The theorem implies immediately that for any positive integer $k$, one of $k+1, k+2, \ldots, 2k$ is a prime (since one of these integers must be divisible by a prime $\geq k+1).$"

Unfortunately, my internet search has not led me to the original paper by Sylvester.

Answer by Igor Rivin for Are there any interesting or lesser known proofs related to Bertrand's Postulate

2012-08-02T00:38:14Z

A stronger version is proved by Jonathan Sondow in this arxiv preprint (which looks like a monthly paper).

Answer by unknown (google) for Are there any interesting or lesser known proofs related to Bertrand's Postulate

2012-08-07T06:46:23Z

Nice question! For a point of view from the perspective of Goldbach's conjecture, perhaps one can consider also Theorem 3.7 of "The Hardy-Littlewood Method", 2nd edition, by R.C. Vaughan.

Answer by Aaron Meyerowitz for Are there any interesting or lesser known proofs related to Bertrand's Postulate

2012-08-07T21:34:47Z

The proofs for $(2n,3n)$ and $(3n,4n)$ are elementary and very pleasing (based on a quick look.) It is known that there is always a prime between $k$ and $\frac{6k}{5}\ $ for $k \gt 24.$ The proof is more involved but do not use anything analytic. Putting $k=2n$ etc gives a prime in $(4n,\frac{24n}{5})$ (except $n=1,2,6$) and hence in $(4n,5n)$ except for $n=1,2$ and one in $(5n,6n)$ except for $n=1$.

Answer by Larry Freeman for Are there any interesting or lesser known proofs related to Bertrand's Postulate

2012-09-06T01:05:33Z

I found an interesting proof today that demonstrates a stronger form of Bertrand's Postulate. I hadn't seen it before:

Abstract. In this paper we give a stronger form of Bertrand's postulate and use it to prove that every positive integer, except 1, 2, 4, 6, and 9, can be written as the sum of distinct odd primes.

http://www.ams.org/journals/proc/1972-033-02/S0002-9939-1972-0292746-6/S0002-9939-1972-0292746-6.pdf

Answer by Armin for Are there any interesting or lesser known proofs related to Bertrand's Postulate

2013-05-03T11:46:59Z

Re-extending Chebyshev’s theorem about Bertrand’s conjecture:

http://link.springer.com/article/10.1007%2Fs11253-008-0034-7