Who first proved the generalization of Bertrand's postulate to (2n,3n) and (3n,4n)?

In Wikipedia's page for Bertrand's postulate, it is said that its (2n,3n) version was proved by El Bachraoui in 2006. Seems likely that it was first proved way before than that! Can anyone point to the first source, or at least to a previous one?

Analogously, Wikipedia said until recently that the (3n,4n) version was due to Andy Loo in 2011. I'm aware of a proof by Denis Hanson in 1973, so I have updated the page with that info, but I don't know if his proof is the first one. Do you know of previous proofs?

• I haven't looked at the proofs but I'm trying to understand how they could possibly be novel. Selberg and Erdős gave elementary proofs of the PNT in 1948. So we already had all these elementary theorems for sufficiently large $n$ seventy years ago, is there some difficulty in making them effective with an elementary argument? – Dan Brumleve Dec 27 '17 at 20:49
• references to this problem are given in arXiv:1212.2785, with the remarkable theorem: The list of integers $k$ for which every interval $(kn, (k + 1)n)$, $n > 1$, contains a prime includes $k = 1,2,3,5,9,14$ and no others – Carlo Beenakker Dec 27 '17 at 21:04
• @CarloBeenakker Looks like a great paper! Nevertheless, the references it gives are the same as the Wikipedia ones (it fails to cite Hanson's work, for example). Besides, it is not for every $k$, just for every $k<10^8$. – Jose Brox Dec 27 '17 at 21:07
• The answers to mathoverflow.net/questions/113840/… have pertinent references. – Tony Huynh Dec 28 '17 at 19:33

I have finally found the following papers and results, which predate Nagura's paper of 1952. I cite them from newest to oldest:

1. (Molsen, 1941):
• For $n\geq 118$ there are primes in $(n,\frac43n)$ congruent to 1,5,7,11 modulo 12.
• For $n\geq 199$ there are primes in $(n,\frac87n)$ congruent to 1,2 modulo 3. This result implies that of Nagura.

K. Molsen, Zur Verallgemeinerung des Bertrandschen Postulates, Deutsche Math. 6 (1941), 248-256. MR0017770

1. (Breusch, 1932):
• For $n\geq 7$ there are primes in $(n,2n)$ congruent to 1,2 modulo 3 and to 1,3 modulo 4.
• For $n\geq 48$ there is a prime in $(n,\frac98n)$. This result implies those of Nagura and Molsen (but not the congruences part).

R. Breusch, Zur Verallgemeinerung des Bertrandschen Postulates, dass zwischen x und 2x stets Primzahlen liegen, Math. Z. 34 (1932), 505–526. MR1545270

1. (Schur, 1929, according to Breusch in the previous paper):
• For $n\geq 24$ there is a prime in $(n,\frac54n)$. This result already implies those of Hanson and El Bachraoui.

I. Schur, Einige Sätze über Primzahlen mit Anwendungen auf Irreduzibilitätsfragen I, Sitzungsberichte der preuss. Akad. d. Wissensch., phys.-math. Klasse 1929, S.128.

• I could not locate Schur's paper. I'd be glad if someone can provide a link! – Jose Brox Dec 28 '17 at 18:44
• Surprising and impressive find. – Ofir Gorodetsky Dec 28 '17 at 21:34

EDIT: Jose Brox, in another answer, had provided references which not noly predate Nagura's result, but they are also stronger.

J. Nagura, already in 1952, proved that the interval $(x,\frac{6}{5}x)$ contains a prime for any $x \ge 25$. The proof appears in the paper "On the interval containing at least one prime number" published in Proc. Japan Acad. Volume 28, Number 4 (1952), 177-181 (see this link). From Nagura's theorem one obtains the results mentioned in your post by choosing $x=2n$ or $x=3n$, and checking what happens for small $n$.

• Scooped me by 2 minutes :) – Igor Rivin Dec 27 '17 at 20:58
• @DanBrumleve The proof uses properties of the Gamma function for a real variable only, as far as I can see. The idea of employing properties of the Gamma function (including Stirling's formula) goes back to Ramanujan's proof of Bertrand's Postulate (1919). – Ofir Gorodetsky Dec 27 '17 at 21:06
• Nice! Do you have any hint that this is indeed the first proof of this kind? – Jose Brox Dec 27 '17 at 21:09
• @DanBrumleve I don't think so... Wikipedia doesn't say that they provide simple proofs, just states "this person proved this fact" (btw, as does Moser's paper linked by Carlo Beenakker in a commentary) – Jose Brox Dec 27 '17 at 21:11
• @JoseBrox According to Google Scholar, there isn't a paper from 1952 or before citing Chebyshev's, Ramanujan's or Erdős's proofs. On the other hand, Google Scholar does not list Nagura's paper in the list of citations, so it is far from perfect. – Ofir Gorodetsky Dec 27 '17 at 21:29