Talk:Bonse's inequality

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[Untitled]

Here are some MathReviews involving Bonse's inequality:

MR0150113 (27 #116) Mamangakis, S. E. Synthetic proof of some prime number inequalities. Duke Math. J. 29 1962 471--473.

A ``synthetic proof (i.e., one not involving the limit concepts of analysis) is given by the author for a theorem from which specializations lead to the following inequalities: $$ \prod_{i=1}^np_i>p_{4n}, n\geq 11; \prod_{i=1}^{4n-9}p_i>p_{4n}{}^4, n\geq 46, $$ where $\displaystyle p_n$ denotes the $n$th prime. These inequalities are sharper than the inequality $[\prod_{i=1}^np_i>p_{n+1}^2]$ of Bonse.

MR0986921 (90f:11007) Sándor, József Über die Folge der Primzahlen. (German) [On the sequence of primes] Mathematica (Cluj) 30(53) (1988), no. 1, 67--74.

Let $p(n)$ be the $n$th prime, where $n$ is a natural number. A strikingly simple and elementary proof due to Bonse, published in 1907, shows that (1) $p(1)p(2)\cdots p(n)\geq p(n+1)^2$ if $n\geq 4$; and with more work but on the same elementary level, Bonse proved (2) $p(1)p(2)\cdots p(n)\geq p(n+1)^3$ if $n\geq 5$.

The present author proves in an even simpler way $$ p(1)p(2)\cdots p(n)\geq p(1)p(2)\cdots p(n-1)+p(n)+p(p(n-2)),\tag 3 $$ $$ p(1)p(2)\cdots p(n-2)p(n-1)^2\geq p(n)+p(p(n-1)-1),\tag 4 $$ both if $n\geq 3$; and then quite easily derives (1), (2) from (3), (4). The approach is quite different from Bonse's.

MR1845702 (2002g:11009) Panaitopol, Laurenţiu(R-BUCHM) An inequality involving prime numbers. Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. 11 (2000), 33--35 (2001).

The Rosser-Schoenfeld and Robin estimates for $\pi(x)$ and $\theta(p_n)$ are used to deduce the following Bonse-type inequality: $p_1p_2\cdots p_n>(p_{n+1})^{n-\pi(n)}$, for all $n\geq 2$. This improves Pósa's inequality in the following form: $p_1p_2\cdots p_n>p^k_{n+1}$ for $n\geq 2k$ ($k\geq 1$ given).

{Reviewer's remark: Many other results of this type are included in the reviewer's monograph with D. S. Mitrinovič and B. Crstici [Handbook of number theory, Kluwer Acad. Publ., Dordrecht, 1996; MR1374329 (97f:11001)].}

MR2318393 (2008g:11147) Betts, Robert J.(1-MAL) Using Bonse's inequality to find upper bounds on prime gaps. (English summary) J. Integer Seq. 10 (2007), no. 3, Article 07.3.8, 7 pp. (electronic).

Using the elementary Bonse inequality $p_{k+1}^2 < p_1p_2\cdots p_k$ (valid for $k\ge 4$), where $p_i$ stands for the $i$-th prime number, the author obtains the weak inequality $$p_{k+1}-p_k< p_k(p_1p_2\cdots p_{k-1}-p_k)/(p_{k+1}-p_k),$$ observing that by the Prime Number Theorem this last quotient is asymptotic to $$p_k(p_1p_2\cdots p_{k-1}-p_k)/log((k+1)^{k+1}k^k).$$

Plclark (talk) 03:49, 15 August 2009 (UTC)[reply]

Say, isn't it supposed to be "p_1,...,p_n,p_(n+1) are the smallest n+1 primes"? Dlivnat (talk) 21:30, 5 February 2010 (UTC)[reply]