Generalizing a theorem of Schur

Jones, Lenny. "Generalizing a theorem of Schur." Czechoslovak Mathematical Journal 64.4 (2014): 1063-1065. <http://eudml.org/doc/269851>.

@article{Jones2014,
abstract = {In a letter written to Landau in 1935, Schur stated that for any integer $t>2$, there are primes $p_\{1\}<p_\{2\}<\cdots <p_\{t\}$ such that $p_\{1\}+p_\{2\}>p_\{t\}$. In this note, we use the Prime Number Theorem and extend Schur’s result to show that for any integers $t\ge k \ge 1$ and real $\epsilon >0$, there exist primes $p_\{1\}<p_\{2\}<\cdots <p_\{t\}$ such that \[ p\_\{1\}+p\_\{2\}+\cdots +p\_\{k\}>(k-\epsilon )p\_\{t\}. \]},
author = {Jones, Lenny},
journal = {Czechoslovak Mathematical Journal},
keywords = {Prime Number Theorem; Schur; Prime Number Theorem; result of Schur},
language = {eng},
number = {4},
pages = {1063-1065},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Generalizing a theorem of Schur},
url = {http://eudml.org/doc/269851},
volume = {64},
year = {2014},
}

TY - JOUR
AU - Jones, Lenny
TI - Generalizing a theorem of Schur
JO - Czechoslovak Mathematical Journal
PY - 2014
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 64
IS - 4
SP - 1063
EP - 1065
AB - In a letter written to Landau in 1935, Schur stated that for any integer $t>2$, there are primes $p_{1}<p_{2}<\cdots <p_{t}$ such that $p_{1}+p_{2}>p_{t}$. In this note, we use the Prime Number Theorem and extend Schur’s result to show that for any integers $t\ge k \ge 1$ and real $\epsilon >0$, there exist primes $p_{1}<p_{2}<\cdots <p_{t}$ such that \[ p_{1}+p_{2}+\cdots +p_{k}>(k-\epsilon )p_{t}. \]
LA - eng
KW - Prime Number Theorem; Schur; Prime Number Theorem; result of Schur
UR - http://eudml.org/doc/269851
ER -