Generalizing a theorem of Schur
Czechoslovak Mathematical Journal, Tome 64 (2014) no. 4, pp. 1063-1065
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In a letter written to Landau in 1935, Schur stated that for any integer $t>2$, there are primes $p_{1}p_{t}$. 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}(k-\epsilon )p_{t}. \]
In a letter written to Landau in 1935, Schur stated that for any integer $t>2$, there are primes $p_{1}$ 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 $p_{1}$ such that \[ p_{1}+p_{2}+\cdots +p_{k}>(k-\epsilon )p_{t}. \]
@article{10_1007_s10587_014_0153_2,
author = {Jones, Lenny},
title = {Generalizing a theorem of {Schur}},
journal = {Czechoslovak Mathematical Journal},
pages = {1063--1065},
year = {2014},
volume = {64},
number = {4},
doi = {10.1007/s10587-014-0153-2},
mrnumber = {3304798},
zbl = {06433714},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.1007/s10587-014-0153-2/}
}
Jones, Lenny. Generalizing a theorem of Schur. Czechoslovak Mathematical Journal, Tome 64 (2014) no. 4, pp. 1063-1065. doi: 10.1007/s10587-014-0153-2
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