| Title:
|
Generalizing a theorem of Schur (English) |
| Author:
|
Jones, Lenny |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
64 |
| Issue:
|
4 |
| Year:
|
2014 |
| Pages:
|
1063-1065 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
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 \[ p_{1}+p_{2}+\cdots +p_{k}>(k-\epsilon )p_{t}. \] (English) |
| Keyword:
|
Prime Number Theorem |
| Keyword:
|
Schur |
| MSC:
|
11A41 |
| MSC:
|
11N05 |
| idZBL:
|
Zbl 06433714 |
| idMR:
|
MR3304798 |
| DOI:
|
10.1007/s10587-014-0153-2 |
| . |
| Date available:
|
2015-02-09T17:41:24Z |
| Last updated:
|
2020-07-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/144161 |
| . |
| Reference:
|
[1] Lehmer, E.: On the magnitude of the coefficients of the cyclotomic polynomial.Bull. Am. Math. Soc. 42 389-392 (1936). Zbl 0014.39203, MR 1563307, 10.1090/S0002-9904-1936-06309-3 |
| Reference:
|
[2] Suzuki, J.: On coefficients of cyclotomic polynomials.Proc. Japan Acad., Ser. A 63 279-280 (1987). Zbl 0641.10008, MR 0931264, 10.3792/pjaa.63.279 |
| . |
