Romanov type problems

Kód výsledku v IS VaVaI

<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F61988987%3A17310%2F18%3AA1901XZM" target="_blank" >RIV/61988987:17310/18:A1901XZM - isvavai.cz</a>

Výsledek na webu

<a href="http://dx.doi.org/10.1007/s11139-017-9972-8" target="_blank" >http://dx.doi.org/10.1007/s11139-017-9972-8</a>

DOI - Digital Object Identifier

<a href="http://dx.doi.org/10.1007/s11139-017-9972-8" target="_blank" >10.1007/s11139-017-9972-8</a>

Jazyk výsledku

angličtina

Název v původním jazyce

Romanov type problems

Popis výsledku v původním jazyce

Romanov proved that the proportion of positive integers which can be represented as a sum of a prime and a power of 2 is positive. We establish similar results for integers of the form n = p + 2(2k) + m! and n = p + 2(2k) + 2(q) where m, k is an element of N and p, q are primes. In the opposite direction, Erdos constructed a full arithmetic progression of odd integers none of which is the sum of a prime and a power of two. While we also exhibit in both cases full arithmetic progressions which do not contain any integers of the two forms, respectively, we prove amuch better result for the proportion of integers not of these forms: (1) The proportion of positive integers not of the form p + 2(2k) + m! is larger than 3/4. (2) The proportion of positive integers not of the form p + 2(2k) + 2(q) is at least 2/3.

Název v anglickém jazyce

Romanov type problems

Popis výsledku anglicky

Romanov proved that the proportion of positive integers which can be represented as a sum of a prime and a power of 2 is positive. We establish similar results for integers of the form n = p + 2(2k) + m! and n = p + 2(2k) + 2(q) where m, k is an element of N and p, q are primes. In the opposite direction, Erdos constructed a full arithmetic progression of odd integers none of which is the sum of a prime and a power of two. While we also exhibit in both cases full arithmetic progressions which do not contain any integers of the two forms, respectively, we prove amuch better result for the proportion of integers not of these forms: (1) The proportion of positive integers not of the form p + 2(2k) + m! is larger than 3/4. (2) The proportion of positive integers not of the form p + 2(2k) + 2(q) is at least 2/3.

Druh

J_imp - Článek v periodiku v databázi Web of Science

CEP obor

—

OECD FORD obor

10101 - Pure mathematics

Projekt

—

Návaznosti

I - Institucionalni podpora na dlouhodoby koncepcni rozvoj vyzkumne organizace

Rok uplatnění

2018

Kód důvěrnosti údajů

S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

Název periodika

RAMANUJAN J

ISSN

1382-4090

e-ISSN

1572-9303

Svazek periodika

47

Číslo periodika v rámci svazku

2

Stát vydavatele periodika

NL - Nizozemsko

Počet stran výsledku

23

Strana od-do

267-289

Kód UT WoS článku

000447277000003

EID výsledku v databázi Scopus

2-s2.0-85041521307