Romanov type problems

Result code in 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>
Result on the web
<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>

Result language
angličtina
Original language name
Romanov type problems
Original language description
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.
Czech name
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Czech description
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Type
J<sub>imp</sub> - Article in a specialist periodical, which is included in the Web of Science database
CEP classification
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OECD FORD branch
10101 - Pure mathematics

Project
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Continuities
I - Institucionalni podpora na dlouhodoby koncepcni rozvoj vyzkumne organizace

Publication year
2018
Confidentiality
S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

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