Abstract:
The conjecture that every sufficiently large even natural number $2n$ can be represented in the form $2n=p+q$, where $p$ and $q$ are prime, is called the binary Goldbach problem, which so far remains unproved. Up to now, it has been established by the Vinogradov method that among the even numbers less than $x$, there are at most $Cx^{0.95}$ exceptional numbers, that is, ones that cannot be represented as the sum of two primes (here $C>0$ is an absolute constant). For the analogous problem of representing a natural number m in the form $m=[ap]+[bq]$, where $p$ and $q$ are prime and $a/b$ is an algebraic number, G. I. Arkhipov and V. N. Chubarikov have proved that the number of $m<x$ not so representable is bounded above by $C(\varepsilon)x^{2/3+\varepsilon}$, where $\varepsilon>0$ is arbitrary. Questions related to this result were considered in the lecture.
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