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It is shown that the absolute constant in the Berry--Esseen inequality fori.i.d. Bernoulli random variables is strictly less than the Esseen constant, if$1\le n\le 5$, where $n$ is a number of summands. This result is got bothwith the help of a supercomputer and an interpolation theorem, which is provedin the paper as well. In addition, applying the method developed by S. Nagaevand V. Chebotarev in 2009--2011, an upper bound is obtained for the absoluteconstant in the Berry--Esseen inequality in the case under consideration, whichdiffers from the Esseen constant by no more than 0.06%. As an auxiliary result,we prove a bound in the local Moivre--Laplace theorem which has a simple andexplicit form. Despite the best possible result, obtained by J. Schulz in 2016,we propose our approach to the problem of finding the absolute constant in theBerry--Esseen inequality for two-point distributions since this approach,combining analytical methods and the use of computers, could be useful insolving other mathematical problems.

On a bound of the absolute constant in the Berry--Esseen inequality for i.i.d. Bernoulli random variables