## 2 About the Residue Number System and Akushsky Core Function  
The Residue Number System (RNS) is a non-positional number system used to represent and process integers [16]. The main properties of RNS were presented in the Chinese Remainder Theorem as a solution to the system of linear comparisons modulo [17]. RNS received its modern representation in the works of [PERSON] [18] and [PERSON] [16]. RNS can also be thought of as performing modular arithmetic on a residue ring. To represent the number \(X\) in RNS, it is necessary to obtain a set of relatively prime numbers \(\{p_{1},p_{2},\ldots,p_{n}\}\), which are called RNS moduli. Then, you need to obtain the remainder by dividing \(X\) by each modulo from the set, resulting in a set of remainders \(\{x_{1},x_{2},\ldots,x_{n}\}\). Thus, RNS is a system of comparisons. \[ \begin{array}{l}x_{1}\equiv X(mod\ p_{1}),\\ x_{2}\equiv X(mod\ p_{2}),\\ x_{n}\equiv X(mod\ p_{n}),\end{array} \]  
whose solution is the original number. This representation of the number has certain advantages. Since a sufficient condition for a correct mapping of a number in RNS is that \(X\in[0,P)\), where \(P=\prod_{i=1}^{n}p_{i}\)--is the RNS basis. To return a number to the positional system, you must use the following formula:  
\[X=\left|\sum_{i=1}^{n}x_{i}\cdot B_{i}\right|_{P} \tag{1}\]  
where \(B_{i}\)--orthogonal basis and \(B_{i}=\left|P_{i}^{-1}\right|_{p_{i}}\cdot P_{i}\) where \(P_{i}=\frac{p}{p_{i}}\), \(\left|P_{i}^{-1}\right|_{p_{i}}\)--inverse multiplicative element of \(P_{i}\) modulo \(p_{i}\). We can represent a number up to 128 bits in four 32 bits, or eight 16 bits, etc. This allows you to control the limit on the size of numbers on 32/64 bit systems. In addition, each pair of remainders \(x_{i}\) and modulus \(p_{i}\) is independent of the others. This allows information to be processed in parallel. In addition, it allows you to use the self-correcting properties of the system. In the 1970s, after describing the basic principles of RNS, [PERSON] proposed a special function for a given number system based on the positional characteristic of a number, calling it the core function, or the Akushsky Core Function (ACF) [19]. During their research, [PERSON] and [PERSON], in parallel with the study of the rank of a number, proposed another positional characteristic of a number, the so-called core of a number. In articles [19; 20], the considered core was presented. Investigations of the core function originate from the so-called Lagrange branch [21] of the low positional system [22], based on the following comparison:  
\[\lambda_{1}x_{1}+\lambda_{2}x_{2}+\cdots\lambda_{n}x_{n}\equiv W(mod\ D)=rD+W,\]  
where \(D\)--low position system modulo, \(r\)--number rank, \(\lambda\)--system coefficients, and \(W\) or \(w_{i}\)--weights. Studies have shown that these weights can take on various values, both negative and greater than the modulus \(D\) itself--which, by analogy with \(P\), is a constraint. Further research by [PERSON] showed that certain restrictions can be set for the values of \(W\). The main characteristic of this function is the so-called range core \(C(P)\). This value is dynamic and, like the number \(X\), must be in the range \(X\in\overline{1,P}\). In [20], [PERSON] and [PERSON] established that, to simplify the practical implementation of the core function, \(C(P)\) is equal to the largest modulo from the set, or their product. Then, if \(0<w_{i}<C(P)\), then the core function is monotonically increasing, which expands the range of applicability of the core function. To determine the positional characteristic of a number represented in RNS using ACF, it is necessary to introduce such a concept as the core of an orthogonal basis--\(C(B_{i})\). Which is located as follows:  
\[C(B_{i})=B_{i}\cdot\frac{C(P)}{P}-\frac{w_{i}}{p_{i}}, \tag{2}\]  
Then you can obtain the positional characteristic using the following formula:  
\[C(X)=\left|\sum_{i=1}^{n}x_{i}\cdot C(B_{i})\right|_{C(P)}. \tag{3}\]  
Formula (3) has similarities with Formula (1).