# Prospects for the Use of Quasi-Mersen Numbers in the Design of Parallel-Serial Processors
## 1 Introduction  
At present, there is considerable interest in increasing the productivity of computing equipment [1, 2, 3, 4]. One approach to solving this problem is the use of parallel-sequential calculations [5, 6], which are implemented, among other methods, using the residue number system (RNS) [7, 8]. In this case, each integer \(x\) is represented as \((x_{1},x_{2},\ldots,x_{n})\), where \(x_{i}=x\ mod\ m_{i}\), \(m_{i};i=1,2,\ldots n\) are integers that form the RNS [9]. The moduli \(m_{i}\) are generally assumed to be pairwise coprime. The advantage of RNS is its ability to independently operate with individual remainders \(x_{i}\), in particular, \(z_{i}=x_{i}+y_{i}\), if \(z=x+y\) (similarly for multiplication). Such properties allow the move from sequential to parallel-sequential computations, since, when \(x<M=m_{1}m_{2}\ldots m_{n}\), each set of residues \((x_{1},x_{2},\ldots,x_{n})\) represents exactly one integer \(x\). Consequently, as long as this inequality is satisfied, computations using integers can be reduced to computations in RNS. From the perspective of computer technology, this limitation is not significant. In any case, the range within which standard processors operate is also limited (e.g., 16-bit adders). The question of choosing a concrete set of moduli \(m_{i}\) from the perspective of computer technology is not trivial. In particular, in the case that all numbers \(m_{i}\) are prime, i.e., \(m_{i}=p_{i}\), the computations corresponding to each component \(x_{i}\) of the number \(x\) are de facto performed in the Galois field \(GF(p_{i})\). Currently, numerous different types of electronic circuits are known that are designed to perform addition and multiplication operations in such fields, i.e., adders and multipliers modulo a prime number [10; 11; 12; 13]. They differ significantly from one another in their design, which, moreover, significantly depends on the choice of the modulus, as is proven, in particular, by report [14]. Moreover, it was noted in [15] that different Galois fields have different specifics. Consequently, there is every reason to assert that the choice of a set of moduli \(m_{i}\) should be determined, first, by the specifics of the problems being solved. The technical implementation of this approach is facilitated by the widespread use of microcircuits with a reconfigurable logical structure [16; 17; 18]. Accordingly, it is permissible to raise the question of developing calculators oriented towards solving a certain range of problems. In [19], it was shown that for calculating digital convolutions, it is convenient to use an RNS corresponding to a set of prime numbers \(p_{i}\), such that \(p_{1}p_{2}\ldots p_{n}+1=P\), where \(P\) is also a prime number. In this case, it becomes possible to reduce the discrete scale corresponding to the result of calculating the convolution to a discrete scale corresponding to the original function, without using fractional values. This paper shows that a serial-parallel processor, comparable in bit depth to a 16-bit binary processor, can be implemented based on an algorithm built on the residue number system. The distinctive feature of this processor is the use of the first four quasi-Mersenne numbers, i.e., prime numbers representable as \(p=2^{k}+1\). The basis of such a processor is an adder modulo a quasi-Mersenne prime number. A specific electronic circuit of such an adder is presented, which can be called a trigger, since it can be in two different states. Its advantage over known analogs is the simplicity of the design. It should be noted that Mersenne primes are currently widely used in information technology. In particular, these numbers are used to generate pseudorandom numbers [20]. Specifically, a pseudorandom number generator called the Mersenne Twister, developed in 1997 by Japanese scientists [PERSON] and [PERSON] [21], is directly based on the use of Mersenne numbers. Some reports demonstrating the expediency of using such numbers for data transmission are presented in the current literature [22; 23], etc.