Factorial Numbers and Their Practical Applications

This paper develops the theory of factorial numbers formed by positional number systems with mixed-base factorial number systems. The fundamentals of the theory of factorial numbers are outlined, a method for obtaining them is given, their range is determined, and noise immunity is assessed. The addition and subtraction operations on factorial numbers are considered, and examples of their implementation are given. Methods for transitioning from positional numbers to factorial numbers and back are proposed. Methods are given for converting factorial numbers into permutations and permutations into factorial numbers. The operations of adding and subtracting permutations using factorial numbers are described. The noise immunity of permutations and the ability to detect and correct errors are assessed. The prospects of using permutations formed by factorial numbers for constructing noise-resistant ciphers are shown.