Abstract.

The problem of multiplicative differentiation has stagnated in recent times. However, multiplicative differentiation has a great theoretical potential at the field of number theory. The current draft of graduation paper introduces multiplicative differentiation; consequently, it provides the history of the research, describes properties and theoretical applications of the mapping. In addition to that, it calls on to pay more attention to number theory, as a subset of modern applied algebra. Finally, the line of activity is given.

Formulation of the problem

Multiplicative differentiation is a specially constructed sphere of number theory, which did not achieve its deserved attention. Since the object of the research is very nonobjective, it was sidelined by theories which have practical applications in economics. The following paper is designed to fill this omission.

Actuality

Number theory is a branch of pure mathematics devoted primarily to the study of the integers. There is no secret to anyone that since ancient times mathematicians were desperate to describe structure (or, better to say, distribution) of integers. That is why theoretical background dedicated to prime numbers was developed. Up to now, a lot of conjectures remain unproved, (such as famous Prime Twins Conjecture), or proved hard (such as Fermat’s Last Theorem).

Unfortunately Number Theory is strictly theoretical (not to consider empirical verification). Besides, sometimes the Conjecture itself is useless from the scientific point of view (Fermat’s Last Theorem is the most obvious example); however the method created for the proof is very demanded (method of trigonometric sums created by Vinogradof for weak Goldbach’s conjecture proof) . That is the reason to reformulate number theory problems to the language of other fields of mathematics. Multiplicative differentiation is a map which associates a natural number with the operation of derivative calculation.

The Aims

The aim of the paper is to consider the basic properties of this map and to show how to generalize the notion to the case of rational and arbitrary real numbers. We make some conjectures and find some connections with classic number theory problems, such as Goldbach's Conjecture and the Twin Prime Conjecture, which basically needs absolutely no introduction . Finally, we solve the easiest associated diferential equations with the usage of programming

and calculate the generating function. After that, we shall make approximating predictions on achieved results. Finally, we try to describe the function which can have similar value as the calculus solution.

Object of the research

The object of the research is a set of natural, integer, rational and real numbers.

Practical value

The anticipated results can help to make progress in proof of fundamental problems of number theory. In addition to that, this work can attract attention to almost forgotten application of number theory.