New types of finite groups and generated algorithm to determine the integer factorization by Excel

Abstract

In this work, new types of finite subgroups are given, they are called prime gaps additive subgroups (S, +p) of Zp and prime gaps multiplication subgroups (S, p) of Zp*. Also, new practical algorithm to track prime factors for any composite numbers is introduced. This new method depends on the general rule for prime numbers that has one variable (n). The new rule is very important because it exactly gives the number of primes for any interval of integer numbers. Also, the large gaps between n-th primes are given in this paper. Next, this formula is used to make application by Excel in order to exhibit the results for any integer number if it is prime or not at polynomial time. By this new method, we can find straightforwardly more than (97.5%) of the composite numbers while the rest of the percentage needs polynomial-time. In addition, we can use this rule to exhibit all prime numbers between the smallest prim number and any number x in sequence list. Finally, by this work the following points are considered; The distribution for any sequence of prime numbers is given, Our results better than Riemann Hypothesis (see Tables 3 and 4), For any large integer we can find its factorization in polynomial time by a classical computer.