## A Set (GAIISUU) of Generalizing Prime Numbers

*Hirotaka Ebisui*

`hirotaka.ebisui@nifty.ne.jp`

**Oval Research Center**

Japan

### Abstract

So far, Prime numbers have been good material of research for people who like Mathematics. And, they consist of infinite numbers, and we can obtain Prime numbers by the sieve of Eratosthenes. Moreover, we have found some large Prime number which is called Mersenne Prime number expessed by 2^n-1(n=6972593), using Computer. For example, 2^107-1=162259276829213363391578010288127

On the other hands, Prime numbers make composite numbers
using multiplicator among two or more pairs of Prime numbers. And their numbers are all different numbers and are different from Prime numbers. This property can be extended to define a set which is called GAIISUU-IKI.

Here, we define GAIISUU as a extension of Prime number like this.

**[DEFINITION 1]**

On any two elements gi, gj of a subset G in Natural
Numbers, we make the sum (Sij)of gi and gj. Then, Sij(=gi+gj) are all different numbers for different pair (i, j), (i<=j)of Natural number, at the same time, Sij are not contained in G.

G is called GAIISUU-IKI with Evaluation number 2 on
ADDITION,
and an element of G is GAIISUU, and Sij is called as GAISUU, and the
SET of Sij is called GAI-IKI.

So far, We obtain some examples of GAIISUU, and some properties and an expectation of G.

And we show a table of GAIISUU-IKI with
Evaluation number 2 on ADDITION, etc.

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