What is the Fundamental Theorem of Prime Factorization? ? Binary and Hexadecimal by 天風 20231127

This is Tenpu. Today, I would like to write about the idea of ​​prime factorization, which the blog owner uses in my hobby research.

Binary number

　First, let's talk about the first method. First of all, do you, readers, know what binary numbers are? Binary is one way to represent numbers. For example, let's represent the numbers 0 to 10 in binary.

0 = 0

1 = 1

2 = 10

3 = 11

4 = 100

5 = 101

6 = 110

7 = 111

8 = 1000

9 = 1001

10 = 1010

　The right side of each is a binary number. You can see that this is made up of 0 and 1 . Here, you can also see that when everything becomes 1, it is carried forward . For example, after 3=11 and 7=111, the numbers are moved up by one digit to 4=100 and 8=1000, respectively.

　At first glance, this concept of binary numbers seems unusual. However, the numbers we usually use, such as the numbers 0 to 9 in the example above, are called decimal numbers. When a decimal number reaches 9, it moves up one digit to 10, and when it reaches 99, it moves up one digit to 100.

　Of course, if we have binary and decimal numbers, we also have ternary and hexadecimal numbers. The idea is the same. For ternary numbers, numbers from 0 to 2 are used, and when all numbers reach 2, the numbers are carried forward; for hexadecimal numbers, numbers from 0 to F are used, and when all numbers become F, they are carried forward.

　Now you may be wondering, "What is F?" In hexadecimal numbers, 10 is represented as A, 11 as B, 12 as C, 13 as D, 14 as E, and 15 as F. This is because, for example, we know that 1F is 1x16+15, but without expressing 15 as F, we don't know if 115 means 1F or 1x16x16+1x16+5.

　Returning to the topic, this idea of ​​n-ary numbers is important in the blog owner's research, and I especially use binary numbers . How to use it is as follows. Represent the numbers 13 and 11 in binary.

13=8+4+0+1 → 1101　

11=8+0+2+1 → 1011

In this case, a decimal number 4 or 8 corresponds to a binary number 1, and a decimal number 0 corresponds to a binary number 0. This 8+4+0+1 is used in the calculation of prime factorization using the blog owner's method.