Author(s): Hussein Rabab
This article shows new properties of positive integers related to the sum of the deviations of the products of dividing the positive number by each of its factors, as well as the positive integer with only three divisors called the perfect square with a square root as a prime number, and the article also shows the building scheme a computer application to find the number of divisors of a number, and to classify the number into prime and composite.
A composite number is that positive integer that can be formed by multiplying two positive numbers less than or equal to it. In addition, it is a positive integer that has at least one divisor other than 1 and itself [1,2]. Every positive integer, prime number, or 1, so the set of constituent numbers is not prime and the number one is not prime because it has only one divisor. For example, the integer 15 is a complex number because it is the result of multiplying two smaller integers of size 3 x 5. The integers 3, 5, and 7 are not composite numbers because they can only be divided by 1 and itself and are called a prime number [1].
The analysis of the number into its factors is from positive integers, without the presence of the remainder as a result of the process of dividing the number by each of these factors, and when all these conditions are met, these numbers are called the divisors of the number [2].
The topic of the current article studies the sum of the deviations of the products of dividing a positive integer by its various divisors. This topic has not been addressed before by scientists and researchers in the field of mathematics. The article also links the number of divisors to any number and gives a new rule for the prime number.
A divisor, or factor, is a number that divides evenly into a larger integer, and the Set up the equation for determining the number of divisors, or factors, in a number [3]. The equation is d(n)=(a+1) (b+1)(c+1) , where a, b, c is equal to the number of divisors in the number n, and , b, and c are the exponents in the prime factorization equation for the number [4].
There are several methods for finding the denominators of a number, including the simple method, which is based on passing all digits from 1 to the given number and checking that the given number is divisible by those digits, and the pairs method, which is based on when binary denominators are written as pairs that are The product of the two numbers in each pair equals the specified number, so that one number is taken from each pair.
You might have less than three or more than three exponents. The formula simply states to multiply together whatever number of exponents you are working with.
If the prime factorization of a number N is N = k 2 x m b x j c , where k, m, and j are different prime numbers, and a, b, c are natural numbers. then the total number of divisors of N is (a+1) (b+1) (c+1) [5].
A prime number is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways of writing it as a product, 1 x 5 or 5 x 1, involve 5 itself. However, 4 is composite because it is a product (2 x 2) in which both numbers are smaller than 4. Primes are central in number theory because of the fundamental theorem of arithmetic: every natural number greater than 1 is either a prime itself or can be factorized as a product of primes that is unique up to their order [6].
The researcher studied and examined many numerical examples in order to reach relationships that link the number system, which constitute the basic building block in mathematics, as the researcher started with the examples listed below.
From below examples
The sum of the absolute differences between the results of dividing a number by its divisors, respectively
The sum of the absolute differences between the results of dividing a number by its divisors, respectively
The sum of the absolute differences between the results of dividing a number by its divisors, respectively
From the above examples we conclude this rule
The sum of the absolute difference between the real and non-real
divisors of a number is -1
The sum of the absolute difference between the denominators of
a real number is (the number divided by 2)-2
Any number whose real and non-real divisors sum to 3 is a perfect
square of a prime number
The sum of the absolute difference between the real and non-real divisors of a number is -1
The sum of the absolute differences between the results of dividing a a positive number by its divisors, respectively is equal to that number _ 1
e., If N an integer positive number, ai is its divisor, n the number
The sum of the absolute difference between the results of dividing a positive integer by a divisor without 1 and N is equal to this number divided by the least divisor of the number _ the least divisor of the number
Finding a computer application in order to determine the number of divisors of the number so that the dividend is a positive integer and the result of the division is an integer, and thus the number that has only two divisors is a prime number, and the number that has three divisors is a composite number and it is also a perfect square (for a prime number) That is, the square root of a prime number. Through this application, we can know the number of denominators of the number, as well as knowing the prime number from the non-prime
First: Enter the number (positive integer) into the application
Second: The application automatically divides the number by all
numbers that are less than or equal to half of the number entered
into the application.
Third: If there is any number that divides the number entered, the
number will write down these numbers starting with the number 1
Fourth: These divisors are counted and 1 is added to them to be
the sum of the divisors of this number, and they appear on the
screen or the calculator through a button whose symbol is f (n).
Fifth: Classify the number through a special button, and its symbol
is c (n), and the number is classified into: prime number, a number
with 3 divisors (a perfect square of a prime), a number with more
than 3 divisors
Figure 1: Shows the Computer Application and its Working Mechanism
The article presented new rules added to the theory of numbers, which benefit scientists and researchers in the field of mathematics and various sciences, and we offer researchers and scientists new rules that they can build on and employ by those interested in programming and new technology, A scheme that programmers can apply practically so that students and scholars can benefit from this application in their studies and applications.
