# Properties of Composites Composed of Two Odd Prime Factors

1. Nov 21, 2005

### Edwin

I have been working on constructing a method to factor composite numbers composed of two odd prime numbers a and b. As a result, I have been experimenting numerically with various patterns to see if I could find some patterns that would reveal information about composites of two prime numbers that I might use in furthering my progress toward my ultimate goal of constructing a general factoring algorithm that involves solving a system of algebraic equations to factor the composite. I have yet to find a method, but have decided to post what I have so far over the next couple of weeks for others with fresh eyes to take a look at and critique. I will start by laying out a series of theorems in the first post, and then in following posts will elaborate on each theorem and attempt to prove them. Any suggestions, ideas, and questions are most welcome at any time. I also ask for ideas on how I may be able to better present these ideas in ways that would be more effective. Thanks for all of your assistance!

Sincerely,

Edwin

Theorem 1)

Every number $$C$$ that is an odd number composed of two prime numbers $$a$$ and $$b$$ such that

$$\begin{gather*}C = a*b\end{gather*}$$

can be written as the sum of n consecutive odd integers.

Theorem 2)

If $$C$$ is an odd number composed of two prime factors $$a$$ and $$b$$, then the number of consecutive odd integers required to add up to $$C$$, is equal to the smaller of the two prime factors of $$C$$.

Theorem 3)

The larger of the two prime factors of $$C$$ is the average to the first and last number in the consecutive sum of odd numbers that add up to $$C$$.

Theorem 4)

If $$C$$ is an odd number that is composed of two prime factors $$a$$ and $$C$$, then there exists atleast one non-arbitrary perfect square that can be used to factor $$C$$ through a finite number of algebraic operations.

Theorem 5)

If $$C$$ is an odd number that is composed of two prime factors $$a$$ and $$b$$, then the non-arbitrary perfect square that can be used to factor $$C$$ via finite number of algebraic operations is the first perfect square acquired by adding a consecutive number of odd integers starting with the number 1 to $$C$$ and is equal to the average of the prime factors of $$C$$ quantity squared.

Example:

77 is a composite $$C$$ composed of prime factors 7 and 11.

77+1 = 78
77+1 +3 = 81

81 is the first perfect square aquired by adding consecutive odd integers to composite number 77, and ((7 + 11)/(2))^2 = 9^2 = 81.

Over the next few posts I will briefly elaborate on each of the above theorems. I will try to do so concisely in a brief post for each of the theorems above that can be read quickly and percieved quickly and clearly.

Best Regards,

Edwin G. Schasteen