Nonunique factorization theory

In summary, the conversation focuses on Nonunique Factorization Theory, a topic in Number Theory. The concept of an Arithmetic Congruence Monoid is mentioned, which is a set of integers that can be factored in multiple ways. Some examples and counterexamples are provided, and the conversation ends with a suggestion to do further research on the topic online.
  • #1
camilus
146
0
Im applying to an REU in San Diego State where the focus will be Nonunique factorization theory but I'm clueless as to what this actually is. Does anybody know anything about this?

The general area of study will be Number Theory, specifically Nonunique Factorization Theory. An Arithmetic Congruence Monoid, or ACM, is a multiplicatively closed subset of the naturals, such as {6, 36, 66, 96, 126, 156, 186, ...} with {1} included for convenience. We are concerned with the multiplicative structure. For example, 66*66=6*726, a factorization into "primes" in two different ways.
 
Physics news on Phys.org
  • #2
I know nothing about this, but brush up of your knowledge of basic number theory, including primes and factorizations, and you should be fine. This is an REU, so they don't expect you do be an expert, especially on a relatively obscure field such as this. It sounds very interesting. Enjoy!
 
  • #3
I have heard of a construction that gives a set of integers that cannot be factorized uniquely. Consider the numbers that have an even number of distinct primes as factors. It is not hard to see that the primes of this set are those integers that have at most two distinct prime factors. Let p and q be two ordinary prime numbers. Then, p^2*q^2=(pq)^2. Since p^2, q^2, and pq have no factors in this set, p^2*q^2 is a counterexample to unique factorization in this set. I suspect that something similar may happen when an even number is replaced with multiples of any integer. Maybe something like this is what you were looking for? You might be looking for something a great deal more advanced. This is just what I've heard that might be related to what you want.

Um, something is happening. I am not reposting this. I think something's going wrong with my computer. I keep clicking edit and i get a new post. SORRY!
 
  • #4
Mathguy15 said:
I have heard of a construction that gives a set of integers that cannot be factorized uniquely. Consider the numbers that have an even number of distinct primes as factors. It is not hard to see that the primes of this set are those integers that have at most two distinct prime factors. Let p and q be two ordinary prime numbers. Then, p^2*q^2=(pq)^2. Since p^2, q^2, and pq have no factors in this set, p^2*q^2 is a counterexample to unique factorization in this set. I suspect that something similar may happen when an even number is replaced with multiples of any integer. Maybe something like this is what you were looking for? You might be looking for something a great deal more advanced. This is just what I've heard that might be related to what you want.

EDIT:SORRY, I did not carefully read your post. You are looking for something specifically about the REU. Sorry for my redundant ramblings.
 
  • #5
Mathguy15 said:
I have heard of a construction that gives a set of integers that cannot be factorized uniquely. Consider the numbers that have an even number of distinct primes as factors. It is not hard to see that the primes of this set are those integers that have at most two distinct prime factors. Let p and q be two ordinary prime numbers. Then, p^2*q^2=(pq)^2. Since p^2, q^2, and pq have no factors in this set, p^2*q^2 is a counterexample to unique factorization in this set. I suspect that something similar may happen when an even number is replaced with multiples of any integer. Maybe something like this is what you were looking for? You might be looking for something a great deal more advanced. This is just what I've heard that might be related to what you want.

SORRY, I did not carefully read your post. You are looking for something specifically about the REU. Sorry for my redundant ramblings.

edit:Sorry for the repost.
 
  • #6
do an internet search for Non Unique factorization. A lot of material is available out there.
 

1) What is nonunique factorization theory?

Nonunique factorization theory is a branch of mathematics that deals with the study of numbers that can be broken down into smaller factors in more than one way. It examines the properties and patterns of numbers that do not have a unique factorization, meaning they can be expressed as a product of prime numbers in more than one way.

2) How is nonunique factorization theory different from unique factorization theory?

The main difference between nonunique factorization theory and unique factorization theory is that in unique factorization theory, every integer can be written as a unique product of prime numbers, whereas in nonunique factorization theory, some integers can have more than one possible factorization. Additionally, unique factorization theory is applicable to all integers, while nonunique factorization theory is mainly concerned with special types of integers, such as Gaussian integers or algebraic integers.

3) What are some examples of numbers that have nonunique factorizations?

Some examples of numbers that have nonunique factorizations include perfect squares (numbers that are the square of an integer), such as 4, 9, and 16, and numbers that have multiple prime factors with the same value, such as 12 (2 x 2 x 3) and 18 (2 x 3 x 3).

4) What are the applications of nonunique factorization theory?

Nonunique factorization theory has applications in various fields, including cryptography, number theory, and algebra. It is also used in the study of quadratic forms and Diophantine equations, which have applications in physics and engineering. Additionally, nonunique factorization theory has connections to other branches of mathematics, such as algebraic geometry and algebraic number theory.

5) What are some open questions in nonunique factorization theory?

Some open questions in nonunique factorization theory include finding a complete characterization of integers that have nonunique factorizations, determining the distribution of these numbers in different sets of integers, and studying the behavior of nonunique factorizations in different algebraic structures. There is also ongoing research in finding new applications of nonunique factorization theory and connections to other areas of mathematics.

Similar threads

  • MATLAB, Maple, Mathematica, LaTeX
Replies
2
Views
2K
  • MATLAB, Maple, Mathematica, LaTeX
Replies
1
Views
2K
Replies
21
Views
3K
Back
Top