The discussion centers around Nonunique Factorization Theory within the context of Number Theory. Participants explore the concept, its implications, and related examples, particularly in relation to an upcoming REU program.
Participants do not reach a consensus on the specifics of Nonunique Factorization Theory, and multiple viewpoints and examples are presented without resolution.
Some discussions involve assumptions about the nature of prime factors and their role in unique versus non-unique factorization, but these assumptions are not universally agreed upon.
Individuals interested in advanced number theory, particularly those exploring concepts related to factorization and its implications in mathematical structures.
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.
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.
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.