Unique factoring over E-Primes criteria

[RJLiberator]

Homework Statement

Let E denote the set of positive even integers. An element p ∈ E is called an E-prime if p cannot be written as a product of two elements of E. Determine a simple criteria for when elements of E can be uniquely factored into a product of E-primes.

Homework Equations

Some E-primes from my understanding: 2, 6, 10, 14, 18, 22, 26, 30, 34, 38, 42, 46, 50
Some terms that can be uniquely factored into e-primes: 4, 12, 20, 28, 44, 52

The Attempt at a Solution

I've spent much more time on this problem then I should have.

I can clearly see that all terms that can be uniquely factored into e-primes must be divisible by 4.
It seems like there is a common ratio of 8 between terms until we get to terms 36 and 60. 36 can be factored into 6*6 and 18*2 while 60 can be factored into 6*10 and 30*2. So these are clearly not unique.

All the terms that can be uniquely factored can not be divisible by 8, that makes sense since 8 = 4*2.

But beyond this, I don't really know what it means by criteria? Should I spit out an equation like 4+8n for n=all integers except 4,7,10,13,...

 

[Samy_A]

Homework Statement

Let E denote the set of positive even integers. An element p ∈ E is called an E-prime if p cannot be written as a product of two elements of E. Determine a simple criteria for when elements of E can be uniquely factored into a product of E-primes.

Homework Equations

Some E-primes from my understanding: 2, 6, 10, 14, 18, 22, 26, 30, 34, 38, 42, 46, 50
Some terms that can be uniquely factored into e-primes: 4, 12, 20, 28, 44, 52

The Attempt at a Solution

I've spent much more time on this problem then I should have.

I can clearly see that all terms that can be uniquely factored into e-primes must be divisible by 4.
It seems like there is a common ratio of 8 between terms until we get to terms 36 and 60. 36 can be factored into 6*6 and 18*2 while 60 can be factored into 6*10 and 30*2. So these are clearly not unique.

All the terms that can be uniquely factored can not be divisible by 8, that makes sense since 8 = 4*2.

But beyond this, I don't really know what it means by criteria? Should I spit out an equation like 4+8n for n=all integers except 4,7,10,13,...

You have established that an e-prime is divisible by 4 and not divisible by 8.
So if e is an e-prime, e=2².f, where f is an odd number.
What condition(s) does f have to satisfy in order for the factoring to be unique? What can you tell about the prime factors of f?

 

[RJLiberator]

You have established that an e-prime is divisible by 4 and not divisible by 8.

Well, the terms that can be factored into e-primes are divisible by 4 and not divisible by 8.
The e-primes are not divisible by 8 as well, but can not be divisible by 4.

So if e is an e-prime, e=2².f, where f is an odd number.

I'm not quite sure this holds. Let's see, E-primes are: 6, 10, 14, 18, but that doesn't quite fit into the equation.

:/

 

[Samy_A]

Yes, I was wrong, sorry.

An e-prime e=2.f, where f is odd.
A product of two e-primes is 4.f.g (f and g odd).
How can you guarantee that this factoring in e-primes is unique?

 

[RJLiberator]

That makes a lot of sense to me.

So since we can define an e-prime to be e = 2*f where f is odd.
And we see that the produce of two eprimes is e*e = 2*f*2*g = 4*f*g

We can guarantee that this factoring is unique when f or g is a prime number and the other is 1.

For example. If we were to let g and f both be prime numbers, say 7 and 5, then 4*7*5 = 140 which does NOT have a unique e-prime factorization as 2*70 (both eprimes) and 10*14 (both eprimes). So f and g can't both be prime numbers.
Now, if we let one of them be prime and the other be, even this also doesn't work. Take for example 13*4*2 we get 104 which does NOT have a unique e-prime factorization.So in conclusion,
products of two eprimes occur when 4*f*g where f is a prime number and g is 1. or more simply, 4*f where f is a prime number.

 

[Samy_A]

That makes a lot of sense to me.

So since we can define an e-prime to be e = 2*f where f is odd.
And we see that the produce of two eprimes is e*e = 2*f*2*g = 4*f*g

We can guarantee that this factoring is unique when f or g is a prime number and the other is 1.

For example. If we were to let g and f both be prime numbers, say 7 and 5, then 4*7*5 = 140 which does NOT have a unique e-prime factorization as 2*70 (both eprimes) and 10*14 (both eprimes). So f and g can't both be prime numbers.
Now, if we let one of them be prime and the other be, even this also doesn't work. Take for example 13*4*2 we get 104 which does NOT have a unique e-prime factorization.So in conclusion,
products of two eprimes occur when 4*f*g where f is a prime number and g is 1. or more simply, 4*f where f is a prime number.

Looks good. f can also be 1.

 

[RJLiberator]

Indeed! Thank you for your help, Samy_A.

 

[Samy_A]

You are welcome. Sorry for the confusion in my first post.