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Unique factoring over E-Primes criteria

  1. Jan 18, 2016 #1

    RJLiberator

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    1. The problem statement, all variables and given/known data
    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.

    2. Relevant 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

    3. 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,...
     
  2. jcsd
  3. Jan 18, 2016 #2

    Samy_A

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    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?
     
  4. Jan 18, 2016 #3

    RJLiberator

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    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.

    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.

    :/
     
  5. Jan 18, 2016 #4

    Samy_A

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    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?
     
    Last edited: Jan 18, 2016
  6. Jan 18, 2016 #5

    RJLiberator

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    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.
     
  7. Jan 18, 2016 #6

    Samy_A

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    Looks good. f can also be 1.
     
  8. Jan 18, 2016 #7

    RJLiberator

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    Indeed! Thank you for your help, Samy_A.
     
  9. Jan 18, 2016 #8

    Samy_A

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