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Small number theory problem

  1. Dec 8, 2014 #1
    1. The problem statement, all variables and given/known data
    This is a problem I had as a margin note in an old notebook that I will recycle. I want write it using LaTeX. Problem is that I also want to write it using "proper" math notation instead of English words.

    Firstly, I got this:
    [tex]\textrm{Proof that }\nexists x, y \in \mathbb{N}^* | x + y + xy = 36[/tex]

    Which seems to be correct. (Should 'x, y' be between parentheses?)

    2. Relevant equations
    None.

    3. The attempt at a solution
    [tex]\begin{gather}
    \begin{aligned}
    x + y + xy
    &= x + xy + y \\
    &= (x + 1) (y + 1) - 1 \\
    &\therefore (x + 1) (y + 1) = 37 \\
    &\therefore x = 0 \lor y = 0
    \notag
    \end{aligned}
    \end{gather}[/tex]

    Now the question: how to finish this? Obviously x = 0 or y = 0 is false as neither x nor y can be 0 as the set of all N but 0 does not include 0. How do I write this?
     
  2. jcsd
  3. Dec 8, 2014 #2

    HallsofIvy

    User Avatar
    Staff Emeritus
    Science Advisor

    That's pretty good but you need to say a bit more about what you are doing. You should, for example, write
    that since 37 is a prime number either x+ 1= 1 and y+ 1= 37 or x+ 1= 37 and y+ 1= 1.

    Then look at the two cases: if x+ 1= 1 then x= 0. But that is a contradiction because x must be a positive integer
    If y+ 1= 1 then y= 0. But that is a contradiction because y must be a positive integer.
     
  4. Dec 8, 2014 #3
    All right, I will write more! But first you need to teach me how (or point me a not too long book / guide / tutorial).

    Which symbols do I use? I remember some '#' for "absurd" or "impossible" on paper, but I have no idea on how to do it here. Neither if that was correct.
     
  5. Dec 8, 2014 #4

    Mark44

    Staff: Mentor

    What I would do is start by assuming that x + xy + y = 36.
    Then x + xy + y + 1 - 1 = 36
    ##\Rightarrow## (x + 1)(y + 1) = 37
    Since 37 is prime, its only factors are 1 and 37, so there's your contradiction.

    As far as I can tell, there's no LaTeX symbol that specifically denotes "contradiction." Some people use # for this purpose. A symbol I've seen for a long time is two arrows with their heads touching, like this: ##\Rightarrow\Leftarrow##.


     
  6. Dec 8, 2014 #5
    Were you going to comment something else then changed your mind?

    It follows that x = 0 or y = 0 because either x + 1 = 1 or y + 1 = 1.

    Will look up that symbol you wrote.
     
  7. Dec 8, 2014 #6

    Mark44

    Staff: Mentor

    At first I didn't understand how you arrived at x = 0 or y = 0, but saw you had explained it following your proof, so I removed my question.
    It's two symbols: # #\Rightarrow\Leftarrow # #
    The spaces between the # characters is to keep the above from rendering...
     
  8. Dec 8, 2014 #7
    I know how to make it, I just want to look it up in a book or Wikipedia to get some usage examples.

    In fact, as you may know, unless you use an image with an obscure filename*, I just need to quote you in order to see your [itex]\LaTeX[/itex] code.

    * Even if it was an image, there are still ways to try to get LaTeX from it.
     
  9. Dec 8, 2014 #8

    Mark44

    Staff: Mentor

  10. Dec 8, 2014 #9
    The best thing I got from this thread. Thanks.
    It makes me sad that Q.E.A. is not widely used (what inclines me towards avoiding it).
    Anyway, this is a picture of what I rendered. Suggestions wanted.
    upload_2014-12-8_15-21-6.png

    As Mark somehow suggested, I added one not-so-obvious extra step.
     
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