1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

How to disprove (m^2)+m+1=(n^2)?

  1. Feb 26, 2017 #1
    1. The problem statement, all variables and given/known data
    "Prove: There exists no ##m,n∈ℕ## such that ##m^2+m+1=n^2##."

    2. Relevant equations


    3. The attempt at a solution
    I basically rewrote it as:

    ##m^2+2m+1=n^2+m##

    or

    ##(m+1)^2=n^2+m##,

    and subtracting ##n^2##, I get

    ##(m+1-n)(m+1+n)=m##.

    Then I divided both sides of the equation by ##m## to get:

    ##(1+\frac{1}{m}-\frac{n}{m})(m+n+1)=1##.

    I then argued that the only way for this to be possible was (1) if both terms on the left were ##1##, which would contradict the fact that ##m## and ##n## were positive integers, because if they were, the second term would be greater than one; or (2) if the two terms were multiplicative inverses of each other. The latter is the one I had trouble with, because it led to some kind of circular reasoning. Basically, I ended up with the exact same problem which I had to once again disprove. Can anyone provide me with any hints on how to approach this proof?
     
  2. jcsd
  3. Feb 26, 2017 #2

    fresh_42

    Staff: Mentor

    I have no idea how to proceed from the point you got stuck. But I think, it could be done by solving ##m^2+m+(1-n^2)=0## for ##m## and then analyze the root. Also consider, that beside the root, the summand ##-\frac{1}{2}## that you get has to vanish, too.
     
  4. Feb 27, 2017 #3

    pasmith

    User Avatar
    Homework Helper

    Set [itex]m^2 + m + 1 - n^2 = (m - m_1)(m - m_2)[/itex]. What must [itex]m_1 + m_2[/itex] be?
     
  5. Feb 27, 2017 #4

    PeroK

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    Here's another idea. It seems to me that adding ##m +1## is not enough to get you from ##m^2## to the next square.
     
  6. Feb 27, 2017 #5
    Thanks for the help. I finished it, by the way.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted