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Proof about product of 4 integers

  1. Jun 28, 2014 #1
    1. The problem statement, all variables and given/known data
    Prove that the product of four consecutive integers is always one less than a perfect square.
    3. The attempt at a solution
    I tried looking at the product [itex] (n-1)(n)(n+1)(n+2)=x^2-1 [/itex]
    but i couldn't seem to get anything useful out of it. I added one to both sides .
    I tried to see if i could some how show that the product of the four integers plus 1 had an odd number of divisors but I couldn't see a way to do it. I did notice that these numbers are never divisible by 2,3,or 4, and alot of the time they produce a prime, all though I didn't check very many.
    And when it wasn't a prime it was divisible by 5. I was trying to think of a way to do this with Pythagorean triples but im not sure.
     
  2. jcsd
  3. Jun 28, 2014 #2

    AlephZero

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    Look harder :smile:. Don't overthink the problem with the other ideas you tried. This doesn't depend on any theorems in number theory!

    It's hard to give a hint without telling you the answer, but think about what you can you do with ##x^2-1##.
     
  4. Jun 28, 2014 #3
    are you talking about factoring it into (x+1)(x-1) I not sure how that helps I did look at that. we do know that x needs to be odd. so x=2y+1 so (2y+1+1)(2y+1-1)=(2y+2)(2y)=4y(y+1) I still dont see how it will work. I guess Ill keep trying stuff. thanks for your post.
     
  5. Jun 28, 2014 #4

    AlephZero

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    Don't bother about x being odd or even. Just factor the left hand side into the same form.
     
  6. Jun 28, 2014 #5
    So we have (n^2-1)(n)(n+2)=x^2-1 I just dont see what this does, I can see how we have similar things on both sides. but it seems like we still have 2 extra terms on the left side.
     
  7. Jun 28, 2014 #6

    AlephZero

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    Try taking different pairs of factors from ##(n-1)(n)(n+1)(n+2)## to get ##(an^2 + bn + c -1)(an^2 + bn + c + 1)##.

    Or, factorize ##(n-1)(n)(n+1)(n+2) + 1## into a perfect square, like ##(an^2 + bn + c)^2##. It's not hard to guess the values of ##a## and ##c##.
     
  8. Jun 29, 2014 #7
    ok thanks for your help, I see it now
     
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