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How is this proof finished? I was told it is proven

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  1. Sep 3, 2015 #1
    Prove that the product of four consecutive natural numbers cannot be the square of an integer.

    So let n be a natural number. So f(n) = n(n+1)(n+2)(n+3)

    n --- 1 --- 2 --- 3 --- 4 --- 5 --- 10
    f(n)-24--120-360-840-1080-17160

    The conjecture I want to prove is F(n) + 1 is always a square.

    n(n+1)(n+2)(n+3) = (n2+3n)(n2+3n+1)

    so

    [(n2+3n-1)+1)][(n2+3n+1)-1]+1

    and because a2-b2 = (a+b)(a-b)

    [(n2+3n+1)2-1)+1] = (n2+3n)2

    So this proves that f(n) + 1 is in fact a square.

    My question is how does this prove that f(n) is NOT a square? I mean it seems obvious, but I am trying to learn to prove things.
     
  2. jcsd
  3. Sep 3, 2015 #2

    TeethWhitener

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    Think about the size of the gap between consecutive square numbers.
     
  4. Sep 3, 2015 #3
    This is exactly what a friends said. The size gap. Ill think on this the rest of the day and see if I can find why that matters. Thanks.
     
  5. Sep 3, 2015 #4

    TeethWhitener

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    Just try a test case: what's the gap between x2 and (x-1)2?
     
  6. Sep 6, 2015 #5
    By the way, n(n+1)(n+2)(n+3) = (n2+3n)(n2+3n+2), so f(n)+1 = (n^2+3n+1)^2
    The argument is correct, however.
     
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