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Math Help - Proof by contradiction help

  1. #1
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    Proof by contradiction help

    The sum of the squares of 3 consecutive integers cannot have remainder -1 on division by 12.

    Can anyone help me prove this by contradiction?
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  2. #2
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    Re: Proof by contradiction help

    The idea of a proof by contradiction is to assume the negation of what you need to prove and derive a contradiction.

    0. Assume that there exists some integer n such that the sum of squares of n, n + 1 and n + 2 has remainder -1 when divided by 12.

    1. Write the sum of squares of n, n + 1 and n + 2. Simplify.

    2. Write the equation saying that the result of step 1 has remainder -1 when divided by 12. Simplify as much as possible.

    3. Consider the cases when n is even and when n is odd and show that the equation from step 2 cannot hold in either case. Show that one side is even and the other is odd, or that one side is divisible by 4 and the other is not, or something like this.

    If you need more help, post the results of steps 1 and 2.

    Edit: Step 3 of my initial solution was incorrect, but it is now updated.
    Last edited by emakarov; December 7th 2012 at 05:13 AM.
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  3. #3
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    Re: Proof by contradiction help

    Using n-1, n, n+1 would probably lead to an easier expression.

    Also, the task will be to show that you reach a contradiction, regardless of the value of n.

    Finding one particular value for n that gives a contradiction will not suffice.
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