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Math Help - Problem 18

  1. #1
    Grand Panjandrum
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    Problem 18

    Two easy ones this week:

    1.Let n be a positive integer. Prove that the numbers n+2 and n^2+n+1 cannot both be perfect cubes.

    2. Which regular n-gons can be inscribed in a non-circular ellipse?

    RonL
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  2. #2
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    Quote Originally Posted by CaptainBlank
    1.Let n be a positive integer. Prove that the numbers n+2 and n^2+n+1 cannot both be perfect cubes.
    n+2=a^3
    n^2+n+1=b^3
    Thus,
    (n^2+n+1)(n+2)=a^3b^3=(ab)^3=m^3
    (n^2+n+1)((n-1)+3)=m^3
    (n^2+n+1)(n-1)+3(n^2+n+1)=m^3
    n^3-1+3n^2+3n+3=m^3
    (n^3+3n^2+3n+1)+1=m^3
    (n+1)^3+1^3=m^3
    Fermat's Last Theorem n=3.
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  3. #3
    Grand Panjandrum
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    Quote Originally Posted by CaptainBlack View Post
    Two easy ones this week:

    1.Let n be a positive integer. Prove that the numbers n+2 and n^2+n+1 cannot both be perfect cubes.
    If both n+2 and n^2+n+1 are both cubes then so is their product, but:

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

    but this is imposible as no two cubes of integers differ by 1.

    2. Which regular n-gons can be inscribed in a non-circular ellipse?
    A regular n-gon can be inscribed in a circle, but the vetices also lie on
    the non-circular ellipse. But a pair of distinct conics intersect at no more
    than four points, so n<=4.

    RonL
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    Quote Originally Posted by CaptainBlack View Post
    A regular n-gon can be inscribed in a circle, but the vetices also lie on
    the non-circular ellipse. But a pair of distinct conics intersect at no more
    than four points, so n<=4.

    RonL
    For some reason it seemed to me you where asking for which regular n-gons are constructable on a non-circular ellipse with a compass and straightedge.
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