Two easy ones this week:

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

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

RonL

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- Feb 5th 2007, 06:00 AM #1

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## Problem 18

Two easy ones this week:

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

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

RonL

- Feb 11th 2007, 10:05 AM #2

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- Feb 12th 2007, 09:14 AM #3

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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?

the non-circular ellipse. But a pair of distinct conics intersect at no more

than four points, so n<=4.

RonL

- Feb 12th 2007, 10:36 AM #4

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