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 AMCaptainBlackProblem 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 AMThePerfectHackerQuote:

Originally Posted by**CaptainBlank**

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. - Feb 12th 2007, 09:14 AMCaptainBlack
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.

Quote:

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 AMThePerfectHacker