Number Theory #1

by djmathman, Aug 5, 2011, 6:57 PM

Problem wrote:
Find all positive primes $n$ such that $n^3+1$ is a perfect square.

We see that $n=2$ works, so we try to find more values that result in a perfect square. We can rewrite this problem as $n^3+1=k^2$ for some integer $k$. Then we have $n^3=k^2-1\implies n^3=(k-1)(k+1)$. Since $n$ is a positive prime, we can consider two cases:

CASE 1:

We let $k-1=1$ and $k+1=n^3$. We see from the first equation that $k=2$, and plugging this into the second equation gives us $3=n^3$, which does not have an integer solution.

CASE 2:

We let $k-1=n$ and $k+1=n^2$. Rewriting the first equation gives us $k=n+1$, and substituting this value into the second equation gives us $n+2=n^2\implies n^2-n-2=0\implies (n-2)(n+1)=0$. Therefore, either $n=2$ or $n=-1$. However, $n$ has to be positive, so the only value that works is $\boxed{2}$.
This post has been edited 1 time. Last edited by djmathman, Aug 6, 2011, 2:15 PM
number theory
Perfect Squares

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Also, a simpler way to do this it to factor the sum of cubes. $$n^3+1=(n+1)(n^2-n+1)$$. We want this quantity to be a perfect square, thus the two factors $(n+1)$ and $(n^2-n+1)$ must be equal. Setting them equal to each other, we have $$n+1=n^2-n+1$$.
$$\implies n^2=2n$$So, the only positive solution is $\boxed{n=2}$.

by sgadekar, Jan 23, 2017, 7:38 PM

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    by balllightning37, Jul 26, 2024, 1:05 AM

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  • waiting for a recap of your amc proposals for this year :D

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  • hello :)

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  • Do you have a link to your main blog that you started after graduating from high school, I couldn't find it. @dj I met you IRL at Awesome Math summer Program several years ago.

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