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- Feb 14, 2012

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Find all prime numbers $p$ such that $2p^3+4p^2-3p+12$ is the fifth power of an integer.

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- Thread starter
- Admin
- #1

- Feb 14, 2012

- 3,909

-----

Find all prime numbers $p$ such that $2p^3+4p^2-3p+12$ is the fifth power of an integer.

-----

Remember to read the POTW submission guidelines to find out how to submit your answers!

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- Feb 14, 2012

- 3,909

\begin{equation*}

\begin{array}{c|c|c|c|c}

n & 0 & 1 & 2 &3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 \\

\hline

n^2 & 0 & 1 & 4 & 9 & 9 & 3 & 3 & 5 & 9 & 1 & 1 \\

n^3 & 0 & 1 & 8 & 5 & 9 & 4 & 7 & 2 & 6 & 3 & 10 \\

n^5 & 0 & 1 & 10 & 1 & 1 & 1 & 10 & 10 & 10 & 1 & 10 \\

f(n) & 1 & 4 & 5 & 5 & 5 & 6 & 9 & 4 & 3 & 7 & 6 \\

\end{array}

\end{equation*}

As one can see from the table, the only remainders upon division by 11 that the fifth power of an arbitrary integer $n$ can give are 0, 1, and 10. On the other hand, integers of the form $f(n)$ give only remainders 1, 3, 4, 5, 6, 7, and 9 upon division by 11, whereby the remainder is 1 only if $n$ is divisible by 11. Consequently, $f(p)$ can be the fifth power of an integer only if $p$ is divisible by 11. As $p$ is prime, the only possibility is $p=11$. And indeed, $f(11)=2\cdot 11^3+4\cdot 11^2-3\cdot 11+12=3125=5^5$.

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