Perfect Power??

Can there exist a number of the form $xyxyxy$ which is a perfect power???

$\textbf{Details and Assumptions:}$ A perfect power means a number of the form $p^{k}$ where $p$ is an integer.

#NumberTheory #CosinesGroup #Goldbach'sConjurersGroup #TorqueGroup #PerfectPower

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Math	Appears as
Remember to wrap math in `$` ... `$` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$

Comments

$\overline{xyxyxy}$

$= \overline{xy} \times 10101$

$= \overline{xy} \times 3 \times 7 \times 13 \times 37$

This implies that for $\overline{xyxyxy}$ to be a perfect power, $\overline{xy}$ must contain a factor of $3 ,7,13$ and $37$ which is not possible as it will cause it to be greater than 2 digits.

Siddhartha Srivastava - 7 years, 1 month ago

Nice job!

Eddie The Head - 7 years, 1 month ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$