14
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What should be the missing number?

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  • $\begingroup$ For the number theorists out there, I'd say this puzzle is rather (ROT13) anepvffvfgvp :P $\endgroup$ – Feeds Sep 29 '18 at 12:15
15
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The answer is

$13139$

Because,

Power $4$ the digits. Summing them will get you to the next number.

$4338$ -> $4^4 + 3^4 + 3^4 + 8^4$ -> $4514$.
$4514$ -> $4^4 + 5^4 + 1^4 + 4^4$ -> $1138$.
...

And yes,

It makes a cycle, :)

There is a nice video too about this: https://www.youtube.com/watch?v=_DpzAvb3Vk4 (by Numberphille).

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  • 2
    $\begingroup$ Very nice answer, and thanks for that referenced video! $\endgroup$ – Phylyp Mar 19 '18 at 12:38

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