MathDB

Let

k

be a given even positive integer. Sarah first picks a positive integer

N

greater than

1

and proceeds to alter it as follows: every minute, she chooses a prime divisor

p

of the current value of

N

, and multiplies the current

N

p^k -p^{-1}

to produce the next value of

N

. Prove that there are infinitely many even positive integers

k

such that, no matter what choices Sarah makes, her number

N

will at some point be divisible by

2018

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